Quasiflats in hierarchically hyperbolic spaces

Speaker: 

Jason Behrstock

Institution: 

CUNY

Time: 

Monday, April 2, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

Hierarchically hyperbolic spaces provide a uniform framework for working with many important examples, including mapping class groups, right angled Artin groups, Teichmuller space, and others. In this talk I'll provide an introduction to studying groups and spaces from this point of view. This discussion will center around recent work in which we classify quasiflats in these spaces, thereby resolving a number of well-known questions and conjectures. This is joint work with Mark Hagen and Alessandro Sisto.

Hodge metric of nilpotent Higgs bundles

Speaker: 

Qiongling Li

Institution: 

Caltech

Time: 

Monday, February 12, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

On a complex manifold, a Higgs bundle is a pair containing a holomorphic vector bundle E and a holomorphic End(E)-valued 1-form. In this talk, we focus on nilpotent Higgs bundles, for example, the ones arising from variations of Hodge structures for a deformation family of Kaehler manifolds. We first give an optimal upper bound of the curvature of Hodge metric of the deformation space of Calabi-Yau manifolds. Secondly, we prove a rigidity theorem of the holonomy of polystable nilpotent Higgs bundles via the non-abelian Hodge theory when the base manifold is a Riemann surface. This is joint work with Song Dai.

Choosing distinct points on cubic curves

Speaker: 

Weiyan Chen

Institution: 

University of Minnesota

Time: 

Tuesday, April 17, 2018 - 3:00pm

Host: 

Location: 

RH 340P

It is a classical topic dating back to Maclaurin (1698–1746) to study certain special points on smooth cubic plane curves, such as the 9 inflection points (Maclaurin and Hesse), the 27 sextatic points (Cayley), and the 72 points "of type 9" (Gattazzo). Motivated by these algebro-geometric constructions, we ask the following topological question: is it possible to choose n distinct points on a smooth cubic plane curve as the curve varies continuously in family, for any integer n other than 9, 27 and 72? We will present both constructions and obstructions to such continuous choices of points, state a classification theorem for them, and discuss conjectures and open questions.

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.

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