Section problems

Speaker: 

Lei Chen

Institution: 

Caltech

Time: 

Monday, December 3, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

In this talk, I will discuss a direction of study in topology: Section problems. There are many variations of the problem: Nielsen realization problems, sections of a surface bundle, sections of a bundle with special property (e.g. nowhere zero vector eld). I will discuss some techniques including homology, Thurston-Nielsen classication and dynamics. Also I will share many open problems. Some of the result are joint work with Nir Gadish, Justin Lanier and Nick Salter.

String topology, Hitchin's integrable system and noncommutative geometry

Speaker: 

Nick Rozenblyum

Institution: 

University of Chicago

Time: 

Monday, April 30, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

A classical result of Goldman states that character variety of an oriented surface is a symplectic algebraic variety, and that the Goldman Lie algebra of free loops on the surface acts by Hamiltonian vector fields on the character variety. I will describe a vast generalization of these results, including to higher dimensional manifolds where the role of the Goldman Lie algebra is played by the Chas-Sullivan string bracket in the string topology of the manifold. These results follow from a general statement in noncommutative geometry. In addition to generalizing Goldman's result to string topology, we obtain a number of other interesting consequences including the universal Hitchin system on a Riemann surface. This is joint work with Chris Brav.

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.

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