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Speaker: Ana Balibanu
Title: Steinberg slices and group-valued moment maps
Abstract: We define a class of transversal slices in spaces which are quasi-Poisson for the action of a complex semisimple group G. This is a multiplicative analogue of Whittaker reduction. One example is the multiplicative universal centralizer of G, which is equipped with the usual symplectic structure in this way. We construct a smooth partial compactification of Z by taking the closure of each centralizer fiber in the wonderful compactification of G. By realizing this partial compactification as a transversal in a larger quasi-Poisson variety, we show that it is smooth and log-symplectic.
Speaker: Patrick Brosnan
Title: Fixed points in toroidal compactifications and essential dimension of covers
Abstract: I will explain joint work with Najmuddin Fakhruddin on fixed points and essential dimension of congruence covers. In particular, I will discuss our main fixed point theorem, which gives a way to prove the existence of fixed points in certain equivariant compactifications of etale covers. Then I will explain the applications to essential dimension, which were motivated by work of Farb, Kisin and Wolfson (FKW) on essential dimension of congruence covers of Shimura varieties. Using our results we give geometric proofs of many (but definitely not all) of the lower bounds on essential dimension, obtained by FKW using arithmetic methods. We also obtain some new results and ask a few questions, and I will discuss these as well.
Speaker: Graham Denham
Title: Bipermutohedral combinatorics
Abstract: In joint work with Federico Ardila and June Huh, we consider a new geometric model for a matroid, which we call the conormal fan. Historically, the Bergman fan, De Concini and Procesi's wonderful compactification, and the permutohedron enjoy close relations. The conormal fan, by way of comparison, comes with a new polytope which we call the bipermutohedron, together with a configuration space and some "bipermutohedral combinatorics" which I will describe in my talk.
Speaker: Nir Gadish
Title: From compactified configurations on graphs to top weight cohomology of the moduli spaces M_{2,n}
Abstract: Configuration spaces of points on a graph are hard to understand topologically. Their one-point compactifications, on the other hand, are proper homotopy invariants and their homology admits a simple algebraic description. Via the machinery of tropical geometry, such compactified configuration spaces detect cohomology of moduli spaces of curves with marked points. I will discuss joint work with C. Bibby, M. Chan and C. Yun, in which we combine this approach with some representation theory magic to compute the top weight rational cohomology of the moduli space M_{2,n} of genus 2 curves with n marked points, along with their action of the symmetric group, for all n < 11.
Speaker: Iva Halacheva
Title: Cacti, De Concini-Procesi's wonderful compactification, and maximal commutative algebras
Abstract: For any semisimple Lie algebra g, De Concini-Procesi’s wonderful model M(g) provides a nice compactification of the complement of the root hyperplanes. The real locus of this model has fundamental group the pure cactus group, closely related to the pure braid group associated to g. In joint work with Kamnitzer, Rybnikov and Weekes, we prove that M(g) indexes a family of maximal commutative subalgebras of U(g), the shift of argument algebras, and for any g-representation, this produces a covering of M(g) with monodromy action also realizable combinatorially. I will describe this construction and time permitting, discuss related results for Gaudin algebras and, from a categorical point of view, perverse equivalences.
Speaker: Ben Knudsen
Title: Extremal stability for configuration spaces
Abstract: We study the longterm behavior of the rational Betti numbers of configuration spaces of manifolds, regarded as functions of the number of particles. According to classical homological stability, the Betti number in fixed dimension is eventually equal to a polynomial in the number of particles, whose degree is bounded in terms of the dimension 0 homology of the manifold. We prove a dual result, namely that the Betti number in fixed codimension is eventually equal to a quasi-polynomial in the number of particles, whose degree is bounded in terms of the codimension 1 homology of the manifold. This talk represents joint work with Jeremy Miller and Philip Tosteson.
Speaker: Laurentiu Maxim
Title: On the topology of aspherical complex projective manifolds
Abstract: I will describe recent progress on the study of the topology of aspherical complex projective manifolds, with an eye towards the Singer-Hopf conjecture and the Bobadilla-Kollar conjecture. (Joint work with Y. Liu and B. Wang.)
Speaker: Eric Ramos
Title: The categorical graph minor theorem and graph configuration spaces
Abstract: Perhaps one of the most well-known theorems in graph theory is the celebrated Graph Minor Theorem of Robertson and Seymour. This theorem states that in any infinite collection of finite graphs, there must be a pair of graphs for which one is obtained from the other by a sequence of edge contractions and deletions. In this talk, I will present work of Nick Proudfoot, Dane Miyata, and myself which proves a categorified version of the graph minor theorem. As an application, we show how configuration spaces of graphs must display some strongly uniform properties. We then show how this result can be seen as a vast generalization of a variety of classical theorems in graph configuration spaces.
Speaker: Botong Wang
Title: Singular Hodge theory of matroids
Abstract: The de Bruijn-Erdos theorem states that the pairwise intersection of $n$ lines in the projective plane is either a single point or consists of at least $n$ points. As its higher dimensional generalization, Dowling and Wilson proposed the Top-heavy conjecture, which states that in a rank $d$ matroid, the number of rank $k$ flats is always less or equal to number of rank $d-k$ flats for $k\leq d/2$. I will give an overview of a proof of the conjecture in the realizable case, which is a joint work with June Huh. The proof uses the singular Hodge theory of certain algebraic varieteis called the Schubert varieties of hyperplane arrangements. I will also give a brief summary of a more recent work, further joint with Tom Braden, Jacob Matherne and Nick Proudfoot, which proves the Top-heavy conjecture in full generality.
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