Projective Geometry, Complex Hyperbolic Space, and Geometric Transitions.

Speaker: 

Steve Trettel

Institution: 

UCSB

Time: 

Monday, May 6, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

The natural analog of Teichmuller theory for hyperbolic manifolds in dimension 3 or greater is trivialized by Mostow Rigidity, so mathematicians have worked to understand more general deformations.  Two well studied examples, convex real projective structures and complex hyperbolic structures, have been investigated extensively and provide independently developed deformation theories.  Here we will discuss a surprising connection between the these, and construct a one parameter family of geometries deforming complex hyperbolic space into a new geometry built out of real projective space and its dual.

 

Sasaki-Einstein manifolds and AdS/CFT correspondence

Speaker: 

Dan Xie

Institution: 

Tsinghua University

Time: 

Monday, February 11, 2019 - 4:00pm

Location: 

RH 340P

Sasakian manifolds are odd dimensional analog of Kahler manifolds,
and it is an interesting question to determine when
a Sasakian manifold admits an Einstein metric. Five dimensional
Sasaki-Einstein (SE)  manifolds play an important role in AdS/CFT
correspondence, which relates a string theory and a quantum field theory. I
will discuss the existence of SE manifolds and its geometric properties
which will be of great interest to AdS/CFT correspondence.

Cohomology of the space of polynomial morphisms on A^1 with prescribed ramifications

Speaker: 

Oishee Banerjee

Institution: 

University of Chicago

Time: 

Monday, April 8, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

In this talk we will discuss the moduli spaces Simp^m_n of degree n+1 morphisms  \A^1_K\to \A^1_K  with "ramification length <m" over an algebraically closed field K. For each m, the moduli space Simp^m_n is a Zariski open subset of the space of degree n+1 polynomials over K up to Aut(\A^1_K). It is, in a way, orthogonal to the many papers about polynomials with prescribed zeroes- here we are prescribing, instead, the ramification data. We will also see why and how our results align, in spirit, with the long standing open problem of understanding the topology of the Hurwitz space.

Representation Stability and Milnor Fibers

Speaker: 

Phil Tosteson

Institution: 

Michigan

Time: 

Monday, May 20, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

The Type  Milnor fiber is the subset of  defined by the equation .  It carries an action of the alternating group and the th roots of unity. We will discuss how tools from representation stability can be used to study the homology of the Milnor fiber for  and determine the stable limit.  This is joint work with Jeremy Miller. 

Solving the Twisted Rabbit Problem using trees

Speaker: 

Rebecca Winarski

Institution: 

University of Michigan

Time: 

Monday, January 28, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

The twisted rabbit problem is a celebrated problem in complex dynamics. Work of Thurston proves that up to equivalence, there are exactly three branched coverings of the sphere to itself satisfying certain conditions. When one of these branched coverings is modified by a mapping class, a map equivalent to one of the three coverings results. Which one?

After remaining open for 25 years, this problem was solved by Bartholdi-Nekyrashevych using iterated monodromy groups. In joint work with Belk, Lanier, and Margalit, we present an alternate solution using topology and geometric group theory that allows us to solve a more general problem.

Algebraic fibrations of Kahler groups

Speaker: 

Stefano Vidussi

Institution: 

UC Riverside

Time: 

Monday, October 22, 2018 - 4:00pm

Location: 

RH 340P

One of the major results in the study of 3-manifolds is the fact that most 3-manifolds have a finite cover that fibers over S1. One may ask what is the counterpart of this result for other classes of manifolds. In this talk we will discuss the case of smooth projective varieties (or more generally Kaehler manifolds) and present some geometric and group-theoretic aspects of  "virtual algebraic fibrations" of their fundamental groups.

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