Colloquia/Fall18: Difference between revisions
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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. | 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)=== | ===February 12: Mihai Putinar (UC Santa Barbara)=== | ||
Revision as of 17:45, 8 January 2015
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.