Let be a closed orientable surface of genus at least 2. We consider three classes of geometric structures on : hyperbolic metrics, singular flat metrics arising from half-translation surfaces, and those obtained via grafting — an operation introduced by Thurston to study complex projective structures. We show that these three families of metrics can be unified within a single connected moduli space. In particular, we prove that half-translation surfaces arise as geometric limits (up to rescaling) of grafted surfaces. Our proof relies on recent results on the orthogeodesic foliation constructed by Calderon and Farre.
We will discuss two results about the existence of closed minimal surfaces in non-compact hyperbolic 3-manifolds. In the first part, we will prove the following dichotomy: either all bounded geometry surface groups admit a minimal surface, or there exists an example with a minimal fibration. In the second part we will discuss minimal surface existence for hyperbolic handlebodies. Based in joint work with Baris Coskunuzer, Zheng Huang and Ben Lowe.
Foundation models are transforming science by offering powerful ways to generalize across domains, scale learning, and generate meaningful structure from data. In this talk, I explore how these models can be used not only to describe patterns in data but also to construct executable, generalizable theories of complex systems. I focus on human behavior as a test case, showing how tools from statistics and the behavioral sciences can be combined with large language models to reverse-engineer cognitive processes. I will present a cognitive foundation model trained on over 200 canonical psychology experiments that predicts human behavior across tasks and generalizes to entirely new ones. This model can be used for scientific discovery via methods of regret minimization. I will also showcase results from a model that goes one step further, mapping from behavioral data directly to the code of underlying cognitive models, allowing for transparent, testable theory generation, and outline a vision of a model that maps directly from data to the best hypothesis generating observations. Taken together, this line of work outlines a path toward scientific theory discovery powered by foundation models, using behavioral science as a proving ground for a broader approach to modeling the world.
In the seminal work of Culler and Shalen (1983) a method is outlined to detect essential surfaces in a three-manifold by studying their SL(2,C)-character variety. The method underscores connections between the theory of incompressible surfaces in three-manifolds, splittings of fundamental groups, group actions on trees, and the geometry of representation varieties. In this talk, we will motivate and then lay a general foundation for this theory in arbitrary characteristic by using the same approach instead over F, an algebraically closed field of positive characteristic. We then apply the theory to a variety of settings. This is joint work with Stephan Tillmann.
We motivate the study of holonomy for connections with skew-symmetric torsion. I will then introduce a Riemannian submersion theorem via reducible holonomy representations of connections with skew torsion. After giving the general theorem I will discuss the most prominent example linking --Sasaki manifolds to quaternionic Kähler spaces.
We consider certain representations of a surface group called convex cocompact coaffine representations. These representations act geometrically on a 3-dimensional convex body in projective space, and are part of a broader (and more difficult) landscape of such geometric actions. From classical cases, we would anticipate an analytic object called a transverse measured lamination to capture the geometry of these representations, however paradoxical examples reveal that a generalization is necessary. We will discuss a nice resolution to these difficulties, and describe a space of affine measured laminations which parametrize the space of convex cocompact coaffine representations. Along the way we make an interesting connection to the dynamics of affine interval exchange transformations. Joint work with James Farre.
Let G be a noncompact simple real Lie group (e.g., G = SL(2,R)), and let Γ be a Zariski dense discrete subgroup of G × G whose intersection with the diagonal subgroup Δ(G) is a lattice. In this talk, we will present some structural constraints on such subgroups Γ and discuss related questions. This is based on ongoing joint work with Sebastian Hurtado.
The spectral radius compactification is the generalization of Thurston’s compactification of Teichmüller space to SL(n,R) Hitchin components obtained by using logarithms of top eigenvalues in place of hyperbolic lengths. Just as Thurston boundary points parametrize R-trees, which are dual to measured laminations, we find that spectral radius boundary points parametrize n-1 dimensional polyhedral spaces which are dual to geodesic currents with particular restrictions on self intersection. The construction is particularly nice for n=3 where often the geodesic current comes in a concrete way from a cubic differential, and the polyhedral space is simply the universal cover of S equipped with a triangular Finsler metric.
A "card" of an unlabelled graph is the subgraph obtained by deleting one vertex. The multiset of all such cards forms a graph's "deck". Kelly and Ulam proposed the Graph Reconstruction Conjecture, which asks if one can determine a graph from its deck (up to isomorphism). We approach this conjecture from an invariant theory perspective.
We show how basic tools from the decorated Teichmüller theory of punctured surfaces can be applied in Diophantine approximation. In particular, we will use this approach to describe the rational numbers with approximation constant at least 1/3 in terms of geodesics in the modular torus.
Let O be a compact reflection n-orbifold whose underlying space is homeomorphic to a truncation n-polytope, i.e. a polytope obtained from an n-simplex by successively truncating vertices. In this talk, I will give a complete description of the deformation space of convex real projective structures on the orbifold O of dimension at least 4. Joint work with Suhyoung Choi and Ludovic Marquis.
This lecture is intended to be an homage to Yuri MANIN who passed away in January 2023 and who contributedto science, and in particular to mathematics, in so many different ways. The book « Mathematics as Metaphor » presents a number of essaysthat he produced and in particular an expanded version of thelecture with this title he gave at the International Congress of Mathematicians held in Kyoto in 1990. The emphasis will be on tthe metaphoric approach of mathematics Yuri MANIN was advocating and on his vision of the interaction between mathematics and physics.
The notion of a "coarse median" on a metric space, introduced by Bowditch, is a non-positive curvature condition that generalizes both metric trees and Euclidean spaces. Answering a question of Bowditch, Haettel showed that the only higher-rank symmetric spaces that admit a "coarse median" are products of rank-one, e.g. H^2 times H^2. In this talk, we will introduce the notion of a "coarse higher median" and show that many irreducible higher-rank symmetric spaces admit such higher medians. This is a work in progress with Grazia Rago.
In this talk, I am aiming to survey some results around linear hyperbolic groups admitting Anosov representations and mention some open questions. I will also discuss some ongoing work with Subhadip Dey on constructions of new examples of hyperbolic groups admitting Anosov representations. No previous knowledge on Anosov representations will be required for this talk.
In this talk, we will present three problems coming from distinct areas of mathematics: Tverberg-Vrećica conjecture (computational and combinatorial geometry), the existence of linearly independent complex line fields on manifolds (algebraic and differential topology), and the calculation of persistent equivariant cohomology of the circle (applied topology). We will discuss how Chern classes arise in the methods used to their resolution and pose several intriguing questions for future research. The talk is based on several works with different groups of coauthors (P. Soberón; B. Schutte; H. Adams, E. Lagoda, M. Moy & A. De Saha).
In his 1994 paper on Moduli of Representations, Carlos Simpson used the scaling action on Higgs fields to show that moduli space of semistable Higgs bundles of fixed rank and degree are connected. There are several contexts in which one might want to study a similar question for Higgs bundles with some kind of additional symmetry. In joint work with Tommaso Scognamiglio (Heidelberg University), we look at the case of Higgs bundles over a real algebraic curve and explain how to extend Simpson’s method in this context, at least when the rank and degree are coprime. We will see in particular, the role played by the modular interpretation of Galois fixed points in order to be able to count the number of connected components of this space as a subset of the moduli space of Higgs bundles.
I will talk about an ongoing project with Beatrice Pozzetti, which focuses on the generalization of translation surface, called 1/n translation surface. I will build different kinds of Finsler metrics with the foliation structure and embed it into the space of geodesic currents.
Dual Artin groups have been recently used to solve the K(\pi, 1) conjecture for rank three Artin groups. The goal of this talk is to define dual Artin groups, give a few examples and results, and mention open questions.
Observing that the notion of disk in is invariant under projective transformations, Kojima, Mizushima and Tan proposed the study of circle packings on surfaces equipped with complex projective structures. The way disks are arranged in a circle packing is described by a triangulation of the surface, called the nerve of the packing. By a celebrated theorem of Koebe and Andreev-Thurston, for any triangulation T of a surface of genus g>1, there exists a unique hyperbolic metric which admits a circle packing with nerve T. However the space of projective structures equipped with a circle packing with a given nerve is a real algebraic set of dimension 6g-6. Kojima Mizushima and Tan conjectured that the natural forgetful map associating to a projective structure equipped with a circle packing the underlying conformal structure is a diffeomorpfismo of this space to the Teichmüller space.
In the talk I will explain some recent developments obtained in collaboration with Mike Wolf.
We study the moduli space of marked hyperbolic surfaces and related objects. Thurston shows that this space can be identified with the space of faithful and discrete representations of a finitely generated group in , up to post-conjugation by . This space is a key connected component of the character variety.
In this seminar we examine degenerations of these representations by studying compactifications of the character variety. In particular we present the real spectrum compactification, its topological properties, and show that it projects continuously onto the oriented compactification of the character variety, defined by Maxime Wolff. To this end, we interpret its boundary points geometrically and associate oriented real trees with them.
In this talk, we will study representations of a hyperbolic group into the isometry group of a Gromov hyperbolic space. We will begin by recalling the definition of quasi-isometrically embedded representations, which naturally give rise to open domains of discontinuity. Next, we will explain Minsky's construction of primitive-stable representations, which led to strictly larger domains of discontinuity.
Finally, we will discuss the properties of the analogous commutator-stable representations.
The cluster modular group is a generalization of the mapping class group of a surface and acts as the symmetry group of a cluster algebra. I will explain an unexpected connection between cluster modular groups and identities between partition q-series like the Rodgers-Ramanujan identities.
