# Mathematics Colloquium

All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.

## Spring 2015

date speaker title host(s)
January 12 (special time: 3PM) Botong Wang (Notre Dame) Cohomology jump loci of algebraic varieties Maxim
January 14 (special time: 11AM) Jayadev Athreya (UIUC) Counting points for random (and not-so-random) geometric structures Ellenberg
January 15 (special time: 3PM) Chi Li (Stony Brook) On Kahler-Einstein metrics and K-stability Sean Paul
January 21 Jun Kitagawa (Toronto) TBA Feldman
January 23 Tentatively reserved for possible interview
January 30 Tentatively reserved for possible interview
February 6 Morris Hirsch (UC Berkeley and UW Madison) Fixed points of Lie group actions Stovall
February 13 Mihai Putinar (UC Santa Barbara, Newcastle University) Quillen’s property of real algebraic varieties Budišić
February 20 David Zureick-Brown (Emory University) Diophantine and tropical geometry Ellenberg
February 27 Allan Greenleaf (University of Rochester) TBA Seeger
March 6 Larry Guth (MIT) TBA Stovall
March 13 Cameron Gordon (UT-Austin) TBA Maxim
March 20 Murad Banaji (University of Portsmouth) TBA Craciun
March 27 Kent Orr (Indiana University at Bloomigton) TBA Maxim
April 3 University holiday
April 10 Jasmine Foo (University of Minnesota) TBA Roch, WIMAW
April 17 Kay Kirkpatrick (University of Illinois-Urbana Champaign) TBA Stovall
April 24 Marianna Csornyei (University of Chicago) TBA Seeger, Stovall
May 1 Bianca Viray (University of Washington) TBA Erman
May 8 Marcus Roper (UCLA) TBA Roch

## Abstracts

### January 12: Botong Wang

#### Cohomology jump loci of algebraic varieties

In the moduli spaces of vector bundles (or local systems), cohomology jump loci are the algebraic sets where certain cohomology group has prescribed dimension. We will discuss some arithmetic and deformation theoretic aspects of cohomology jump loci. If time permits, we will also talk about some applications in algebraic statistics.

#### Counting points for random (and not-so-random) geometric structures

We describe a philosophy of how certain counting problems can be studied by methods of probability theory and dynamics on appropriate moduli spaces. We focus on two particular cases:

(1) Counting for Right-Angled Billiards: understanding the dynamics on and volumes of moduli spaces of meromorphic quadratic differentials yields interesting universality phenomenon for billiards in polygons with interior angles integer multiples of 90 degrees. This is joint work with A. Eskin and A. Zorich

(2) Counting for almost every quadratic form: understanding the geometry of a random lattice allows yields striking diophantine and counting results for typical (in the sense of measure) quadratic (and other) forms. This is joint work with G. A. Margulis.

### February 12: Mihai Putinar (UC Santa Barbara)

#### Quillen’s property of real algebraic varieties

A famous observation discovered by Fejer and Riesz a century ago is the quintessential algebraic component of every spectral decomposition result. It asserts that every non-negative polynomial on the unit circle is a hermitian square. About half a century ago, Quillen proved that a positive polynomial on an odd dimensional sphere is a sum of hermitian squares. Fact independently rediscovered much later by D’Angelo and Catlin, respectively Athavale. The main subject of the talk will be: on which real algebraic sub varieties of $\displaystyle{ \mathbb{C}^n }$ is Quillen theorem valid? An interlace between real algebraic geometry, quantization techniques and complex hermitian geometry will provide an answer to the above question, and more. Based a recent work with Claus Scheiderer and John D’Angelo.