The sequential topological complexities (TCs) of a space are integer-valued homotopy invariants that are motivated by the motion planning problem from robotics. After outlining their definitions, I will discuss the sequential TCs of aspherical spaces and describe how they can be investigated by purely algebraic means. Sectional categories of subgroup inclusions are straightforward generalizations of this algebraic setting and we will discuss how to obtain lower bounds on their values using elementary homological algebra and discuss consequences for sequential TCs. This is joint work with Arturo Espinosa Baro, Michael Farber and John Oprea.
Asymptotic representation theory deals with questions of probabilistic flavor about representations of groups of growing size. For classical Lie groups there are two distinguished regimes of growth. One of them is related to representations of infinite-dimensional groups, and the other appears in random tilings problems. In the talk I will discuss differences and similarities between these two settings.
It is an interesting open question whether the full group of a hyperfinite equivalence relation (or its not so distant relative, the unitary group of the hyperfinite II_1-factor) can contain a discrete non-abelian free subgroup. Motivated by this, we will discuss various constraints on general countable subgroups in the full group, the nature of the action, and on the measure of fixed point sets, that imply non-discreteness. It turns out that an important condition here is the so-called MIF property (mixed identity-free) which had been studied and used in various other contexts. The talk is based on the recent joint work with Alessandro Carderi, Andreas Thom and Robin Tucker-Drob.
Gromov hyperbolic groups were introduced by Gromov in the 80s as a generalization of the class of fundamental groups of closed negatively curved Riemannian manifolds. A fundamental open question in geometric group theory is whether all hyperbolic groups are residually finite, and if such examples exist, they are necessarily non-linear. In this talk, I will describe a construction of the first known examples of non-linear hyperbolic groups which are residually finite. This is joint work with Nicolas Tholozan.
There are several concepts of "self-similar spaces", which can be assigned properties like fractal dimensions or L²-invariants. However, the word "invariant" is hard to justify unless we can find suitable maps between these spaces to compare them. So what exactly should the "self-similar category" be? We'll explore several approaches and their successes and failures.
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. I will focus on a novel construction of these optimal Lipschitz maps using the orthogeodesic foliation construction. This is joint work with Yair Minsky and Or Landesberg featuring joint work with Aaron Calderon.
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).
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).
Consider the closed convex hull K of a monomial curve given parametrically as (tm1,…,tmn), with the parameter t varying in an interval I. We show, using constructive arguments, that K admits a lifted semidefinite description by O(d) linear matrix inequalities (LMIs), each of size ⌊n2⌋+1, where d=max{m1,…,mn} is the degree of the curve. On the dual side, we show that if a univariate polynomial p(t) of degree d with at most 2k+1 monomials is non-negative on ℝ+, then p admits a representation p=t0σ0+⋯+td−kσd−k, where the polynomials σ0,…,σd−k are sums of squares and deg(σi)≤2k. The latter is a univariate positivstellensatz for sparse polynomials, with non-negativity of p being certified by sos polynomials whose degree only depends on the sparsity of p. Our results fit into the general attempt of formulating polynomial optimization problems as semidefinite problems with LMIs of small size. Such small-size descriptions are much more tractable from a computational viewpoint.
https://arxiv.org/abs/2303.03826
I will briefly recall concepts of CAT curvature, some of its geometric and topological consequences, and then discuss three examples: toric varieties, iterated blowups and ramified coverings. In the first two existence of CAT0 metric is established in some generality, ramified coverings are somewhat problematic, but perhaps most interesting.
Seminar webpage: https://www.math.uni-leipzig.de/cms2/en/institut/seminare/acseminar/
We give a comprehensive structural analysis of amenable subrelations of a treed measure-class-preserving equivalence relation. The main philosophy is to understand the behavior of the Radon-Nikodym cocycle in terms of the geometry of the amenable subrelation within the treeing. This allows us to extend structural results that were previously only known in the measure-preserving setting, e.g., we show that every nowhere smooth amenable subrelation is contained in a unique maximal amenable subrelation. Two of the main ingredients are an extension of Carrière and Ghys's criterion for nonamenability, along with a new Ping-Pong-style argument we call the "Paddle-ball lemma" that we use to apply this criterion in our setting. This is joint work with Anush Tserunyan.
Let $\mathcal{V}$ be a hypersurface in a $n$--dimensional projective space The Hessian map is a rational map from $\mathcal{V}$ to the projective space of symmetric matrices that sends $p\in \mathcal{V}$ to the Hessian matrix of the defining polynomial of $\mathcal{V}$ specialized at $p$. The Hessian correspondence is the map that sends a hypersurface to Zariski closure of its image through the Hessian map. In this paper, we study this correspondence for the cases of hypersurfaces of degree $3$ and $4$. We prove that, for degree $3$ and $n=1$, the map is two to one, and that, for degree $3$ and $n\geq 2$, and for degree $4$, the Hessian correspondence is birational. Moreover, we provide effective algorithms for recovering a hypersurface from its image through the Hessian map for degree $3$ and $n\geq 1$, and for degree $4$ and $n$ even.
I will explain that all affine isometric actions of higher rank simple Lie groups and their lattices on arbitrary uniformly convex Banach spaces have a fixed point. This vastly generalises a recent breakthrough of Oppenheim. Combined with earlier work of Lafforgue and of Liao on strong Banach property (T) for non-Archimedean higher rank simple groups, this confirms a long-standing conjecture of Bader, Furman, Gelander and Monod. As a consequence, we deduce that box space expanders constructed from higher rank lattices are superexpanders.