I would like to present an approach to proving some rigidity results in higher rank which was developed by U. Bader and A. Furman. It relies on Furstenberg's boundary theory and structural results for algebraic groups, and in particular their Weyl groups. Using these ideas, one can prove statements of the form "if this group G has a large (generalized) Weyl group, then any action on this class of spaces is elementary". I will give examples of such phenomena, and if time allows it, I will mention an ongoing project where "this space" is a (possibly exotic) affine building.
In this talk we will consider lightlike geodesics in Lorentzian manifolds. We will focus on the model space of conformal Lorentzian geometry, namely the Einstein Universe. It is known that the space of these geodesics is a contact manifold. In dimension three we will describe this contact structure thanks to the Engel structure with a split carried by the space of pointed lightlike geodesics. We will also discuss the four-dimensional case. In this framework, we will present a way to see that the space of lightlike geodesics is equipped with a Lorentzian CR structure in addition of the contact structure.
Given a flow on a manifold, we may define the flow group to be the subgroup of the fundamental group generated by the almost flow loops, namely by using based loops that are obtained by coning to a base-point flow segments that start and end in a fixed contractible neighbourhood of that base-point. Under mild hypothesis, we prove that the flow group equals the fundamental group. As a special case, the flow group result holds for the diagonal (Teichmüller) flow on moduli spaces of abelian and quadratic differentials. As an application, we prove simplicity of Lyapunov exponents for the plus and minus Kontsevich—Zorich cocycles, a generalisation both in statement and approach of the result for abelian strata by Avila—Viana. In the process, we also answer affirmatively several questions by Yoccoz regarding Rauzy diagrams for interval exchange maps. This is (variously) joint work with Arana-Herrera, Bell, Delecroix, Gutierrez-Romo and Schleimer.
We introduce the notions of asymptotic rank and injective hulls before investigating a coarse version of Dress’ 2(n+1)-inequality characterising metric spaces of combinatorial dimension at most n. This condition, referred to as (n,δ)-hyperbolicity, reduces to Gromov's quadruple definition of δ-hyperbolicity for n=1. The ℓ∞ product of n δ-hyperbolic spaces is (n,δ)-hyperbolic and, without further assumptions, any (n,δ)-hyperbolic space admits a slim (n+1)-simplex property analogous to the slimness of quasi-geodesic triangles in Gromov hyperbolic spaces. Using tools from recent developments in geometric group theory, we look at some examples and show that every Helly group and every hierarchically hyperbolic group of asymptotic rank n acts geometrically on some (n,δ)-hyperbolic space. Joint work with Urs Lang.
The operation of taking the universal cover, despite being fundamental in the context of topology and differential geometry, remains rather mysterious in the context of complex algebraic geometry. One of the main difficulties is that the universal cover of a smooth complex algebraic variety is merely a complex manifold usually lacking any algebraic structure. Moreover, it is usually ‘’wild’’ in a certain sense. This is made precise by the Kollár-Pardon Conjecture, which says that the universal cover of a smooth complex projective variety is almost never a semialgebraic subset of an algebraic variety, except for several classical cases.
I will discuss the context of the Kollár-Pardon Conjecture and the known results in its direction. After that, I will explain how one can see Kollár-Pardon Conjecture as a special case of a general Meta-Conjecture, which can be applied to a huge set of various geometric situations and discuss its version for Riemannian manifolds.
Enumerative invariants of Calabi-Yau manifolds are most naturally organized in terms of partition functions of physical theories. Higher genus Gromov-Witten invariants of CY threefolds correspond to the expansion coefficients of a series in a formal parameter which corresponds to the topological string coupling. This series is however only asymptotic. I will show how the analysis of finite difference equations and Borel summation reveals the piecewise analytic structure behind the asymptotic expansion. The resulting Stokes jumps of the piecewise analytic structure encode another set of enumerative invariants of the threefold, namely Donaldson-Thomas invariants.
I will give an introduction to the study of the dynamics of polygonal billiards. This talk has no dynamical pre-requisites and it will include an introduction to the relevant ideas and history from the field of dynamical systems. Our objective is explain why it is useful to study dynamics on moduli spaces of abelian differentials.
The holonomy representation of a projective surface is a representation of its fundamental group in PGL(3,R), the group of linear transformations of the projective plane. It lifts to SL(3,R) and the aim of the talk is to describe its deformation space in SL(4,R), after composing with the standard embedding of SL(3,R) in SL(4,R). I plan to discus the general case of an irreducible representation in SL(n,R) deformed in SL(n+1,R), which includes the Barbot component when n=2.
A classical problem in complex algebraic geometry is understanding the topology of closed complex submanifolds of complex projective space, so-called smooth complex projective varieties, and, more generally, of compact Kähler manifolds. Two natural topological invariants to consider are the fundamental group and its profinite completion; the latter is also known as the algebraic fundamental group. In this talk I will address the following questions: When is the fundamental group of a compact Kähler manifold uniquely determined by its profinite completion? And, when does the profinite completion even determine the homeomorphism type of the underlying manifold? In particular, I will explain positive answers to both questions in the case of a direct product of fundamental groups of closed hyperbolic Riemann surfaces. This talk is based on joint work with Hughes, Py, Spitler, Stover and Vidussi.
I will discuss a class of integrable systems introduced by Goncharov and Kenyon (1107.5588 [math.AG]), whose phase spaces are cluster varieties. These systems share many features with the moduli spaces of framed local systems that are central to Fock–Goncharov theory of higher-rank Teichmüller spaces. For instance, they admit coordinate systems derived from configurations of flags, although infinite ones. Cluster integrable systems appear to be the cluster varieties underlying the generalization of spectral networks, known as exponential networks, introduced in the context of mirror symmetry.
Eigenvalues of transfer operators, known as Pollicott-Ruelle resonances provide insight into the long-term behaviour of the underlying dynamical system, in particular determining its exponential mixing rates.
I will present a complete description of Pollicott-Ruelle resonances for a class of rational Anosov diffeomorphisms on the two-torus. This allows us to show that every homotopy class of two-dimensional Anosov diffeomorphisms contains (non-linear) maps with the sequence of resonances decaying stretched exponentially, exponentially or having only trivial resonances.
In this talk, I will introduce a new approach to the problem of classification of transitive Anosov flows in dimension 3 up to orbital equivalence. To every transitive Anosov flow on a 3-manifold, we can associate a group action on a bifoliated plane characterizing completely the original flow up to orbital equivalence. During my thesis, I proved that all the information of the previous action can be stored inside a combinatorial object, called a geometric type, and thus that geometric types can be used to classify Anosov flows in dimension 3. In this talk, I will explain how one constructs geometric types for any Anosov flow and I will also mention some recent applications of this classification method in the theory of Anosov flows.
Given a closed surface S of genus at least 2 and an integer d>2, the Teichmüller space T(S) of S embeds in the space of representations of the fundamental group of S in PSL(d,R), modulo conjugation, by postcomposing with the irreducible representation of PSL(2,R) in PSL(d,R). The connected component of T(S) is the Hitchin component H(S).
A decade ago, Bridgeman, Canary, Labourie and Sambarino constructed on H(S) several metrics, called pressure metrics, that extend the Weil--Petersson metric on T(S). Their construction uses a caracterisation of the WP metric in terms of hyperbolic lengths due to Thurston and Wolpert. We will describe the pressure metrics on certain subspaces of H(S) that are transverse to T(S), consisting of deformations through the famous bending procedure. This will allow us, in genus>2, to show that T(S) is distorted in H(S), in the sense that there is a sequence of points in T(S) whose distance to the origin diverge for the WP metric, but remains bounded for the pressure metrics.
The Virasoro algebra is the symmetry algebra of two-dimensional conformal theories. It is the universal central extension of the Witt algebra, the Lie algebra of vector fields on the circle. Several constructions allow to deduce the Virasoro algebra from the Witt algebra, most notably the Feigin-Fuks construction. I'll discuss a more general framework of this construction and apply it to a q-deformed Witt algebra, giving a new deformation of the Virasoro algebra.
Frieze patterns were introduced by Coxeter in the 1970s who, with Conway, established a correspondence between frieze patterns and triangulated polygons. It turned out later that this object is very rich in connections with different fields in mathematics. We use hyperbolic geometry to provide a classification of positive integral friezes on marked bordered surfaces. This is a joint work with Pavel Tumarkin.
In his thesis, Andrey Markov classified the unimodular indefinite binary quadratic forms whose values on the integer lattice stay farthest away from zero. This is closely linked to the classification of the worst approximable irrational numbers. The main tool in Markov's classical theory have always been continued fractions. Geometrically, they describe the symbolic dynamics of geodesics in the Farey triangulation of the hyperbolic plane. This talk will be about a new geometric approach that eliminates the complicated symbolic dynamics of continued fractions in favor of considering not only the Farey triangulation but all ideal triangulations of the modular torus. We will see how Markov's classification problem boils down to the question: How far can a geodesic that crosses a triangle stay away from the vertices? This geometric approach can also be used to classify the worst approximable rational (sic!) numbers.
Motivated by the study of higher Teichmuller theory, Guichard and Wienhard generalized Lusztig's notion of total positivity and introduce theta positivity. The totally nonnegative part of flag varieties has been studied extensively for many years, and many remarkable properties have been etablished. In this talk, I will discuss the analogues for a family of theta nonnegative part of Lagrangian Grassmannian.
