Let $X$ be a hyperbolic surface of genus $g$. Maryam Mirzakhani proved that the number of simple closed geodesics of length less then $L$ in $X$ grows polynomially in $L$. She also showed that the coefficient of the highest term of asymptotics is related to the Thurston measure of the set of measured laminations which has length less then $1$ in $X$. Recently, this was generalized by several authors to the count of mapping class group orbits of geodesic currents. In my talk I will explain these results and will talk about weighted count of mapping class group orbits. I will also talk about two different ways to normalise Thurston measure and how are these normalisation different from each other.

Spatial graphs are (piecewise-linear) graphs embedded in $S^3$. In joint work with S. Friedl, L. Munser and Y. Santos Rego, we show that there exists an algorithm to test whether two spatial graphs are isomorphic. The proof relies on a theorem of Matveev in piecewise-linear $3$-manifold topology, which will be black-boxed, and on the existence of a canonical decomposition of spatial graphs into pieces to which Matveev's theorem applies.

An arrangement of hyperplanes is called formal provided all linear dependencies among the defining linear forms of the hyperplanes are generated by ones corresponding to intersections of codimension two. This notion turns out to be necessary at the one hand for the apshericity of the complement of a complex arrangement due to work by Falk and Randell. One the other hand it is also necessary for the freeness of the module of logarithmic vector fields thanks to a result by Yuzvinsky. In joint work with T. Möller and G. Röhrle we extend the above line of results by showing that the combinatorial property of factoredness implies formality. Furthermore, we study formality with respect to the standard arrangement constructions of restriction and localization and comment on the behavior of the stronger property of k-formality introduced by Brandt and Terao.

The $3$-strand braid group is isomorphic to the knot group of the trefoil knot. The Coxeter presentation of the symmetric group on $3$ letters is obtained from a presentation of the knot group where generators are meridians (or Dehn twists) by declaring meridians to have order two. We study all possible quotients of torus knot groups obtained in this way, that is, by requiring meridians to have finite order. We thereby obtain a three-parameter family of groups, which are infinite in general, and behave like complex reflection groups of rank two. In particular they can be endowed with a "reflection group" structure in some sense. These groups include Coxeter's truncations of the three strand braid group, as well as all finite complex reflection groups of rank two with a single conjugacy class of reflecting hyperplanes. We classify these reflection groups, show that they have a cyclic center, and that they are central extensions of alternating subgroups of Coxeter groups of rank three. We will also give a few open questions about these groups.

We study algebras of modular forms on unitary groups of signature $(n,1)$. We give a sufficient criterion for the ring of unitary modular forms to be freely generated in terms of the divisor of a modular Jacobian determinant. We use this to prove that a number of rings of unitary modular forms associated to Hermitian lattices over the rings of integers of $Q(\sqrt{d})$ for $d=−1,−2,−3$ are polynomial algebras without relations. This is joint work with Haowu Wang.

Let $F$ be a non-Archimedean local field with a ring of integers $O$ and a finite residue field $k$. In contrast to the well-understood representation theory of the finite groups of Lie type $GL_n(k)$ or of the locally compact groups $GL_n(F)$, representations of groups $GL_n(O)$ are considerably less understood. For example, the uniqueness of the Whittaker model is well known for the complex representations of both $GL_n(k)$ and $GL_n(F)$. In this talk, we will discuss the parallel questions for $GL_n(O)$. We will prove the uniqueness of the Whittaker model for representations of $GL_n(O)$ and describe its irreducible constituents. This talk is based on joint work with Shiv Prakash Patel.

Heckenberger introduced the Weyl groupoid of a finite dimensional Nichols algebra of diagonal type. We replace the matrix of its braiding by a higher tensor and present a construction which yields further Weyl groupoids. This is joint work with Tobias Ohrmann.

The representation theory of a reductive algebraic group $G$, defined over an algebraically closed field of characteristic $p>0$, is governed to a large extent by the alcove geometry with respect to the $p$-dilated shifted action of its affine Weyl group $W$ on the weight lattice of $G$. More specifically, the action of $W$ on the weight lattice gives rise to a decomposition of the category $\mathrm{Rep}(G)$ of finite-dimensional rational $G$-modules into so-called linkage classes, one for each weight in the closure of a fixed alcove. Furthermore, it often suffices to understand one specific linkage class in order to obtain information about all other linkage classes via Jantzen's translation principle. This strategy is (a priori) not well-suited for studying tensor products of $G$-modules because, in general, the linkage classes are not closed under tensor products. The aim of this talk is to describe a method for overcoming this problem. We define a tensor ideal of singular $G$-modules and consider the quotient of $\mathrm{Rep}(G)$ by this tensor ideal. It turns out that both the decomposition of $\mathrm{Rep}(G)$ into linkage classes and the translation principle are well-behaved with respect to tensor products in the quotient category. This makes it possible to combinatorially describe the structure of tensor products of $G$-modules, up to singular direct summands, in terms of the affine Weyl group.

We discuss foundational questions in real algebraic geometry going back to Hilbert and Artin from a new perspective, namely that of projective geometry. This has led to new insights into classical questions in real algebraic geometry as well as new questions and results in projective algebraic geometry. The connection between real questions about nonnegativity and invariants complex algebraic geometry is based on the special role that spectrahedra play for sums of squares of polynomials. This is based on joint work with Greg Blekherman, Greg Smith, and Mauricio Velasco.

The modular isomorphism problem asks whether the isomorphism type of the modular group algebra of a $p$-group $G$ over a field of characteristic $p$ determines the isomorphism type of $G$. It was explicitly mentioned in a survey by Richard Brauer in 1963, and was the only classical version of the isomorphism problem for group rings which had resisted a solution, though it received considerable attention. Several partial positive solutions were obtained imposing very strong conditions on the group $G$, for instance the one of being metacyclic. In a joint work with Leo Margolis and Ángel del Rı́o, we solve the modular isomorphism problem in the negative by exhibiting a series of pairs of non-isomorphic $2$-groups with isomorphic modular group algebras over every field of characteristic $2$. These groups are two-generated with cyclic derived subgroup. We will also discuss the (lack of) posi- bility of obtaining, in a naive way, analogues to the counterexamples for $p>2$ verifying this condition.

An $n$-person game is specified by $n$ tensors of the same format. We view its equilibria as points in that tensor space. Dependency equilibria are defined by linear constraints on conditional probabilities, and thus by determinantal quadrics in the tensor entries. These equations cut out the Spohn variety, named after the philosopher who introduced dependency equilibria. The Nash equilibria among these are the tensors of rank one. We study the real algebraic geometry of the Spohn variety. We characterize the payoff regions and their boundaries using oriented matroids, and we develop the connection to Bayesian networks in statistics.