Finitely generated sequences of linear subspace arrangements

Speaker: 

Nir Gadish

Institution: 

University of Chicago

Time: 

Monday, March 19, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

Hyperplane arrangements are a classical meeting point of topology, combinatorics and representation theory. Generalizing to arrangements of linear subspaces of arbitrary codimension, the theory becomes much more complicated. However, a crucial observation is that many natural sequences of arrangements seem to be defined using a finite amount of data.

In this talk I will describe a notion of 'finitely generation' for collections of arrangements, unifying the treatment of known examples. Such collections turn out to exhibit strong forms of stability, both in their combinatorics and in their cohomology representation. This structure makes the appearance of representation stability transparent and opens the door to generalizations

Spherical twists and projective twists in Fukaya categories

Speaker: 

Weiwei Wu

Institution: 

University of Georgia

Time: 

Monday, March 5, 2018 - 4:00pm

Location: 

RH 340P

Seidel's Lagrangian Dehn twist exact sequence has been a
cornerstone of the theory of Fukaya categories.  In the last decade,
Huybrechts and Thomas discovered a new autoequivalence in the derived
cateogry of coherent sheaves using the so-called "projective objects", which
are presumably mirrors of Lagrangian projective spaces.   On the other hand,
Seidel's construction of Lagrangian Dehn twists as symplectomorphisms can be
easily generalized to Lagrangian projective spaces.  The induce
auto-equivalence on Fukaya categories are conjectured to be the mirror of
Huybrechts-Thomas's auto-equivalence on B-side.  

This remains open until recently, and I will explain my joint work with
Cheuk-Yu Mak on the solution to this conjecture using the technique of
Lagrangian cobordisms.  Moreover, we will explain a recent progress, again
joint with Cheuk-Yu Mak, on pushing this further to Lagrangian embeddings of
finite quotients of rank-one symmetric spaces, leading to another new class
of auto-equivalences, which are different from the classical spherical
twists only in coefficients of finite characteristics.

Subdivisional spaces and configuration spaces of graphs

Speaker: 

Gabriel Drummond-Cole

Institution: 

POSTECH IBS-CGP

Time: 

Monday, January 22, 2018 - 4:00pm to 5:00pm

Location: 

RH 340P

Configuration spaces of manifolds are often studied using the local model of configurations of Euclidean space. Configuration spaces of graphs have been studied as rigid combinatorial objects. I will describe a model for configuration spaces of cell complexes which combines the best features of both of these traditions, along with some applications in the homology of the configuration spaces of graphs. This is joint work with Byunghee An and Ben Knudsen.

Ordering actions on hyperbolic metric spaces

Speaker: 

Carolyn Abbott

Institution: 

UC Berkeley

Time: 

Monday, February 5, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

Every group admits at least one action by isometries on a hyperbolic metric space, and certain classes of groups admit many different actions on different hyperbolic metric spaces (in fact, often uncountably many).  One such class of groups is the class of so-called acylindrically hyperbolic groups, which contains many interesting groups, such as mapping class groups, Out(F_n), and right-angled Artin and Coxeter groups, among many others.  In this talk, I will describe how to put a partial order on the set of actions of a given group on hyperbolic spaces which, in some sense, measures how much information about the group the action provides.  This partial order defines a "poset of actions" of the given group.  I will then define the class of acylindrically hyperbolic groups and give some structural properties of the resulting poset of actions for such groups.  In particular, I will discuss for which (classes of) groups the poset contains a largest element.

Cohomology of arithmetic groups and characteristic classes of manifold bundles

Speaker: 

Bena Tshishiku

Institution: 

Harvard University

Time: 

Monday, January 8, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

A basic problem in the study of fiber bundles is to compute the ring H*(BDiff(M)) of characteristic classes of bundles with fiber a smooth manifold M. When M is a surface, this problem has ties to algebraic topology, geometric group theory, and algebraic geometry. Currently, we know only a very small percentage of the total cohomology. In this talk I will explain some of what is known and discuss some new characteristic classes (in the case dim M >>0) that come from the unstable cohomology of arithmetic groups. 

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