Let be a finitely generated associative algebra, in general noncommutative. A double Poisson bracket on in the sense of Van den Bergh arXiv:math/0410528 is a bilinear map from to , subject to certain conditions. Van den Bergh showed that any such bracket induces a Poisson structure on the space of -dimensional representations of the algebra for any . We propose an analog of Van den Bergh's construction, which produces Poisson structures on certain subspaces of the representation spaces . We call these subspaces the involutive representation spaces. They arise by imposing an additional symmetry condition on -- just as the classical groups from the series B, C, D are obtained from the general linear groups (series A) as fixed point sets of involutive automorphisms. The talk is based on a joint paper with Grigori Olshanski arXiv:2310.01086.
The Anosov representations of a hyperbolic group into a rank one Lie group are exactly those that are convex-cocompact. They have some important properties, for example, a representation of a finitely generated group into PO(1,n) is convex-cocompact if and only if its orbit maps are quasi-isometric embeddings into the associated symmetric space, the n-dimensional hyperbolic space. Moreover, the set of convex-cocompact representations into PO(1,n) is open in the space of all the representations in PO(1,n) (this is called the stability of convex-cocompact representations). In this talk, we will be interested in some infinite-dimensional representations, namely representations into PO(1,infinity), where the same properties for convex-cocompact representations remain true. In particular, the stability property allows the use of deformation techniques to produce other convex-cocompact representations of a given group.
We study cluster structures on spinor helicity and momentum twistor varieties which describe the kinematic spaces of massless scatterings in physics. Both varieties are certain partial flag varieties. We exhibit embeddings of the corresponding cluster algebras into cluster algebras of sufficiently large Grassmannians and show how the former is obtained by freezing certain cluster variables from the latter. This is joint work with Lara Bossinger.
I will discuss topological obstructions for a closed four-manifold that admits a (real) properly convex projective structure. I will present a vanishing result on the Pontryagin classes, that in dimension four provides information on the geography problem for properly convex projective manifolds, and implies the solution to a question of Yves Benoist on the (in)compatibility of real convex projective structures and complex hyperbolic structures. Time permitting, I will also discuss the construction of examples that realize all possible positive values of the Euler characteristic. Based on joint work with Stefano Riolo and Leone Slavich.
Let S be a closed oriented surface, denote with Teich(S) its Teichmuller space, and with Hit_3(S) the Hitchin component of the SL(3,R)-character variety, endowed with the Labourie-Loftin complex structure. The well-known Bers Theorem provides a biholomorphism between Teich(S)xTeich() and an open subset of the SL(2,C)-character variety corresponding to quasi-Fuchsian representations. In this talk, we present an analog result for Hit_3(S). In fact, we introduce a notion of complex affine spheres in , which extend the classical (real) affine spheres in , and use them to prove that the natural inclusion of the diagonal of Hit_3(S)x{Hit_3()} inside the character variety of SL(3,C) extends to a unique local biholomorphism on a "big" open subset of Hit_3(S)xHit_3() which in particular contains Hit_3(S)xTeich() and Teich(S) x Hit_3(). As a consequence of this approach, we also show that the Goldman symplectic form is compatible with the Labourie-Loftin complex structure, determining a pseudo-Kahler structure on Hit_3(S). This is joint work with Nathaniel Sagman.
The K-theoretic Gromov-Witten (KGW) theory studies enumerative invariants of varieties through counting nodal curves. It naturally arises in the framework of 3D mirror symmetry, a duality between mirror holomorphic symplectic varieties, of which (type-A) flag varieties are among the first examples. In this talk, we study the genus-zero KGW theory of flag varieties from the perspective of Givental cone. We prove a reconstrction theorem of the cone, and discuss several implications. The proof involves torus fixed point localization and abelian/non-abelian correspondence.
Structure-preserving discretization in the field of geometry is the paradigm of discrete differential geometry. In some aspects, the discrete theory turns out to be even richer than its smooth counterpart. It focuses on developing constructive methods. The well-established theory of discrete conformal maps and circle patterns is related to discrete integrable models of mathematical physics and has found applications in geometry processing. We present their generalizations beyond the conformal limit: decorated discrete conformal maps and ring patterns, which share the corresponding existence and uniqueness statements. The theory and construction methods are based on convex variational principles related to hyperbolic geometry.
I'll explain how to define a natural family of Axiom A flows on non-empty domains of discontinuity for general Anosov representations. This is ongoing joint work with Daniel Monclair and Andrew Sanders.
Global hyperbolicity is the strongest causality condition in Lorentzian geometry. It encapsulates both the non-existence of causal loops and of causal paths going to the edge of the spacetime in finite time and is of interest to both physicists and mathematicians. In non-Lorentzian pseudo-Riemannian geometry of signature (p,q), i.e such that p and q are greater than 1, there does not exist any obvious generalization of global hyperbolicity as there does not exist any causality for non-positive paths. This stems for the connectedness of the space of negative vectors in when q is greater than 1. My goal in this talk will be to introduce a way to generalize global hyperbolicity in higher signatures as well as other concepts in Lorentzian geometry.
A discrete group of conformal automorphisms of the Riemann sphere is called a Kleinian group. Anosov Kleinian surface groups are quasi-conformal deformations of Fuchsian surface groups; they preserve a Jordan curve cutting the sphere into two disks on which the action of the group is nice. Bers proved that quasi-Fuchsian surface group representations are determined uniquely by the pair of conformal structures at infinity, giving a natural parameterization by a product of Teichmüller spaces. We study the higher rank analogue: hyperconvex Anosov surface group representations into PSL(d,C), introduced by Pozzetti—Sambarino—Wienhard. We define a natural map into a product of Teichmüller spaces of Riemann surface foliations and prove the following analogue of a famous theorem of Bowen from the 70’s: The Hausdorff dimension of the limit set of a fully hyperconvex surface subgroup into PSL(d,C) is equal to 1 iff it is conjugated into PSL(d,R). This is joint work with Beatrice Pozzetti and Gabriele Viaggi.
In this talk, after a brief historical review, I will explain how to sample a random hyperbolic surface. After that, we will focus on the length spectrum of a random surface. More precisely, we will investigate short closed geodesics on a random hyperbolic surface of large genus. It turns out that the lengths of these geodesics are distributed in exactly the same way as those of the short cycles in a large random graph. This is a joint work with Simon Barazer and Alessandro Giacchetto.
The goal of this talk is to present tiling billiards and interval exchange transformations.
A tiling billiard is a dynamical system whose trajectories are in the broken lines in the Euclidean plane. There are studied since about a decade, motivated by metamaterials with negative index of refraction. I will give an overview of what we know about tiling billiards.
An IET is a map defined on an interval, which is piecewise a translation. I will explain how, in certain cases, we can study a tiling billiards with an IET with flips.
A representation is said to be almost-Fuchsian if it admits a nice equivariant map with strong geometric properties. In turn, it implies that the representation is convex-cocompact, hence discrete and faithful.
In this talk we will explain how to build almost-fuchsian representations in the group PU(2,1) with Toledo invariant close to being maximal. These will admit equivariant holomorphic almost-fuchsian immersions in the complex hyperbolic space.
A polygonal surface in the pseudo-hyperbolic space is a complete maximal surface bounded by a lightlike polygon with finitely many vertices. We give several characterizations of these surfaces. Polygonal surfaces are characterized by finiteness of their total curvature and by asymptotic flatness.
The goal of the talk is to explain some constructions coming from nonpositive curvature geometry to establish the equivalence between being polygonal and having finite total curvature.
A metric space satisfies a Euclidean isoperimetric inequality for -spheres, if every -sphere bounds a ball with . Every CAT(0) space satisfies Euclidean isoperimetric inequalities for -spheres with the sharp constant . Moreover, if such inequalities hold with a constant strictly smaller than , then has to be Gromov hyperbolic. In particular, a sharp isoperimetric gap appears. In the talk I will focus on the case , namely fillings of 2-spheres by 3-balls.
This is based on joint work with Drutu, Lang and Papasoglu.
Finite mapping class group orbits are both a rare and fascinating phenomenon of dynamics on character varieties. The first non-trivial examples go back to Fuchs and Picard. You won't find them in (higher) Teichmüller spaces; you'll have to explore the "other" components to stumble across interesting examples. This talk will focus on the most recent developments about representations of punctured spheres. We'll see how a result by Corlette--Simpson tells us exactly where to look for finite orbits and how it can be used to complete the classification of finite orbits for representations into SL(2,C), confirming Tykhyy's Conjecture. All of this is joint work with Samuel Bronstein.
A characteristic of moduli spaces of parabolic bundles on Riemann surfaces is their dependence on a set of real parameters, which gives rise to wall-crossing phenomena in the study of their birational geometry. In a less explicit way, this dependance also occurs in their natural Kähler metrics, and it is compelling to express this dependence as a suitable "Torelli theorem" answering the question: to what extent does the Kähler metric (an analytic invariant) determine the stability conditions used to define the moduli problem?
In this talk I will describe a strategy to address this variation problem in terms of complex-analytic techniques arising in the study of the spectral geometry of resolvents. I will also explain how these techniques can be applied to the study and classification of gravitational instantons of ALG type. This is work in progress, joint with Hartmut Weiss.
The goal of the talk is proving a conjecture of Claude Roger about the universal central extension of the Lie algebra of volume-preserving vector fields. In the beginning we will briefly review the notion of central extensions of Lie algebras and their link to Chevalley-Eilenberg-cohomology. We will then proceed to Rogers conjecture, which lies in the (continuous) infinite-dimensional setting. To solve it we will need a combination of analytical and geometrical methods, and maybe even a bit of representation theory.
Based on an ongoing collaboration with Bas Janssens and Cornelia Vizman.
