There is a strong, but limited, analogy between mapping class groups of finite-type surfaces and lattices in Lie groups. In light of this analogy, a natural problem asks to classify all possible homomorphisms between mapping class groups. The first part of the talk will be devoted to reviewing what is known about this problem, as well as highlighting the main ideas in the various proofs. As an important special case, we will see that mapping class groups of finite-type surfaces are co-Hopfian: every injective endomorphism is an automorphism. In the second part, we will construct (uncountably many) examples of infinite-type surfaces whose mapping class groups fail to be co-Hopfian. This is joint work with Chris Leininger (Rice) and Alan McLeay (Luxembourg).

I will introduce a new class of multivariate rational functions associated with hyperplane arrangements, viz. Hilbert--Poincar\'e series. As I shall explain, these series essentially solve natural counting problems arising in the theory of hyperplane arrangements. I will report on a general self-reciprocity result and explicate a number of connections with other, more classical Hilbert- (and Ehrhart-)series from combinatorial algebra. No prior expert knowledge will be required. This is joint work with Josh Maglione.

Stringy Hodge numbers are certain generalizations, to the singular setting, of Hodge numbers. Unlike usual Hodge numbers, stringy Hodge numbers are not defined as dimensions of cohomology groups. Nonetheless, an open conjecture of Batyrev's predicts that stringy Hodge numbers are nonnegative. In the special case of varieties with only quotient singularities, Yasuda proved Batyrev's conjecture by showing that the stringy Hodge numbers are given by orbifold cohomology. For more general singularities, a similar cohomological interpretation remains elusive. I will discuss a conjectural framework, proven in the toric case, that relates stringy Hodge numbers to motivic integration for Artin stacks, and I will explain how this framework applies to the search for a cohomological interpretation for stringy Hodge numbers. This talk is based on joint work with Matthew Satriano.

The topological zeta function is a singularity invariant associated with a hypersurface, introduced by Denef and Loeser. We explain how the wonderful log resolutions of De Concini-Procesi can be used to transfer the definition from hyperplane arrangements to matroids. By analysing the proof we can then also go beyond the classical definition and introduce combinatorial variations that do not have a geometric counterpart.

If two infinite groups have the same set of finite quotients, are they isomorphic or at least commensurable? This question is particularly intriguing for lattices in simple Lie groups. We show that in most higher rank Lie groups, there exist lattices with the same finite quotients that are not commensurable. But surprisingly, three exceptional Lie groups exhibit profinite rigidity: in the complex groups of type $E_8$, $F_4$, and $G_2$, the set of finite quotients determines the commensurability class of a lattice. Joint work with Steffen Kionke.

A triple of finite groups $(H,M,K)$, usually written $H > M < K,$ is called a primitive amalgam if $M$ is a subgroup of both $H$ and $K$, and each of the following holds: (i) Whenever $A$ is a normal subgroup of $H$ contained in $M$, we have $N_K(A)=M$; and (ii) whenever $B$ is a normal subgroup of $K$ contained in $M$, we have $N_H(B)=M$.

Primitive amalgams arise naturally in many different contexts across pure mathematics, from Tutte's study of vertex-transitive groups of automorphisms of finite, connected, trivalent graphs; to Thompson's classification of simple $N$-groups; to Sims' study of point stabilizers in primitive permutation groups, and beyond. In this talk, we will discuss some recent progress on the central conjecture from the theory of primitive amalgams, called the Goldschmidt–Sims conjecture. Joint work with László Pyber.

Surfaces of infinite type, such as the plane minus a Cantor set, occur naturally in dynamics. However, their mapping class groups are much less studied and understood compared to the mapping class groups of surfaces of finite type. Many fundamental questions remain open. We will discuss the mapping class group G of the plane minus a Cantor set, and show that any nontrivial G-action on the circle is semi-conjugate to its action on the so-called simple circle. Along the way, we will discuss some structural results of G to address the following questions: What are some interesting subgroups of G? Is G generated by torsion elements? This is joint work with Danny Calegari.

