Combinatorial surfaces can be seen as the limit of bordered hyperbolic surfaces with large boundaries, and we can study the problem of counting multicurves on them.For a given combinatorial structure, multicurves correspond to integral points in a polytope defined in terms of ribbon graphs, and the asymptotic number of multicurves of very large length is controlled by the volume of those polytopes. This volume is a function on the moduli space of bordered surfaces, and motivated by questions from random geometry, one can ask for which s is this function $L^s$. The analog of this question can be posed in the hyperbolic case, and the answer is: $s \leq 2$. In the hyperbolic case, recent results of Arana-Herrera show $L^2$. In the combinatorial case, we show using methods from convex geometry and analysis of divergences from subgraphs that the answer is $s < s_{g,n}$ for an explicit $s_{g,n}$ depending on the genus $g$ and the number of boundaries $n$. This leads to various (analytic and arithmetic) questions around such enumeration problems, which can be compared to questions about Feynman integrals. This is a joint work with Charbonnier, Delecroix, Giacchetto and Wheeler.
A Möbius strip is chiral: It has a right-handed and a left-handed version, where one cannot be deformed into the other. Any space that embeds into $R^d$ is not chiral in $R^{d+1}$. I will explain that general non-embeddabilty results may be extended to chirality results. In fact, one can use the combinatorics of triangulations of a space $X$ to lower bound the topology of the space of embeddings of $X$ into Euclidean space. This chirality phenomenon can also be explored in combinatorics. Here chirality translates into chromatic mixing for graphs, and this viewpoint gives a generalization of Lovász's general topological lower bound for the chromatic number to additionally prohibit chromatic mixing.The topological part is joint with Michael Harrison, the combinatorial part is joint with Gunmay Handa.
In this talk, I will present the results of joint work, partially still in progress, with Hulek-Liese and Dutour Sikirić - Hulek about a certain compactification of the moduli space of polarized K3 surfaces of degree 2d. The construction of the compactification is due to Gross-Hacking-Keel-Siebert and uses the birational geometry of the Dolgachev mirror family. It is a toroidal compactification in the sense of Mumford if 2d=2, or rather semi-toroidal in the sense of Looijenga for higher d. For 2d=2, we obtain very explicit information by counting the maximal cones, respectively the rays of the toric fan in question. These results are obtained by counting so-called curve structures, which are combinatorial objects associated to the various birational models of the mirror family.
In this talk, I will discuss the cases of exponential varieties, such as considered in the paper "Exponential Varieties" by Mateusz Michałek, Bernd Sturmfels, Caroline Uhler, Piotr Zwiernik and show its relation to Quantum Field theory, via the structure of Frobenius manifolds and F-manifolds.
Equations of hypertree divisors on the Grothendieck-Knudsen moduli space of stable rational curves, introduced by Castravet and Tevelev, appear as numerators of scattering amplitude forms for n massless particles in N=4 Yang-Mills theory in the work of Arkani-Hamed, Bourjaily, Cachazo, Postnikov and Trnka. We re-interpret and generalize leading singularities of MHV scattering amplitude forms as probabilistic Brill-Noether theory: the study of statistics of images of n marked points on a Riemann surface under a random meromorphic function. This leads to a beautiful physics-inspired geometry for various classes of algebraic curves: smooth, stable, hyperelliptic, real algebraic, etc.
We consider the approach of replacing trees by multi-indices as an index set of the abstract model space $\mathsf{T}$, which was introduced to tackle more classes of singular stochastic partial differential equations. We show that this approach is consistent with the postulates of regularity structures of Hairer when it comes to the structure group $\mathsf{G}$ (the purpose of which is the re-centering of the model when passing from one to another base point). In particular, $\mathsf{G}\subset{\rm Aut}(\mathsf{T})$ arises from a Hopf algebra $\mathsf{T}^+$ and a comodule $\Delta\colon\mathsf{T}\rightarrow \mathsf{T}^+\otimes\mathsf{T}$, which are intertwined in a specific way. In fact, this approach, where the dual $\mathsf{T}^*$ of the abstract model space $\mathsf{T}$ naturally embeds into a formal power series algebra, allows to interpret $\mathsf{G}^*\subset{\rm Aut}(\mathsf{T}^*)$ as a Lie group arising from a Lie algebra $\mathsf{L} \subset{\rm End}(\mathsf{T}^*)$ consisting of derivations on this power series algebra. These derivations in turn are the infinitesimal generators of two actions on the space of pairs (nonlinearities, functions of space-time mod constants). These actions are shift of space-time, and tilt by space-time polynomials. The Hopf algebra $\mathsf{T}^+$ arises from a coordinate representation of the universal enveloping algebra ${\rm U}(\mathsf{L})$ of the Lie algebra $\mathsf{L}$. The coordinates are determined by an underlying (however not entirely closed) pre-Lie algebra structure of $\mathsf{L}$. Strong finiteness properties, which are enforced by gradedness and the restrictive definition of $\mathsf{T}$, allow for this dualization in our infinite-dimensional setting.We also argue that our structure is compatible with the tree-based structure in case of the generalized parabolic Anderson model. More precisely, we construct an endomorphism between the abstract model spaces (which is neither onto nor one to one) that lifts to an endomorphism between the Hopf structures. This is joint work with Pablo Linares and Markus Tempelmayr.