A group G is said to be equationally Noetherian if every system of equations over G has the same solution set as some finite subsystem. his property, introduced in the 1990s in the context of algebraic geometry over groups, has found its way into logic over groups and geometric group theory. In this talk, based on recent joint work with Monika Kudlinska, I will explain why all extensions of a finitely generated free group by the infinite cyclic group are equationally Noetherian.
We show that for a generic Riemannian metric on a compact manifold of dimension at least three all closed geodesics do not have self-intersections. Similar results are possible for geodesic loops and orthogonal geodesic chords. An orthogonal geodesic chord on a manifold with boundary is a geodesic segment connecting points on the boundary and hitting the boundary orthogonally.
There is a storied history to using complexes built from curves on surfaces to understand mapping class groups of surfaces and moduli spaces. The most famous is Harvey's curve complex, which has a vertex for each isotopy class of essential simple closed curve and a simplex when curves can be realized disjointly. This complex is also related to the intersections of the boundary strata of the Deligne-Mumford compactification. Masur and Minsky famously proved that the curve complex is hyperbolic, which has had far-reaching implications for the coarse geometry of the mapping class group and Teichmüller space. In this talk, I will report on ongoing work with Jacob Russell in which we build similar complexes for certain moduli spaces of flat cone structures and analyze their coarse geometry.
Measured Property (T) is a generalization of group Property (T) to graphings. An example of a graphing is obtained from a measure-preserving action of a group on a standard probability space by drawing the Cayley graph of said group in each orbit. Further examples of graphings include, for instance, and very roughly speaking, regraphings of this previous graph by local or random algorithms.
This talk is an invitation to the following question: what geometries can appear in graphings with measured Property (T)? For instance, one cannot obtain graphings whose components are acyclic or planar. This will be discussed in the talk, together with all the necessary background. In the group case, it follows from Thurston's Geometrization Conjecture that an infinite Property (T) group cannot appear as the fundamental group of a compact 3-manifold. How does this fact translate to the measured setting, and what further geometric obstructions to measured Property (T) can occur?
This talk is based on joint work with Łukasz Grabowski and Sam Mellick, as well as on an on-going project with Antoine Poulin.
The notion of a double Poisson algebra was introduced independently in 2007-08 by M. Van den Bergh and by W. Crawley-Boevey, P. Etingof, and V. Ginzburg. A double Poisson bracket on a non-commutative algebra generalizes the classical Poisson bracket for the algebra of smooth functions on a manifold.
Poisson structures on spaces of surface group representations into Lie groups were introduced by W. Goldman in 1986 and have since been studied by many researchers. In particular, V. Fock and A. Goncharov described a Poisson structure on spaces of surface group representations into split Lie groups using their X- and A-coordinates.
In my talk, I will generalize this construction by introducing a double quasi-Poisson structure on the space of representations of the fundamental group of a marked surface S into GL_n(A) for a non-commutative algebra A without polynomial identities. I will demonstrate that this bracket arises from the so-called partial non-abelianization procedure and a double bracket that is naturally associated with an appropriate ramified covering of S.
This is joint work with M. Gekhtman.
We develop a general deformation principle for families of Riemannian metrics on smooth manifolds with possibly non-compact boundary, preserving lower scalar curvature bounds. The principle is used to strengthen boundary conditions from mean convex to totally geodesic or doubling. The deformation principle preserves other geometric properties such as completeness and a given quasi-isometry type.
As an application, we prove non-existence results for Riemannian metrics with uniformly positive scalar curvature and mean convex boundary, including some investigation of the Whitehead manifold.
Character varieties are spaces which parametrise the deformations of representations of the fundamental group of a surface into a Lie group.
The symmetry group of the surface, the mapping class group, naturally act on these space, and this action has interesting dynamical properties.
In this talk I will consider a very particular case: the relative character varieties of the once-punctured torus with values in SL(2,C).
In this case, the spaces are affine algebraic surfaces given by an explicit equation, the famous family of Markov surfaces, and the action of the mapping class group is given by explicit polynomial transformations.
I will describe the stationary measures for this action: these are measures that describe the statistical distribution of the orbits of the group action.
The complex hyperbolic space is the complex counterpart of real hyperbolic geometry, and is the simplest instance of a negatively curved Kähler-Einstein manifold. Similarly to the real case, the Riemannian geometry of the complex hyperbolic space is in one-to-one correspondance with the conformal geometry of its boundary at infinity, which is a strictly pseudoconvex Cauchy-Riemann (CR) structure. This correspondance has proven useful to the study of complex domains as well as that of Kähler manifolds.
In this talk, we consider a complete, non-compact almost Hermitian manifold whose geometry at infinity is locally modelled on that of the complex hyperbolic space. Under purely geometric considerations, we will prove that such a manifold admits a natural compactification at infinity by a strictly pseudoconvex CR structure.
After starting with an overview of the classification of locally homogeneous geometric structures on manifolds, I will describe a particular example of how this general classification problem leads to a class of dynamical systems arising from mapping class group actions on character varieties.
Given a bounded domain U within affine space, one can ask if there exists a discrete group of projective automorphisms acting freely and properly with compact quotient on U. In that case we say that the domain is divisible. In the early 2000s, Benoist conducted an extensive examination of these divisible domains, unearthing a rich variety of ``exotic'' examples. Similar questions arise in other geometric contexts. For instance, in conformal Riemannian geometry, many bounded domains of Euclidean space can be divided by a group of conformal diffeomorphisms. However, in broader geometric frameworks (general flag manifolds), Limbeek and Zimmer conjecture a rigidity result : there should be very few examples of bounded divisible domains. In this presentation, I will show such a rigidity result for pseudo-Riemannian conformal geometry. Specifically, I will establish that any bounded divisible domain of Minkowski space - the pseudo-Riemannian counterpart to Euclidean space - is a diamond. This is joint work with Blandine Galiay.
Given an edge-weighted graph drawn on a torus (or a cylinder), one gets an associated Liouville integrable system. These are known as "cluster integrable systems" because they are preserved by local moves on the graph which correspond to cluster algebra mutations. Goncharov and Kenyon studied these systems in relation with the dimer model, interpreting the Hamiltonians in terms of perfect matchings, and Gekhtman-Shapiro-Vainshtein described them in terms of directed paths. We will study a noncommutative version of these cluster integrable systems, where the edge weights on the graph are formal noncommutative variables. In joint work with Shapiro and Arthamonov, we study a certain non-commutative Poisson structure on this space, and give results analogous to the commutative case, including an R-matrix formula for the brackets, and a set of commuting Hamiltonians which can be interpreted in terms of paths.
The hyperbolic 3-manifold associated to a Fuchsian representation of a surface group admits a fibration over the surface with geodesic fibers that extends to a fibration of the conformal boundary. This also holds for almost Fuchsian representations, but not in general for quasi-Fuchsian representations.We will discuss an analog of this picture for representations of surface groups into . Among these representations there exist a union of connected components containing only discrete and faithful representations, called maximal representations. We will consider fibrations of the projective model of the symmetric space of by projective codimention subspaces. These subspaces are described by pencils of quadrics, and we will characterize maximal representations by the existence of such a continuous fibration, satisfying some additional properties.
The second lecture will be about the novel advances in cluster algebra applications to geometry:We consider the symplectic groupoid: pairs (B,A) with A unipotent upper-triangular matrices and being such that are also unipotent upper-triangular matrices. We explicitly solve this groupoid condition using Fock–Goncharov–Shen cluster variables and show that for B satisfying the standard semiclassical Lie–Poisson algebra, the matrices B, A, and satisfy the closed Poisson algebra relations. Identifying entries of A and with geodesic functions on a closed Riemann surface of genus , we are able construct the geodesic function for geodesic joining two halves of the Riemann surface. We thus obtain the complete cluster algebra description of Teichmüller space of closed Riemann surfaces of genus two . We discuss also the generalization of our construction for higher genera. (based on joint paper ArXiv:2304.05580 with Misha Shapiro).
Geometric deep learning extends deep learning to incorporate information about the geometry and topology of data, especially in complex domains like graphs. Many existing methods in this field rely on message passing. However in this talk we will take a new approach by combining the theory of differential k-forms in Euclidean space with the geometric information of graphs and complexes given by embeddings of node coordinates. This method offers interpretability and geometric consistency without the use of message passing.
The first lecture is the introduction to cluster algebras (an object introduced by Fomin and Zelevinsky in around year 2000) and related Poisson and quantum algebras of geodesic functions (hyperbolic cosines of half-lengths of geodesics in Poincare uniformization of Riemann surfaces with holes) A.K.A. Goldman brackets. The cluster variables therefore provide the parameterization of Teichmuller spaces T_{g,s} of Riemann surfaces of genus g with holes. This is based on papers by V.V.Fock, M.Shapiro, M.Mazzocco and myself in years 1999-2010.
A representation of a surface group into a Lie group of Hermitian type G is called maximal if it maximizes the Toledo invariant. In a 2017 joint work with N. Treib, we proved that in the case of surfaces with non-empty boundary, all such representations could be obtained by a Ping-pong construction in the Shilov boundary of G. For the specific case of Sp(2n,R), maximal representations admit open cocompact domains of discontinuity in the projective space, and we used the Ping-pong construction to obtain explicit fundamental domains for this action bounded by quadric hypersurfaces. In joint work with Pier-Olivier Rodrigue, we construct similar domains for the action of maximal representations in SO(2,n) on the grassmannian of isotropic 2-planes (also called photon space, as it parametrizes photons in the Einstein Universe). We also prove that the same hypersurfaces can be used to bound fundamental domains for representations obtained by the inclusion of a classical Schottky group in SO(1,n) into SO(2,n) and their small deformations.
