Bredon cohomology is one of the most prominent cohomology theories for spaces with an action of a group. It appears as presheaf cohomology on the orbit category $\mathcal O_G$ (over open subgroups). At the same time, discrete $R[G]$-modules (which are right $R[G]$-modules with open stabilizers) were introduced to study the finiteness properties of totally disconnected and locally compact groups. In this project, we equip the orbit category $\mathcal O_G$ with the atomic Grothendieck topology $J_\mathrm{at}$ and show that the category of sheaves on $(\mathcal O_G,J_{\mathrm{at}})$ is the same as the category of discrete $\mathbb Z[G]$-modules. Moreover, we describe a semigroup operation on $\mathrm{dis}(\mathcal Z)$, a discrete $G$-module with $\mathrm{dis}(\mathcal Z)^U=\mathbb Z[U\backslash G]$. This makes $\mathrm{dis}(\mathcal Z)$ a nonunital ring, and $R-\mathrm{mod}$ for $R=\mathrm{dis}(\mathcal Z)$ are exactly the discrete $G$-modules. This project is a joint work with Ilaria Castellano.

In 1972, while investigating the theory of monomer-dimer systems, Heilmann and Lieb proved that the matching polynomial of a graph is real-rooted. This seminal result spurred the application of the geometry of polynomials to algebraic combinatorics. In the spirit of the Heilmann-Lieb theorem, we consider the set of non-nesting rook placements on a skew Ferrers board and probe the distributional properties of its generating polynomial. Surprisingly, the answers are governed by a new matroidal structure that we dub the rook matroid. We will discuss some of the structural properties of the rook matroid in relation to other well-known classes of matroids. We also consider a poset-theoretic perspective of this problem and in doing so, make progress on a conjecture of Brenti (1989) on the log-concavity of P-Eulerian polynomials. In particular, this completes the story of the Neggers-Stanley conjecture in the case of naturally labeled posets of width two. This is joint work with Per Alexandersson.

Let $\Gamma$ be a Hausdorff topological group and $\Lambda$ an open commensurated subgroup of $\Gamma$. A TDLC completion (short for totally disconnected locally compact completion) of $(\Gamma,\Lambda)$ is a pair $(G,\phi)$ where: $G$ is a TDLC group, $\phi\colon \Gamma\to G$ is a continuous homomorphism with dense image and $\Lambda=\phi^{-1}(L)$ for some compact open subgroup $L\subseteq G$. One important example is given by Schlichting completions. Recently, Bonn and Sauer investigated how the compactness properties behave under Schlichting completion. They showed that, from this point of view, the Schlichting completion of a pair $(\Gamma,\Lambda)$ acts precisely as if it were the quotient of $\Gamma$ by $\Lambda$ [BS24]. Motivated by this result, José Pedro Quintanilha and I set out to explore whether a similar phenomenon holds for the $\Sigma$-sets [BHQ24a,BHQ24b]. In this talk, I will present the main result along with some applications (if time permits).

[BS24] Laura Bonn and Roman Sauer. On homological properties of the Schlichting completion. Preprint, arXiv:2406.12740

[BHQ24a] Kai-Uwe Bux, Elisa Hartmann, and José Pedro Quintanilha. Geometric invariants of locally compact groups: the homological perspective. Preprint, arXiv:2411.13272

[BHQ24b] Kai-Uwe Bux, Elisa Hartmann, and José Pedro Quintanilha. Geometric invariants of locally compact groups: the homotopical perspective. Preprint, arXiv:2410.19501

We introduce a family of "Rascal posets" R(k,n-k) containing (n choose k) elements. The undirected Hasse diagram of R(k,n-k) is isomorphic to Young's lattice on integer partitions in a rectangle, but the orientation is different. We show that R(k,n-k) is a graded poset of height n-k containing k^{n-k} maximal chains and we derive an explicit formula for rank-selected multichains. The poset R(k,n-k) has two interesting geometric interpretations. First, its order complex is the type C alcove triangulation of a dilated simplex. Second, it is the poset of bounded regions in the cyclic hyperplane arrangement. This is related to work of Karp and Williams on the m=1 amplituhedron.

Volume polynomials are a class of Lorentzian polynomials coming from algebraic geometry. They measure the self intersection number of linear combinations of positive divisors in projective varieties. Their coefficients satisfy additional inequalities, and the support of a (realizable) volume polynomial is the set of bases of an algebraic polymatroid. We show that Aluffi's covolume polynomials are precisely the polynomial differential operators preserving volume polynomials. This provides us with a sufficiency criterion for linear operators preserving volume polynomials. We furthermore explore applications to algebraic matroids. This is joint work with Huh, Michałek, Süß and Wang.

We study the centralizer of a parabolic subalgebra in the Hecke algebra associated with an arbitrary (possibly infinite) Coxeter group. While the center and cocenter have been extensively studied in the finite and affine cases, much less is known in the indefinite setting. We describe a basis for the centralizer, generalizing known results about the center. Our approach combines algebraic techniques with geometric tools from the Davis complex, a CAT(0)-space associated to the Coxeter group. As part of the construction, we classify finite partial conjugacy classes in infinite Coxeter groups and define a variant of the class polynomial adapted to the centralizer.

Endowed with the Chabauty topology, the space of subgroups of any infinite countable group G is a closed subspace of the Cantor set, equipped with an action by homeomorphisms given by the G-conjugation. We are interested in the dynamics induced by this action on closed G-invariant subspaces. The largest closed subspace without isolated point is an example of such subspace called the perfect kernel of G. In an acylindrically hyperbolic context, Hull, Mynasyan and Osin demonstrated strong mixing properties (namely µ-mixing for a suitable measure µ on G, a strengthening of high topological transitivity). We uncover a radically different situation in the case of non metabelian Baumslag-Solitar groups. For the decomposition of the perfect kernel introduced by Carderi, Gaboriau, Le Maître and Stalder, who proved high topological transitivity on each piece, we show that the conjugation is even µ-mixing in the case of unimodular Baumslag-Solitar groups. On the contrary, when the group is non unimodular, there exists a continuum of measures µ for which the action is µ-mixing only on a single piece of the partition.

Invariant random subgroups, which are probability measures on sets of subgroups, have been an important subject of research in the last 15 years and can be thought of as point stabilizers of probability measure preserving (p.m.p.) actions. We are introducing elementwise conservative (e.c.) actions as generalizations of p.m.p. actions, and boomerang subgroups as "generic instances" of e.c. actions. I will talk about what these objects are, what they have in common and what tells them apart. Topics that we will adress in this talks are Margulis' normal subgroup theorem and the Stuck-Zimmer rigidity thorem, Poincaré recurrence, non-singular dynamics, Schreier graphs and critical exponents or growth rates. No prior knowledge on the subject is required. Based on joint works with Yair Glasner and Tobias Hartnick.