Artin (or Artin-Tits) groups are generalizations of braid groups that are defined using a finite set of generators $S$ and relations $abab\cdots=baba\cdots$, where both words of the equality have the same length. Although this definition is quite simple, there are very few results known for Artin groups in general. Classic problems as the word problem or the conjugacy problem are still open. In this talk, we study a problem concerning a family of subgroups of Artin groups: parabolic subgroups. These subgroups have proven to be useful when studying Artin groups (for example, they are used to build interesting simplicial complexes), but again, we do not know much about them in general. Our problem will be the following: Given two conjugate elements of a parabolic subgroup $P$ of an Artin group $A$, are they conjugate via an element of $P$? This is called the conjugacy stability problem. In 2014, González-Meneses proved that this is always true for braids, that is, geometric embeddings of braids do not merge conjugacy classes. In an article with Calvez and Cisneros de la Cruz, we gave a classification for spherical Artin groups and proved that the answer to the question is not always affirmative. In this talk, we will explain how to give an algorithm to solve this problem for every Artin group satisfying two properties that are conjectured to be always true.

An isomorphism between two hermitian unitals is proved, and used to treat isomorphisms of classical groups that are related to the isomorphism between certain simple real Lie algebras of types A and D (and rank 3).

We study the initial degenerations of the type-A flag varieties, and show how they are related to flag matroid strata and subdivisions of flag matroid polytopes. As applications, we give a complete proof of a conjecture of Keel and Tevelev on log canonical compactifications of moduli spaces of hyperplanes in projective space in general position, and study the Chow quotient of the complete flag variety by the diagonal torus of the projective linear group. This is based on joint work with Olarte and Luber.

Associated with a marked elliptic root system, we introduce the elliptic period domain, on which elliptic Weyl group is acting as a reflection group. The system of reflection hyperplanes form, so called, an elliptic arrangement. The elliptic Weyl group acts regular and proper discontinuously on the complement of the elliptic arrangement, and the regular orbit space of the elliptic Weyl group is described as a complement of elliptic discriminant loci in the invariant space. Similar to the classical finite Coxeter arrangement case, we ask the topology (homotopy type) of the elliptic discriminant complement. We have partial answers 1. and 2. to the question: 1. The fundamental group, called an elliptic Artin group, is presented by homogeneous relations defined on the elliptic diagram. Then, we introduce an elliptic Artin monoid using the same homogeneous relations. 2. Using the non-cancellative property of the elliptic Artin monoid, we construct second homotopy classes of the elliptic discriminant complement, and conjecture that the classes are non-vanishing and they generate the full second homotopy group. In particular, 2. should imply that an expectation that "the discriminant complement may be an Eilenberg-MacLane space" may not hold.

Digital communications relies heavily on the usage of different types of codes. In general, a $d$-code Y is a finite subset in a metric space such that $d$ is the minimum distance that can occur between two distinct elements of $Y$. A central problem is to derive upper bounds for the size of codes. We will first introduce Delsarte's linear programming method, which is a powerful tool to obtain such bounds for different types of codes. Some codes that reach these bounds are related to different types of Steiner systems. We will focus on codes and Steiner systems in polar spaces, which consist of the totally isotropic subspaces of a finite vector space equipped with a nondegenerate form. Using bounds for codes in polar spaces, derived by Delsarte's method, we will give an almost complete classification of Steiner systems in polar spaces. Namely, we will show that such Steiner systems can only exist in some corner cases.

The Thompson’s group F is one of the most famous and very important groups related to many areas of mathematics. The open question about amenability of this group can be expressed in terms of spectra of Markov operators of random walks on F. Moreover, amenability of F would imply amenability of all Schreier graphs of F. This motivates studying properties of these graphs. Schreier graphs associated to the action of the Thompson group F on the interval [0,1] were described by Dmytro Savchuk. In this talk I will present results of a joint work with Rostislav Grigorchuk on spectral and related properties of the family of these graphs, including: - spectra and spectral measures of Laplace type operators associated to these Schreier graphs; - relation to corresponding Koopman representations of F; - the asymptotics of the return probabilities of the simple random walk on these graphs.

Pierpaola Santarsiero: Title: The symmetric geometric rank of symmetric tensors Abstract: Inspired by recent work of Kopparty-Moshkovitz-Zuiddam and motivated by problems in combinatorics and hypergraphs, we introduce the notion of symmetric geometric rank of a symmetric tensor. This quantity is equal to the codimension of the singular locus of the hypersurface associated to the tensor. In this talk, we will first learn fundamental properties of the symmetric geometric rank. Then, we will study the space of symmetric tensors of prescribed symmetric geometric rank, which is the space of homogeneous polynomials whose corresponding hypersurfaces have a singular locus of bounded codimension. This is joint work with J. Lindberg. Paul Breiding: Title: Euclidean Distance Degree and Mixed Volume Abstract: The Euclidean Distance Degree (EDD) of an algebraic variety V counts the number of complex critical points of the distance function from V to a generic fixed point outside of V. The BKK-Theorem (Bernstein, Kushnirenko, and Khovanskii) says that the number of complex zeros of a generic sparse polynomial system is equal to the mixed volume of the Newton polytopes of the polynomials. In this talk I will explain why for a generic sparse polynomial f the EDD of the hypersurface f=0 equals the mixed volume of the Lagrange multiplier equations for the EDD. This has impact on using polynomial homotopy continuation for computing the ED-critical points. And it provides new formulas for the EDD.

Groups of surfaces isogenous to a higher product of curves can be characterised by a purely group-theoretic condition, which is the existence of a so-called Beauville structure. Such groups are also called Beuaville groups, and have attracted much attention in recent years; for instance it was shown that all finite non-abelian simple groups, apart from one, are Beauville groups. Gül and Uria-Albizuri showed that quotients of the periodic Grigorchuk-Gupta-Sidki groups, GGS-groups for short, that act on the $p$-adic tree, are Beauville groups. They thus provide the first link between Beauville groups and groups acting on rooted trees. We extend their result by showing that quotients of infinite periodic GGS-groups, that act on the $p^n$-adic tree, also admit Beauville structures for all primes $p$ and positive integers $n$. This is joint work with Elena Di Domenico and Şükran Gül.

A long-standing open problem in the theory of arrangements of hypersurfaces is to classify which arrangement complements are aspherical. This problem has been a major driver of the research on arrangements of hyperplanes in vector spaces, where asphericity is known for some notable classes such as reflection arrangements and for "fiber-type" arrangements. In this talk I will state the problem, review some of its history, and discuss recent advances in the case of arrangements in connected abelian Lie groups (including complex tori and products of elliptic curves). I will state a combinatorial condition on the partially ordered set of intersections of such an arrangement that gives a combinatorially determined class of aspherical toric and elliptic arrangements. Time permitting I will detail some open problems of combinatorial and topological nature revolving around this class of posets. (This is joint work with Christin Bibby.)

Fix d > 2. Is the number of combinatorial types of triangulations of S^d with N d-simplices bounded by an exponential function of N? Physicists working on discrete models of quantum gravity are interested in this question with d=3. Gromov popularized this question and suggested (together with A. Nabutovski) the following more combinatorial variation of the problem: For fixed L (say L=100), evaluate the number of connected 3-regular graphs with N vertices for which the fundamental group is normally generated by cycles of length at most L. In this talk we will discuss the precise relation between the original problem and Gromov and Nabutovski's variation (both of which are still wide open). I will explain how to use these ideas to construct a family of triangulated 3-spheres for which the dual graphs are an expander family. Based on joint work with Karim Adiprasito.