In this talk we will discuss some recent work on groups which connect self-similar and Higman-Thompson groups to big mapping class groups via "braiding". We will explain some results on the topological finiteness properties of the resulting groups, which are topological generalizations of the algebraic properties of being finitely generated and finitely presented. The talk will involve recent joint works with Xiaolei Wu (Fudan) and Matthew Zaremsky (Albany).
I will discuss the following recent result that was conjectured by Brunella:
Let be a compact complex manifold of dimension . Let be a codimension one holomorphic foliation on with ample normal bundle. Then every leaf of accumulates to the singular set of . This result was obtained in collaboration with M. Adachi.
Classical mapping class groups, i.e. for surfaces of finite type, are well-studied objects: they are discrete groups expressing the symmetries of the surface. When we turn our attention to surfaces of infinite type, the situation changes drastically: In particular, the mapping class groups are now "big" in the sense that they are uncountable, not finitely generated, and not even compactly generated.
This means that we loose some of the tools that are used in the classical case but also that we can ask many new questions, for example such coming from the theory of Polish groups: Are big mapping class groups automatically continuous? Amenable? When considering the conjugacy action of a big mapping class group on itself, can there be comeager, dense, or at least somewhere dense orbits?
In this talk, I will give a short introduction to surfaces of infinite type and big mapping class groups and then answer some of the new questions, based on joint work with Jesús Hernández Hernández, Michael Hrušák, Israel Morales, Manuel Sedano, and Ferrán Valdez.
I will introduce a new algorithm to parallelise the computation of persistent homology of 2D alpha complexes. The algorithm distributes the input point cloud among the cores which then compute a cover based on a rectilinear grid. I will show how to compute the persistence Mayer–Vietoris spectral sequence from these covers and how to obtain persistent homology from it. For this, I introduce second-page collapse conditions and explain how to solve the extension problem. I will give a short overview of an implementation in C++ using Open MPI and discuss some experimental results. Finally, I will give an outlook on the three-dimensional case and some applications.
In this talk, we discuss the classification of hyperbolic Coxeter polyhedral, that is, hyperbolic polyhedra whose dihedral angles are integer submultiples of \pi. We give an overview of the main classification results and discuss the few known examples in higher dimensions. We present a method to construct hyperbolic Coxeter polyhedra with mutually intersecting facets and non-zero dihedral angles. We provide a new Coxeter polyhedron in dimension 9 and complete the classification of hyperbolic Coxeter polyhedra with mutually intersecting facets and dihedral angles \pi/2, \pi/3 and \pi/6.
In 2010, Bonahon and Wong constructed the 2d quantum trace map, connecting two different quantizations of the SL_2(C) character variety of an ideally triangulated hyperbolic surface S. The classical limit of this map expresses the trace of the holonomy of a closed immersed curve in S as a Laurent polynomial in the (square roots of the) shear coordinates for the hyperbolic structure defined with respect to the triangulation. In this talk, we will discuss a new construction of a 3d quantum trace map for ideally triangulated 3-manifolds, focusing on the geometric aspects. This is joint work with Sunghyuk Park.
Profinite groups arise in many guises in mathematics, notably as the quotients of compact Hausdorff groups by the connected component of the identity. It is well known that the cardinality of an infinite profinite group cannot be less than 2ℵ0, the cardinality of the continuum. In 2019, Jaikin-Zapirain and Nikolov proved moreover that each infinite profinite group has at least 2ℵ0 conjugacy classes. After a brief introduction to profinite groups we discuss the consequences of restricting the number of conjugacy classes of elements of various types (such as p-elements or elements of infinite order). In particular, every finitely generated profinite group with fewer than 2ℵ0 conjugacy classes of elements of infinite order is finite.
Despite the apparent simplicity of the questions, the answers seem to depend on results such as the classification of the finite simple groups (and their automorphism groups) and work of Zel’manov on Lie algebras associated with profinite p-groups.
In this lecture I will define a family of finite measures on the moduli space of genus g curves. The measures are defined via a construction analogous to that of the Weil-Petersson metric, using the
extra data of a spin structure. In fact, the measures arise naturally out of the super Weil-Petersson metric defined over the moduli space of super curves. The total measure gives the volume of the moduli space of super curves and can be calculated in many examples.
A real affine structure on a surface is an atlas of charts on a topological surface, taking values in R^2 and whose transitions maps are real affine transformations. I will first try to give a variety of examples to gain familiarity with the concept and then discuss classical questions such as existence and geometry of their moduli space, mapping class group actions etc.
My ultimate goal will be to draw a conjectural picture of the geometry of the moduli space based on information coming from the dynamics of the geodesic foliations of affine surfaces.
The Theorem on the existence of three simple closed geodesics on every Riemannian 2-sphere has a “colorful” history and many applications on closed geodesics. I will give an introduction to the problem and outline a proof for the reversible Finsler case. Finally I will comment on some applications for closed geodesics. The talk is based on a collaboration with De Philippis, Marini, and Mazzucchelli.
We address G.D.Mostow, L.Bers and S.L.Krushkal questions on uniqueness of conformal/spherical CR structures on the sphere at infinity of non-compact (Hermitian) symmetric rank one spaces compatible with the action of a discrete isometry group. We construct such non-rigid discrete isometry groups whose quotients have infinite volumes and whose non-trivial deformations are induced by equivariant homeomorphisms of the (Hermitian) symmetric space with bounded horizontal distortion. This non-rigidity is related to non-ergodic dynamics of our discrete isometry group actions on the limit set which could be the whole sphere at infinity.
Integral probability measures (IPMs) are a family of distances between distributions which return the largest deviation under a specified function class. Many of the central problems in data analysis reduce to comparing distributions, and in most cases an IPM is used.
This talk focuses on coresets; these are small and discrete representations of a large data set or a continuous distribution so that the coreset can be used as proxy, and certain measurements are guaranteed to not deviate too far. Interesting coreset questions ask for a given error tolerance, how small can one make the discrete coreset. Such coresets are one of the main tools for scaling machine learning to massive data sets with guaranteed approximation results.
In this talk, we develop coresets for IPMs when the function class is geometrically defined. This has algorithmic applications to many topics including linear classification, kernel density estimates, Kolmogorov-Smirnov distances, and spatial scan statistics. We will conclude with a deeper dive into how to apply these to spatial scan statistics, and how they provide incredibly scalable algorithms without sacrificing statistical power.
In this talk, we explore the expected topology (measured via homology) and local geometry of two different models of random subcomplexes of the regular cubical grid: percolation clusters, and the Eden Cell Growth model. We will also compare the expected topology that these average structures exhibit with the topology of the extremal structures that it is possible to obtain in the entire set of these cubical complexes. You can have a look at some of these random structures here (https://skfb.ly/6VINC) and start making some guesses about their topological behavior.
We discuss some of our current understanding of Lipschitz maps of least Lipschitz constant between two compact negatively curved Riemannian manifolds and maximal stretched measures arising from Thurston’s length ratio metrics. This is joint work in progress with Gerhard Knieper.
We introduce the notion of extended positive representation of a Fuchsian group into a semisimple Lie group with a -positive (in the sense of Guichard--Wienhard) structure. Examples of such representation include Fock-Goncharov's positive framed representations, maximal representations and classical -positive representations. We show that extended positive representations are divergent and extended geometrically finite (in the sense of Weisman).
Given a second countable locally compact group, G, the set of all subgroups S(G) may be endowed with the Chabauty topology, under which it is a compact space. In general, it is difficult to understand the full topology of the space S(G), and the complete homeomorphism type is known only in very few cases. In the first part of the talk we will give an introduction to the Chabauty topology. Then we will study limits of different kinds of subgroups in SL(n, Qp) using the geometry of the Bruhat–Tits building, and the action of the groups on the building. This will give us insight into parts of S(SL(n, Qp)). We will focus on limits of groups of involutions in SL(2, Qp) (joint with Corina Ciobotaru), and time permitting, discuss limits of other subgroups, based on joint work with Ciobotaru and Alain Valette.
Lagrangian Floer homology is a Hamiltonian isotopy invariant of Lagrangian pairs in a symplectic manifold that counts pseudo-holomorphic curves with boundary conditions on the Lagrangians, and it may be thought of as an infinite-dimensional version of Morse theory. The classical Lagrangian PSS isomorphism relates the Lagrangian Floer homology of the pair to the singular homology of . In this talk we will explain the promotion of this isomorphism to an equivalence of stable homotopy types — the Lagrangian Floer homotopy type of the pair is equivalent to the stable homotopy type of . Although this is the expected result, there is no written account of such an equivalence.
Artin groups form a large family of groups that contains many other interesting families, such as free groups, free abelian groups, braid groups, etc. Despite a lot of effort, these groups remain quite mysterious in general. In this talk, I will explain how one can recover a lot of information about some Artin groups by looking at their actions on a fitting non-positively curved simplicial complex. In particular, I will present how one might recover from such an action various results of quasi-isometric rigidity for the group.
In this talk, we will introduce Bowditch representations of the free group of rank two (defined by Bowditch in 1998) along with primitive-stable representations (defined by Minsky in 2010) acting on Gromov hyperbolic spaces. Minsky first introduced primitive-stable representations in PSL(2,C) to construct an open domain of discontinuity in the character variety. We will discuss the equivalence between primitive-stable and Bowditch representations. We will also introduced simple-stable representations of a surface group and give a similar result in the case of the four-punctured sphere.
