# Spring 2021

Date Speaker Title Abstract
Jan 12 Christian Zickert (Maryland) 3:35p Polylogarithms We give a survey of polylogarithms and their appearances in various areas of mathematics including hyperbolic geometry, invariants of 3-manifolds, motivic cohomology, and cluster algebras.
Jan 19 Ignat Soroko (LSU) Groups of type FP: their quasi-isometry classes and homological Dehn functions There are only countably many isomorphism classes of finitely presented groups, i.e. groups of type $$F_2$$. Considering a homological analog of finite presentability, we get the class of groups $$FP_2$$. Ian Leary proved that there are uncountably many isomorphism classes of groups of type $$FP_2$$ (and even of finer class FP). R.Kropholler, Leary and I proved that there are uncountably many classes of groups of type FP even up to quasi-isometries. Since `almost all' of these groups are infinitely presented, the usual Dehn function makes no sense for them, but the homological Dehn function is well-defined. In an on-going project with N.Brady, R.Kropholler and myself, we show that for any even integer $$k\ge4$$ there exist uncountably many quasi-isometry classes of groups of type FP with a homological Dehn function $$n^k$$. In particular there exists an FP group with the quartic homological Dehn function and the unsolvable word problem. In this talk I will give the relevant definitions and describe the construction of these groups. Time permitting, I will describe the connection of these groups to the Relation Gap Problem.
Jan 26
Feb 2 Jason DeBlois (Pittsburgh) High-density packings of hyperbolic surfaces I will give a survey-ish talk about some problems related to packing complete hyperbolic surfaces of finite area with equal-radius metric disks. For example: what is the maximal density of such packings, and how does one characterize, count, and otherwise understand the geometry of those surfaces that admit maximal-density packings? I will list several open problems and attempt to draw connections to things that other people are interested in.
Feb 9 Daniele Alessandrini (Columbia) Non commutative cluster coordinates for Higher Teichmüller Spaces In higher Teichmuller theory we study subsets of the character varieties of surface groups that are higher rank analogs of Teichmuller spaces, e.g. the Hitchin components and the spaces of maximal representations. Fock-Goncharov generalized Thurston's shear coordinates and Penner's Lambda-lengths to the Hitchin components, showing that they have a beautiful structure of cluster variety. Here we apply similar ideas to Maximal Representations and we find new coordinates on these spaces that give them a structure of non-commutative cluster varieties, in the sense defined by Berenstein-Rethak. This is joint work with Guichard, Rogozinnikov and Wienhard.
Feb 16 Yu-Chan Chang (Emory) Dehn functions and abelian splittings of Bestvina--Brady groups from their defining graphs. The Dehn function of a finitely generated group gives an upper bound of the complexity of the word problem on that group, and the Bestvina--Brady groups have been proved to satisfy quartic Dehn functions. In the first part of the talk, I will discuss a class of Bestvina--Brady groups whose Dehn functions can be identified from their defining graphs. One of the cases within our discussion of the Dehn functions is when the Bestvina--Brady groups split over $$\mathbb{Z}$$. In the second part of the talk, I will discuss some non-trivial splittings of Bestvina--Brady groups over abelian subgroups.
Feb 23 Irene Pasquinelli (Bristol) Deligne-Mostow lattices and branched covers of line arrangements We will talk about lattices in the group PU(n,1) of holomorphic isometries of complex hyperbolic space. Constructing lattices in PU(n,1) has been one of the major challenges of the last decades. In particular, we will talk about a well known class that is that of the Deligne-Mostow lattices. In the first part of the talk, I will introduce the complex hyperbolic space and its space of isometries. I will then explain some equivalent ways of constructing the Deligne-Mostow lattices. Among these, I will concentrate on the construction by Bartel-Hirzebruch-Hoefer, which uses branched covers, ramified over a line arrangement in complex projective 2-space. Finally I will explain to you how this interpretation gives hope towards a complex equivalent of the hybridisation technique.
Mar 2 Darren Long (UCSB) Zariski dense surface groups in SL(2k+1,Z) I'll introduce some of the history of thin groups and discuss a proof that there are Zariski dense surface groups in SL(2k+1,Z).
Mar 9
Mar 16 Corey Bregman (Southern Maine) 2:05p Outer Space for Right-angled Artin Groups Right-angled Artin groups (RAAGs) span a range of groups from free groups to free abelian groups. Thus, their (outer) automorphism groups range from Out(F_n) to GL(n,Z). Automorphism groups of RAAGs have been well-studied over the past decade from a purely algebraic viewpoint. To allow for a more geometric approach, one needs to construct a contractible space with a proper action of the group. I will present joint work with Ruth Charney and Karen Vogtmann in which we construct an analogue of Culler-Vogtmann’s Outer Space for arbitrary RAAGs, and discuss further directions of study.
Mar 23 Harry Bray (George Mason) Volume-entropy rigidity for convex projective manifolds I will discuss joint work with Constantine, building on joint work with Adeboye and Constantine, on a volume-entropy rigidity result for finite volume strictly convex projective manifolds in dimension at least 3. The result is a Besson-Courtois-Gallot type theorem, using the barycenter method. As an application, we get a uniform lower bound on the Hilbert volume of a finite volume strictly convex projective manifold of dimension at least 3.
Mar 30 Christian El Emam (Luxembourg) Immersions of surfaces into SL(2,C) as an approach to transition geometry We will discuss SL(2,C) equipped with its global complex killing form, which turns out to be a very interesting and regular space: in fact, SL(2,C) can be seen as a complex analog of the Riemannian notion of "space form". Moreover, every 3-dimensional pseudo-Riemannian space form of constant curvature -1, such as H^3, AdS^3, and -S^3, embed in a canonical isometric way into SL(2,C), providing a complex formalism to "transit" from one geometry to the other. As an application, the theory of immersions of surfaces into SL(2,C) generalizes the usual theory of immersions into pseudo-Riemannian space forms, and a holomorphic variation of the immersion data into SL(2,C) - passing for instance from an immersion into H^3 to one into AdS^3 - provides a holomorphic variation of its monodromy. This is joint work with Francesco Bonsante.
Apr 6 Anindya Chanda (FSU) Pujal’s Conjecture And Its recent Development A monster problem in the study of partially hyperbolic dynamics is to classify partially hyperbolic maps. Pujal’s conjecture served as one the principal motivation towards the classification problem for the last twenty years. Though the conjecture has been proved to be false in the general case, it is still relevant in its new form in more specific contexts. In our talk we will survey the recent developments, open problems and new examples of partially hyperbolic maps in dimension 3 made in the last few years.
Apr 13

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