This is joint work with Mikael de la Salle.
A compact group $A$ is called an amalgamation basis if, for every way of embedding $A$ into compact groups $B$ and $C$, there exist a compact group $D$ and embeddings $B\to D$ and $C\to D$ that agree on the image of $A$. Bergman in a 1987 paper studied the question of which groups can be amalgamation bases. A fundamental question that is still open is whether the circle group $S^1$ is an amalgamation basis in the category of compact Lie groups. Further reduction shows that it suffices to take $B$ and $C$ to be the special unitary groups. In our work, we focus on the case when $B$ and $C$ are the special unitary group in dimension three. We reformulate the amalgamation question into an algebraic question of constructing specific Schur-positive symmetric polynomials and use integer linear programming to compute the amalgamation. We conjecture that $S^1$ is an amalgamation basis based on our data. This is joint work with Michael Joswig, Mario Kummer, and Andreas Thom.
Given a group action on a measure space X, one can generate a C*-algebra by considering bounded operators on L^2(X) that have "finite dynamical propagation". This construction is inspired by the definition of Roe algebras of metric spaces, and was originally motivated by the study of the Roe algebras of certain metric spaces of dynamical origin (warped cones).
Rather than using C*-algebras, one would usually study group actions on measure spaces in terms of von Neumann algebras. It is then an interesting fact that this C*-algebra does carry meaningful information about the dynamical system. For example, it be used to recognise strong ergodicity (this property is a strong negation of Zimmer amenability). This fact can be proved by investigating spectral properties of some Markov operators which are of independent interest.
This talk will be an introduction to this circle of ideas.
A group is said to satisfy the Tits Alternative if any of its finitely generated subgroups either contains a free nonabelian group or is virtually (that is, has a finite index subgroup which is) solvable. Intuitively the alternative says that every such subgroup is either (respectively) "very big" or "very small". It is believed that groups acting properly on nonpositively curved spaces have this property. One important notion of nonpositive curvature for metric spaces is the CAT(0) property. Trees are 1-dimensional CAT(0) spaces, and it is relatively easy to show that groups acting properly on trees satisfy the Tits Alternative. For higher dimensions, the problem has been open. Together with Piotr Przytycki we showed that groups acting properly on 2-dimensional CAT(0) complexes satisfy the Tits Alternative. In particular, this proves the Tits Alternative for a few classical families of groups, including some Artin groups and some automorphism groups of affine spaces coming from algebraic geometry. I will present the result, its motivations, consequences, and elements of the proof.
In a joint work with Emiliano Ambrosi, we study the real topology of totally real semistable degenerations, with certain technical conditions on the special fiber X0, and we give a bound for the individual real Betti numbers of a smooth fiber near 0 in terms of the complex geometry of X0. The main ingredient is the use of real logarithmic geometry, which allows to work with degenerations which are not necessarily toric and hence to go beyond the case of tropically smooth degenerations. This, in particular, generalizes previous work of Renaudineau-Shaw, obtained via tropical techniques, to a more general setting.
I will discuss non-trivial identities and mixed identities for finite groups. I will be interested in their minimal length for various finite simple groups or families thereof.
Welche Eigenschaften der endlichen Gruppen können mit Formeln erster Stufe charakterisiert werden? Seit 20 Jahren weiß man, dass Auflösbarkeit eine solche Eigenschaft ist. Neulich wurde bewiesen, dass Nilpotenz und Perfektkeit nicht durch Formeln erster Stufe erkennbar sind. Um dass zu beweisen, muss man das Universum der endlichen Gruppen verlassen und Überlegungen über gewisse unendlichen Gruppen (z.B. Ultraprodukte) miteinbeziehen.
Which properties of the finite groups can be characterized with formulas of the first order? It has been known for 20 years that solvability is such a property. Recently it has been proved that nilpotency and perfection are not recognizable by first-order formulas. To prove this, you have to leave the universe of finite groups and include considerations about certain infinite groups (e.g. ultraproducts).
There are several ways of computing the degree of varieties, but the most basic one is to intersect your variety with a line. We will be doing the same in the tropical world. With the techniques developed we will compute the likelihood degeneracy degree of a matroid and recover a result from Agostini et al.In this talk, we will introduce the Bergman fan of a matroid, our tropical variety of interest, study its combinatorics, and identify its degree with the number of so called nbc bases.
Weyl's law describes the asymptotic behavior of the counting function for the discrete or cuspidal spectrum of the Laplace operator on a Riemannian manifold. In this talk, we consider locally symmetric manifolds, which may be compact or non-compact. The Weyl asymptotic is well known, even with an explicit power saving in the remainder term, in the case of compact locally symmetric manifolds and in many other cases. In the case of non-compact manifolds, we only expect Weyl's law to hold in the arithmetic case, and Lindenstrauss-Venkatesh (2007) indeed used Hecke operators (which only exist for arithmetic groups) to establish a general version of the law for the cuspidal spectrum. They did not provide any additional information on the magnitude of the remainder term. We show how to use Hecke operators and Arthur's trace formula to prove Weyl's law for the cuspidal spectrum with a power saving in the remainder term. This is joint work with Erez Lapid and Jasmin Matz.