In 2013, C. Rossi established the existence and the uniqueness of the maximal extension of a globally hyperbolic conformally flat spacetime. Her proof, however, has the unsatisfactory feature that it relies crucially on the axiom of choice, specifically through Zorn's lemma, and does not provide a description of the maximal extension. In this talk, I will propose a constructive proof that avoids the need of Zorn's lemma. The key idea is that any simply-connected globally hyperbolic conformally flat spacetime M can be embedded into a bigger conformally flat spacetime E(M), called enveloping space of M, containing all the Cauchy-extensions of M, and in particular the maximal extension. It turns out that M and its Cauchy-extensions can be well described in M, a description that I will specify. In particular, the description of the maximal extension in E(M) involves the concept of eikonal functions, coming from PDE theory. I will illustrate this construction with examples and conclude with discussing some interesting consequences that arise from it.
The classical Minkowski problem asks for the construction of a strictly convex surface with specified Gaussian curvature and has already been extended to the Minkowski space by Bonsante and Fillastre. Through the use of affine differential geometry, one can generalize the Minkowski space to the setting of the affine space containing an open proper convex cone. That new setting gives birth to an equivariant affine Minkowski problem: can one introduce and fix the curvature of a convex surface invariant under the action of an affine deformation of a subgroup of SL(n,R) dividing our convex cone? I will show how that problem reduces to a Monge-Ampère equation and can be solved, generalising results from the work of Barbot, Béguin and Zeghib, of Bonsante and Fillastre, and of Nie and Seppi.
I will also explain how that work can be anchored in much bigger task of generalising the theory of flat Lorentzian spacetimes and its tools (causality, Cauchy surfaces and development, cosmological time, etc), which are linked to Teichmüller theory through the fundamental work of Mess, to a theory of affine spacetimes then linked to higher Teichmüller theory.
Artin groups are a generalization of braid groups and are closely related to Coxeter groups. Thanks mainly to a result of Van der Lek, a family of subgroups of Artin groups called parabolic subgroups are central to their study. One question regarding parabolic subgroups is whether the intersection of parabolic subgroups is a parabolic subgroup. This question is wide open, but the answer is known in some cases. An article by Cumplido, Martin and Vaskou introduced a geometric strategy for approaching it. In this talk we will show how to use this strategy to study it in the two-dimensional case. To do so, we will introduce systolic-by-function complexes, which are a generalization of systolic complexes, and use their non-positive curvature to get a positive answer to the question.
The space of representations of the fundamental group of a closed surface to PSL(n,R) contains a distinguished connected component consisting entirely of discrete and faithful representations: the Hitchin component. For n = 2, this component is identified with the Teichmüller space of the surface and the study of its compactifications has led to many influential results. The aim of this talk is to present the Hitchin component, its real spectrum compactification and how we can interpret its boundary points geometrically as representations to PSL(n,F), where F is a closed real field, which admit positive limit maps in flag varieties over F.
Anti-de Sitter space is a spacetime of constant negative curvature, playing a similar role for spacetimes as hyperbolic space does for Riemannian manifolds. Cauchy hypersurfaces in a spacetime are a way to describe space at an instant of time, motivated by physics. By taking levelsets of the cosmological time function, Barbot constructed Cauchy hypersurfaces in GH-regular domains in Anti-de Sitter space. In this short introductory talk I will speak about how we can construct new Cauchy hypersurfaces using the cosmological time function and crowns.
Any closed, flat Riemannian manifold is finitely covered by the torus, by Bieberbach’s classical theorem. Similar classifications have been pursued for closed, Riemannian conformally flat manifolds, as well as for closed, flat Lorentzian manifolds. I will discuss recent and ongoing work with Nakyung Lee and Bill Goldman to classifying closed, Lorentzian conformally flat manifolds when they have nilpotent holonomy.
Hyperbolic groups form an important class of finitely generated groups that has attracted much attention in Geometric Group Theory. We call a group of finiteness type if it has a classifying space with finitely many cells of dimension at most , generalising finite generation and finite presentability, which are equivalent to types and . Hyperbolic groups are of type for all and it is natural to ask if their subgroups inherit these strong finiteness properties. In recent work with Py, we used methods from Complex Geometry to prove that for every there is a hyperbolic group with a subgroup of type and not . This answers an old question of Brady and produces many finitely presented non-hyperbolic subgroups of hyperbolic groups. In this talk we will explain this result and present other recent progress on constructing coabelian subgroups of hyperbolic groups with exotic finiteness properties. This talk is based on joint works with Kropholler, Martelli--Py, and Py.
Bowen and Margulis independently proved in the 70s that closed geodesics on compact hyperbolic surfaces equidistribute towards the measure of maximal entropy. From a homogeneous dynamics point of view, this measure is the quotient of the Haar measure on PSL(2, R) modulo some discrete cocompact sugroup.
In a joint work with Jialun Li, we investigate the higher rank setting of this problem by taking a higher rank Lie group (like SL(d, R) for d ≥ 3) and by studying the dynamical properties of geodesic flows in higher rank : the so-called Weyl chamber flows and their induced diagonal action. We obtain an equidistribution formula of periodic tori (instead of closed orbits of the geodesic flow).
A slope p/q is characterising for a knot K in the 3-sphere if the oriented homeomorphism type of the manifold obtained by performing Dehn surgery of slope p/q on K uniquely determines the knot K. Sorya showed that for any knot K, there exists a constant C(K) such that any slope p/q with |q|≥C(K) is characterising for K. However, the proof of the existence of C(K) in the general case is non-constructive, which naturally evokes the question of how to compute explicit values for C(K). In this talk, I will explore methods for finding C(K) in the case where K is a knot of hyperbolic type (meaning that the JSJ decomposition of its complement has a hyperbolic outermost JSJ piece). I will begin with the simplest case, in which K is a hyperbolic knot; time permitting, I will also discuss some ongoing joint work with Patricia Sorya on the more general case.
We construct open domains of discontinuity for Anosov representations acting on some homogeneous spaces, including (pseudo-Riemannian) symmetric spaces. This generalizes work of Kapovich-Leeb-Porti on flag spaces. Our results complement those of Gueritaud-Guichard-Kassel-Wienhard, who constructed proper actions of Anosov representations. For Zariski dense Anosov representations with respect to a minimal parabolic subgroup acting on some symmetric spaces, we show that our construction describes the largest possible open domains of discontinuity. This is joint work with Florian Stecker.
In topological data analysis (TDA), functors from a poset category to vector spaces, called persistence modules, are a central object of investigation. Persistence modules can arise from data via several constructions, and often encode complex geometric information on the data. Finding informative and computable descriptors of persistence modules is crucial to extract this information and use it in data analysis.
A recent trend in TDA is to describe persistence modules over finite posets via relative homological invariants. Relative homological algebra extends constructions of standard homological algebra by redefining the notion of projective module, which depends on the choice of a family of “basic” modules. In this talk, we study relative resolutions, which are a way of approximating a given persistence module by a sequence of persistence modules that are projective relative to the chosen family. Under certain conditions, the multiplicities of the basic persistence modules in a relative resolution are unique, and define invariants called relative Betti diagrams. Using Koszul complexes, relative Betti diagrams can be computed in a simple and local way, avoiding the computation of the entire relative resolution.
The talk is based on a joint work with Wojciech Chachólski, Isaac Ren, Martina Scolamiero, and Francesca Tombari.
Real uniformly distributed sequences have been widely studied due to their connections to number theory and other fields of mathematics as well as their practical applications. In this talk, we review some of the most important concepts within this field. However, the notion of uniform distribution does not only make sense for real numbers but also in a p-adic setting. There are interesting parallels and differences to the real case which will be the main focus of the talk. The presentation will be kept accessible for a broad audience.
A Reeb graph is a graphical representation of a scalar function on a topological space that encodes the topology of the level sets. Reeb graphs and their variants are popular tools in topological data analysis and visualization. In this talk, I will discuss theoretical advances in studying Reeb graphs and their variants, as well as their applications in data mining and machine learning. In particular, I will discuss novel constructions of Reeb graphs that incorporate the use of a measure, which capture robust topology in data.
Sometimes the topology of a closed 3-manifold is dominated by a single closed surface together with some gluing data. A general challenge is to extract from such data information about the topology and the geometry of the manifold keeping in mind questions like: When does the manifold satisfy the hypothesis of the hyperbolization theorem of Thurston and Perelman? Is it possible to give a formula that turns the combinatorial data of the gluing into geometric invariants of the hyperbolic metric? In this talk, I will explain some recent joint work with Peter Feller and Alessandro Sisto addressing these questions for hyperbolic Heegaard splittings (a Heegaard splitting is a decomposition of a closed 3-manifold into two handlebodies intersecting along their common boundary, the so-called Heegaard surface). In particular, we are able to detect and control the length of some very short closed geodesics in the manifold (giving some novel information about the thick-thin decomposition of the manifold).
The classical Shannon-McMillan-Breiman (SMB) theorem states that for an ergodic measure preserving transformation on a probability space, the rate of uncertainty measured by mathematical entropy can be observed in almost every orbit of the system.
We explain how to use an abstract skew product construction in order to prove such results for measure preserving actions of negatively curved groups, with refinements being taken along random almost geodesics. Joint work with Amos Nevo.
In this talk we will investigate the maximal possible intersection of closed curves of a given length on a surface, and we will specifically focus on the case of translation surfaces. One way to approach this question is to consider the so-called (algebraic) interaction strength (here denoted KVol). After having introduced a few notions related to KVol as well as translation surfaces, we will give a method to compute KVol on SL_2(R)-orbits of Bouw-Möller surfaces with a single singularity, which are instances of particularly symmetric translation surfaces (their Veech group are triangle groups).