In this talk, I will define real phase structures on matroid fans and prove that a real phase structure is cryptomorphic to providing an orientation of the underlying matroid. Real phase structures can be extended to general tropical varieties and we can define the real part. In the matroid setting, this yields the topological representation of an oriented matroid in the sense of Folkman and Lawrence. In addition, the real part determines a homology class in a real toric variety and the conditions for real phase structures can be thought of the real analogues of Minkowski weights of fans. This is joint work in progress with Johannes Rau and Arthur Renaudineau. Lastly, I will propose a definition of the first Stiefel-Whitney class of a matroid. This is a homological class which is zero if and only if the matroid is orientable. Determining whether a matroid is orientable NP-complete, so determining whether or not the class is non-zero can be expected to be very difficult in general.
Over 50 years ago, Hasse proved that the set of prime numbers dividing at least one integer of the form $2^n+1$ has density 17/24. This result has later been interpreted as a statement about the properties of 2 as an element of the multiplicative group of non-zero rational numbers. This point of view eventually led to the development of the so-called Kummer theory of commutative algebraic groups. After a general introduction to the subject, which has connections to many classical problems in number theory, I will discuss some more recent results in which the multiplicative group is replaced by an abelian variety, with a particular focus on the case of elliptic curves.
Based on joint work with Antonella Perucca and Sebastiano Tronto.
A widespread principle in real algebraic geometry is to find and use algebraic certificates for geometric statements. This covers for example writing a globally nonnegative polynomial as a sum of squares or expressing a polynomial with only real zeros as the minimal polynomial of a symmetric matrix. In the first part of the talk I will survey some classical results in this direction. I will give a brief introduction to real del Pezzo surfaces and explain how they fit into the aforementioned context. In the second part, I will use the geometry of real del Pezzo surfaces to construct certain Ulrich bundles and explain how this can be used to obtain representations as sums of squares or characteristic polynomials.
The problem of understanding $H(\mathcal M_g)$ is undoubtedly classical but despite the great effort of a lot of mathematicians, we know a little about it. One source of classes (tautological classes) in $H(\mathcal M_g)$ comes from the elements of the Chow group of $\mathcal M_g.$ On another hand, the computation of the Euler characteristic of $\mathcal M_g$ by J. Harer and D. Zagier shows that tautological classes form a very small part of $H(\mathcal M_g).$ Namely there should be a lot of non-tautological classes and specifically a lot of odd degree classes for even $g.$ However the first cohomology class of odd degree was found only in 2005 by O. Tommasi.
After an overview of some known results about the cohomology of $\mathcal M_g$, I will explain the certain approach which should shed some light on where we should look for non-tautological classes in $H(\mathcal M_g).$ This approach is based on the diamond ribbon graph complex, which was introduced by S. Merkulov and T. Willwacher and has roots in the deformation theory of the involutive (diamond) Lie bialgebras. The talk is based on the work in progress.
In this talk, I will present a completely algebro-geometric proofs of a theorem by T. Shiota, and of a theorem by I. Krichever, characterizing Jacobians of algebraic curves among all irreducible principally polarized abelian varieties. Shiota's characterization is given in terms of the KP equation. Krichever's characterization is given in terms of trisecant lines to the Kummer variety. I will treat the case of flexes and degenerate trisecants. The basic tool that I will use is a theorem asserting that the base locus of the linear system associated to an effective line bundle on an abelian variety is reduced. This result will allow me to remove all the extra assumptions that were introduced in the theorems by E. Arbarello, C. De Concini, G.Marini, and O. Debarre, in order to achieve algebro-geometric proofs of the results above. This is a joint work with E. Arbarello and G. Pareschi.
The Diamond Lemma is a result indispensable to those studying associative (and other types of) algebras defined by generators and relations. In this talk, I will explain how to obtain a new approach to this celebrated result through the homotopical algebra of associative algebras: we will see how every multigraded resolution of a monomial algebra leads to "its own" Diamond Lemma, which is hard-coded into the Maurer-Cartan equation of its tangent complex. For the reader familiar with homotopical algebra, we hope to provide a conceptual explanation of a very useful but perhaps technical result that guarantees uniqueness of normal forms through the analysis of "overlapping ambiguities". For a reader familiar with Gröbner bases or term rewriting theory, we hope to offer some intuition behind the Diamond Lemma and at the same time a framework to generalize it to other algebraic structures and optimise it. This is joint work with Vladimir Dotsenko (arXiv:2010.14792).