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 (Purdue)	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 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)	TBA	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.

January 14: Jayadev Athreya

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 [math]\displaystyle{ \mathbb{C}^n }[/math] 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.