Let S be a compact, connected, oriented surface of genus g with one boundary curve, and for n>=0 denote by F_n(S) and C_n(S), respectively, the ordered and the unordered configuration spaces of n distinct points in S. The mapping class group Mod(S) of isotopy classes of boundary-fixing homeomorphisms of S acts naturally on the homology of F_n(S) and C_n(S) with coefficients in any ring R. I will discuss what the kernel of this action is, emphasizing how the answer depends on n in the ordered case, but (almost) does not depend on n in the unordered case. This combines joint work with Jeremy Miller and Jennifer Wilson, and join work with Andreas Stavrou.
I will present some criteria (necessary or sufficient) for the action on the affine space of a group Gamma of affine transformations to be proper. This is joint work with Fanny Kassel.
The main of these criteria links properness of action to the divergence of a parameter called the Margulis invariant. This invariant measures roughly the translation part of an affine transformation, but in a way that is invariant by conjugation.
This link was already known in some special cases (and has often been exploited to construct proper actions). We tried to establish it in as general setting as possible. We proved it in particular if Gamma has some suitable Anosov property (with respect to some natural parabolic subgroup, that depends on the affine group we are working in).
I will possibly also evoke some other invariants similar to the Margulis invariant, that could lead to criteria that work in even more general settings.
Let us consider a tiling of the Euclidean plane by polygons. We play the billiard on this tiling in the following way. A trajectory goes in straight line in each tile. When it reaches a boundary between two tiles, it crosses the boundary and is refracted in the new tile. We get a zigzaging trajectory in the plane. Our goal is to understand what behaviour it can have: Is it periodic or not? bounded or not? If not, how does it go to infinity?
I will first present this dynamical system, and explain how to study it with interval exchange transformations with flips, which are piecewise isometries of the circle.
This will allows me to state a result of my PhD thesis about deviations from asymptotic direction of some tiling billiards trajectories and to explain its proof.
The study of surface group representations into a Lie group is described by the character variety. I shall present a new approach to study character varieties, by so-called Fock bundles. Although similar to Higgs bundles, the new theory does not need a complex structure on the underlying surface. A Fock bundle induces a higher complex structure and gives a connection between these geometric structures and the Hitchin component. Joint work with Georgios Kydonakis and Charlie Reid.
Kahn and Markovic proved the Surface Subgroup conjecture for closed hyperbolic 3-manifolds more than ten years ago. The surface subgroup they constructed can be as close as possible to Fuchsian. However, a closed hyperbolic 3-manifold can also have surface subgroups far away from being Fuchsian. Our result states that provided any genus-2 quasi-Fuchsian group Γ and cocompact Kleinian group G, then for any K>1, one can find a surface subgroup H of G that is K-quasiconformally conjugate to a finite index subgroup F
Multifarious phenomena in quantum field theories are driven or accompanied by extended field configurations such as topological defects. Defects are also expected to play a prominent role for universal self-similar dynamics in systems far from equilibrium. Yet, in simulations of the paradigmatic O(N) vector model, they often appear vastly suppressed in correlations. Persistent homology can provide complementary information to correlations and reveals signatures of the different N-dependent topological defects as I will show. The Betti numbers of sublevel sets of local energy densities contain clear indications for the self-similar scaling of the defects, which has been hard to detect so far.
This application-oriented talk is based on joint work with Viktoria Noel.
Entire functions with finitely many singular values share many dynamical features with polynomials. This is even more the case in the setting of postsingularly finite (psf) functions, here there is not only a finite set of singular values, but every singular value also has a finite orbit.
I will give a brief overview of the dynamics of psf entire functions and show how to approximate psf entire functions by psf polynomials. The main tool is a combinatorial convergence of topological models related to Thurston's classification of branched covers.
This is based on joint work with Malavika Mukundan and Nikolai Prochorov.
The fundamental group of an n-dimensional closed hyperbolic manifold admits a natural isometric action on the hyperbolic space H^n. If n is at most 3 or the manifold is arithmetic of simplest type, then the group also admits many geometric actions on CAT(0) cube complexes. I will talk about a joint work with Nic Brody in which we approximate the asymptotic geometry of the action on H^n by the actions on these complexes, solving a conjecture of Futer and Wise. The main tool is a codimension-1 generalization of the space of geodesic currents introduced by Bonahon. In the 3-dimensional case, we also use some results about minimal surfaces in hyperbolic 3-manifolds.
For S a closed surface of genus at least 2, Labourie proved that every Hitchin representation of pi_1(S) into PSL(n,R) gives rise to an equivariant minimal surface in the corresponding symmetric space. He conjectured that uniqueness holds as well (this was known for n=2,3), and explained that if true, then the space of Hitchin representations admits a mapping class group equivariant parametrization as a holomorphic vector bundle over Teichmuller space.
In this talk we will discuss the analysis and geometry of minimal surfaces in symmetric spaces, and explain how certain large area minimal surfaces give counterexamples to Labourie’s conjecture. This is all joint work with Peter Smillie.
We present variations of an argument of Shalen for demonstrating linearity of certain amalgamated products of linear groups. Applications include (low-dimensional) linearity of some 3-manifold group amalgams. This is joint work in progress with Konstantinos Tsouvalas.
Goldman Conjectured that for components of the PSL(2,R)-character variety of surface groups which do not contain holonomies of hyperbolic structures, the action by the mapping class group should be ergodic. Recently, March\'e and Wolff completely described the genus 2 case, wherein the euler number zero representations surprisingly split into two ergodic components. In new work with James Farre and Peter Smillie, we use a geometric interpretation of the tangent data to reducible representations within the euler class 0 component to give positive evidence towards Goldman's conjecture in the case where genus excedes two and euler number is zero.
A uniform lattice in SL_n(R) with n at least 3 does not admit non-trivial deformations in SL_n(R), thanks to Margulis super-rigidity. However, such a lattice has a natural action on the Furstenberg boundary. Then, a natural question is whether the lattice can be deformed in this (huge) homeomorphism group of the Furstenberg boundary. We prove a semi-conjugacy rigidity result for such deformations. Surprisingly, we also show that no such rigidity holds for the visual boundary of the symmetric space SL_n(R)/SO_n. This is joint work with Chris Connell, Thang Nguyen, and Ralf Spatzier.
I will explain work in progress on a new class of non-commutative cluster algebras which generalize the work of Berenstein and Retakh on non-commutative surfaces. Time permitting, I will briefly discuss the connection with A-type cluster coordinates (Fock-Goncharov coordinates) for decorated representations into Spin(p,q) and the relation to Theta-positivity.
We will give a short overview of the Nielsen-Thurston classification problem on finite-type surfaces and then move to the big setting and see what we know. The objective is to try and push some more geometric approaches to the classification that have not been widely touched in the infinite-type setting with some examples.
Learning numerical representations of graphs is a fundamental problem in machine learning, with traditional Euclidean approaches often falling short in capturing important features such as hierarchies. Motivated by this limitation, there has been growing interest in developing geometric representation learning frameworks that better reflect the properties of target graphs, such as embedding hierarchies in hyperbolic spaces. In this talk, we discuss two such frameworks that go beyond hyperbolic geometry. This talk is based on work with Federico Lopez, Max Riestenberg, Michael Strube, Steve Trettel, and Wei Zhao.
The semistable Higgs bundle moduli space comes naturally equipped with a C*-action and a C*-invariant proper fibration projecting onto a vector space called the Hitchin base. A C*-fixed point is called very stable if it does not admit any other nilpotent Higgs bundle in its upward flow apart from itself. The classification of connected components in the nilpotent cone containing very stable C*-fixed points is a topic of current research. In this talk I will speak about Hecke transformations of Higgs bundles and demonstrate how they can be utilized to study very stable C*-fixed points in the Higgs moduli space.
Due to the isomonodromic property, the space of solutions of the Painleve equations can be parameterized by the monodromy data. Namely, each of the equations can be associated with an affine cubic that is usually called the monodromy surface. Various examples of non-abelian analogs of the Painleve equations have arisen in recent years. In this talk we will discuss how to derive a non-commutative analog of the monodromy surface for them. We will consider the second Painleve equation as an example.
There is a long tradition of using probabilistic methods to solve geometric problems. I will present one such result. Namely, I will show that if the bottom of the spectrum of the Laplacian on a hyperbolic n manifold M is equal to that of its universal cover (or equivalently the fundamental group has exponential growth rate at most (n-1)/2) then M has points with arbitrary large injectivity radius.
Selberg introduced a fundamental domain for discrete groups acting on SL(n,R)/SO(n) which is convex in the projective model. He showed that for uniform lattices, the domain is a finite-sided convex polyhedron. We give sufficient criteria for Anosov subgroups to admit finite-sided Dirichlet-Selberg domains. This is joint work with Colin Davalo.
The character variety associated to a surface group and a semisimple Lie group comes naturally equipped with the so called Goldman symplectic form, and hence one can study Hamiltonian flows on it. One particular instance of such flows are twist flows along simple closed curves, which correspond to „length functions“. I will talk about ongoing work, where instead of considering functions on a single simple closed curve, I look at functions on multiple simple closed curves.
Every horocycle in a closed hyperbolic surface is dense, and this has been known since the 1940's. We study the behavior of horocycle orbit closures in Z-covers of closed surfaces, and obtain a fairly complete classification of their topology and geometry. The main tool is a solution of a surprisingly delicate geometric optimization problem: finding an optimal Lipschitz map to the circle and the associated lamination of maximal stretch. This is joint work with Yair Minsky and Or Landesberg.