Artin groups are a generalization of braid groups first defined by Tits in the 1960s, and which remain the subject of active research. The Artin monoid is the monoid with the same generators and relations as in the Artin group, and many of the questions that remain open for Artin groups are solved for Artin monoids. This motivates the study of the connection between Artin monoids and Artin groups. In a recent paper, Rachael Boyd, Ruth Charney and I investigate the geometric relationship between the monoid and the group. In this talk, I will first discuss Artin monoids, how they differ from Artin groups, and why Artin monoids are interesting objects to study. Then I will outline the construction of the Deligne complex for an Artin monoid. Finally I will discuss some of the geometric properties of this complex that we have derived, including the fact that the Deligne complex of a monoid is always contractible, with a locally isometric embedding into the Deligne complex of the group.

I will define two new constructions of finitely generated simple left orderable groups (in recent joint work with Hyde and Rivas). Among these examples are the first examples of finitely generated simple left orderable groups that admit a minimal action by homeomorphisms on the Torus, and the first family that admits such an action on the circle. I shall also present examples of finitely generated simple left orderable groups that are uniformly simple (these were constructed by me with Hyde in 2019). And present new examples that, somewhat surprisingly, have infinite commutator width.

A (convex) polytope $P$ is inscribable if there is a combinatorially equivalent polytope $P'$ with all vertices contained in a sphere. This notion relates to the combinatorics of ideal hyperbolic polytopes and Delaunay subdivisions. Steinitz showed that not every polytope is inscribable and Rivin gave a complete and effective characterization for $3$-dimensional polytopes. There is no such characterization in dimensions $4$ and up.

I will discuss the related notion of normally inscribable polytopes: A polytope $P$ is normally inscribable there is a continuous deformation of $P$ to an inscribed polytopes that keeps faces parallel. Normal inscribability, it turns out, reveals a fascinating interplay of algebra, geometry, and combinatorics. I will explain connections to a deformation theory of ideal hyperbolic polytopes and Delaunay subdivisions, routed particle trajectories, and reflection groupoids. Reflection groups and their associated reflection arrangements play a distinguished role, as does Grünbaum's quest for simplicial hyperplane arrangements. This is based on joint work with Sebastian Manecke.

Motivated by a question of Dehornoy-Digne-Godelle-Krammer-Michel on a submonoid of Artin's braid group, we give a new Garside structure for torus knot groups. In a specific case, these groups are extensions of Artin's braid group, and the new Garside monoid surjects onto DDGKM's monoid, allowing one to deduce properties of the latter. This allows to partially answer the above mentioned question, and to give a conjectural finite presentation for DDGKM's monoid. In a second part we give some evidence that torus knot groups may be considered as "braid groups" of a family of (in general infinite) complex reflection groups.

In finite group theory, the normalizers of non-trivial p-subgroups are called $p$-local subgroups and play traditionally a big role ($p$ a prime). The $p$-local structure of a finite group $G$ is also closely related to the Bousfield-Kan $p$-completion of the classifying space of $G$. Saturated fusion systems and associated linking systems are categories modelling the $p$-local structure of finite groups. In particular, linking systems contain the algebraic information that is needed to study $p$-completed classifying spaces of fusion systems. Every linking system corresponds to a group-like structure called a locality. I will give an introduction to the subject and mention some of the recent results I have obtained, some in joint work with Andrew Chermak and some in joint work with Justin Lynd.

The theory of $EG$, the class of elementary amenable groups, has developed steadily since the class was introduced constructively by Day in 1957. At that time, it was unclear whether or not $EG$ was equal to the class $AG$ of all amenable groups. Highlights of this development certainly include Chou's article in 1980 which develops much of the basic structure theory of the class $EG$, and Grigorchuk's 1985 result showing that the first Grigorchuk group $\Gamma$ is amenable but not elementary amenable. In this talk we report on work where we demonstrate the existence of a family of finitely generated subgroups of Richard Thompson’s group $F$ which is strictly well-ordered by the embeddability relation in type $\varepsilon_0+1$. All except the maximum element of this family (which is $F$ itself) are elementary amenable groups. In this way, for each $\alpha<\varepsilon_0$, we obtain a finitely generated elementary amenable subgroup of F whose EA-class is $\alpha+2$. The talk will be pitched for an algebraically inclined audience, but little background knowledge will be assumed. Joint work with Matthew Brin and Justin Moore.