It is interesting to ask which word hyperbolic groups arise as Anosov subgroups of semisimple Lie groups. One way to restrict the class of groups is to study how the boundary map sits inside the boundary of the Lie group. Cluster algebras provide a useful tool to understand the topology of these boundaries.
A question of Misha Kapovich asks whether SL_3(Z) contains a subgroup isomorphic to the free product Z^2\ast Z. Motivated by this question, I am going to discuss a characterization of divergent Z^2 subgroups of SL_3(R). This characterization, combined with results of Oh, shows that a Zariski-dense discrete subgroup Γ of SL_3(R) contains a regular Z^2 if and only if Γ is commensurable to a conjugate of SL_3(Z). This is joint work with Sami Douba.
I will talk about Riemann-Hilbert-Birkhoff correspondence between general meromorphic G-connections and Stokes representations, and some ideas on the nonabelian Hodge correspondence.
Let M be a closed hyperbolic 3-manifold and let Gr(M) be its 2-plane Grassmann bundle. We will discuss the following result: the weak-* limits of the probability area measures on Gr(M) of pleated or minimal closed connected essential K-quasifuchsian surfaces as K goes to 1 are all convex combinations of the probability area measures of the immersed closed totally geodesic surfaces of M and the probability volume (Haar) measure of Gr(M).
The stable, unstable (and central) distributions of (partially) hyperbolic dynamics are a priori only Hölder continuous, and several works seem to suggest that their lack of regularity is in fact the only obstacle to their rigidity. Concerning contact-Anosov flows, successive works of Ghys (in dimension three) and Benoist-Foulon-Labourie (in higher dimensions) have for instance proved that the smoothness of the stable and unstable distributions forces the system to be algebraic. In this talk, I will present an analog rigidity result for three-dimensional volume-preserving partially hyperbolic diffeomorphisms whose stable, unstable and central distributions are smooth, and whose stable-unstable plane field is a contact distribution.
I'll discuss joint work with Sami Douba, Balthazar Flechelles, and Feng Zhu, in which we prove that every hyperbolic group acting geometrically on a CAT(0) cube complex admits Anosov representations.
Dyer groups are a family of groups which includes both Coxeter groups and right-angled Artin groups. The goal is to introduce this family, give some properties and some complexes associated to them.
We discuss past work at the intersection of quantum topology and higher Teichmüller theory. One of the guiding philosophies is "Fock-Goncharov duality".
Thurston’s metric is defined on Teichmüller space, which contains very rich geometry. I am trying to mimic the definition and generalize to other spaces, for example, flat cone metrics and positively ratioed representations.
A group equivariant operator transforms measurements of data into other measurements while respecting certain group actions on them. If such operators are also non-expansive, then we talk about GENEOs. GENEOs have been introduced to model neural networks and inject geometrical knowledge about the data, encoded by the group action, to be preserved by the machine. However, data rarely satisfy rigid symmetries due to noise, incompleteness or symmetry-breaking features, and hence the group axioms may not be satisfied. Thus, we propose a topological model to encode partial equivariance in neural networks. To this end, we introduce a class of operators, called P-GENEOs, that change data expressed by measurements, respecting the action of certain sets of transformations, in a non-expansive way. We then study the spaces of measurements, whose domains are subject to the action of certain self-maps, and the space of P-GENEOs between these spaces.
How do you email a Torus? As a parallelogram with opposite sides identified, as an elliptic curve, or as a product of circles embedded in 3-space? We will discuss some presentations of Riemann surfaces, difficulties in moving between the presentations and interesting open problems that arise when considering the moduli space of Riemann surfaces.
In this talk I will discuss about possible generalizations of the notion of equivariant pleated surfaces in the context of higher rank Lie groups, giving a brief overview of what we presently know (based on joint works with G. Viaggi for representations in SO(2,n), and with S. Maloni, G. Martone, and T. Zhang in PSL(d,C)) and what we would like to learn more about.
Linear submanifolds of strata of holomorphic abelian differentials are rare objects. Up to today, only few series in low dimension are known. The classification of linear submanifolds is an open problem, and the question about the existence of examples in higher dimension is still unanswered.
One approach to their study and a possible classification is to analyze their closure inside as suitable compactification of the ambient stratum. I will describe the phenomena encountered at the boundary for the example of the Gothic locus, one of the few exceptional linear submanifolds recently discovered.
As an application of the compactification, I will describe the intersection theoretic framework to compute invariants of linear submanifolds (for example the Masur-Veech volume and Siegel-Veech constants) that we implemented as a SageMath package.
In recent decades, discrete complex analysis has seen remarkable advancements, despite its relatively late inception compared to its continuous counterpart. While continuous complex analysis boasts a long-established history, the discrete analog has emerged more recently and remains an active area of research and exploration.
This presentation focuses on a pivotal aspect of Riemann surfaces: their period matrices. After providing a concise introduction to the linear theory of discrete Riemann surfaces, we introduce the discrete period matrix—a direct counterpart to its continuous counterpart. We also delve into the larger complete discrete period matrix, whose calculation is based on a broader class of discrete holomorphic differentials, thereby giving more information on the underlying discrete Riemann surface.
Our discussion extends to the convergence of discrete period matrices when a sequence of increasingly fine discretizations is applied to a given compact polyhedral surface. We outline the proof strategy for the convergence of these discrete period matrices to the period matrix of the original surface, and elucidate the limits of the blocks within the complete discrete period matrix.
In closing, we explore a discretization of real Riemann surfaces and their period matrices, a collaborative effort with Johanna Düntsch. As one expects, we find that the corresponding discrete period matrices exhibit the same symmetrical properties as their continuous counterparts, with the complete discrete period matrices following a similar pattern.
A translation surface is a collection of polygons with edge identifications given by translations. In spite of the simplicity of the definition, the space of translation surfaces has connections to different areas of math such as the moduli space of Riemann surfaces and rational billiards in the plane.
In this talk we consider the unstable foliation, that locally is given by changing horizontal components of period coordinates, which plays an important role in study of translation surfaces, including their deformation theory and in the understanding of horocycle invariant measures. We show that measures of "large dimension" equidistribute and give an effective rate. An analogous result in the setting of homogeneous dynamics is crucially used in the recent effective equidistribution results of Lindenstrauss-Mohammadi and Lindenstrauss--Mohammadi--Wang.
No prior knowledge of the words in the title is assumed.
The theme of this lecture centers around two famous books, one by Hermann Weyl on “The Classical Groups: Their Invariants and Representations” (1939), another the six-volume work by I.M. Gelfand et al. on “Generalized Functions” (1960’s). I will discuss some broad ideas related to the theme of these two books, as well as some recent development in classical groups and their smooth representations.
The compactification of the moduli space of super Riemann surface has been studied by Deligne (1987) and more recently by Donagi and Witten (2015). We aim to extend those results by constructing a moduli space of stable maps.
Let M be a super Riemann surface with holomorphic distribution D and N a symplectic manifolds with compatible almost complex structure J. We call a map Φ: M-> N a super J-holomorphic curve if its differential maps the almost complex structure on D to J. Such a super J-holomorphic curve is a critical point for the superconformal action and satisfies a super differential equation of first order. Using component fields of this super differential equation and a transversality argument we construct the moduli space of super J-holomorphic curves as a smooth subsupermanifold of the space of maps M->N. The reduced space of the moduli space coincides with the moduli space of J-holomorphic curves from the reduction of M to N.
Gaussian heat kernel bounds play a fundamental role in geometric analysis. We present recent results on explicit Gaussian upper bounds for non-compact manifolds depending on locally uniform Ricci curvature integral assumptions. Furthermore, we discuss generalizations of integral curvature bounds in terms of the so-called Kato class. If time allows, topological applications of those heat kernel upper bounds will be given.
We start giving an overview of Bakry-Emery and Ollivier curvature and corresponding gradient estimates which require global curvature bounds. We then introduce the perpetual cutoff method to give gradient estimates requiring the curvature bound not everywhere. As an application, we give a sharp distance bound to the set where no curvature bound is assumed.
We study the asymptotic behavior of solutions to subcritical Yamabe equation in a punctured ball. We prove that an admissible solution to this equation with a non-removable isolated singular point is asymptotic to a radial solution. Then we are able to obtain higher order expansion of solutions using analysis of the linearized operators. These results generalize earlier pioneering work of Caffarelli, Gidas and Spruck.
On a class of closed oriented Riemannian manifolds with lower bounds on Ricci curvature and injectivity radius we study the parameter given by a characteristic number over the volume of the manifold. The domain of this parameter is endowed with the Benjamini-Schramm (BS) topology. The BS-topology is a weak notion of convergence, that originated from graph theory, where it is insensively studied, but also has applications in various other areas. Using an integral characterization (actually, a generalization of Gauss-Bonnet formula) one can show that the parameter explained is continuous and has a continuous extension to the completion of its domain in the BS topology. From the known fact that this completion is compact one can derive that the parameter is testable in constant time.
Damek-Ricci spaces are some one-dimensional extensions of Heisenberg groups. We prove that there does not exist non-geodesic biharmonic curves in a four-dimensional Damek-Ricci space although there exist such curves in three-dimensional Heisenberg groups.
In this talk, we give a new estimate for the eigenvalues of the Dirac operator on a compact spin manifold in terms of an appropriate endomorphism of the tangent bundle associated with an eigenspinor . We then show that, for isometric immersions and Riemannian flows (local Riemannian submersions), the limiting case could be achieved. In this case the tensor is identified with the second fundamental form of the immersion while it is identified with the O'Neill tensor of the flow.
