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The existence of Kahler-Einstein metrics on Kahler manifolds is a basic problem in complex differential geometry. This problem has connections to other fields: complex algebraic geometry, partial differential equations and several complex variables. I will discuss the existence of Kahler-Einstein metrics on Fano manifolds and its relation to K-stability. I will mainly focus on the analytic part of the theory, discuss how to solve the related complex Monge-Ampere equations and provide concrete examples in both smooth and conical settings. If time permits, I will also say something about the algebraic part of the theory, including the study of K-stability using the Minimal Model Program (joint with Chenyang Xu) and the existence of proper moduli space of smoothable K-polystable Fano varieties (joint with Xiaowei Wang and Chenyang Xu). | The existence of Kahler-Einstein metrics on Kahler manifolds is a basic problem in complex differential geometry. This problem has connections to other fields: complex algebraic geometry, partial differential equations and several complex variables. I will discuss the existence of Kahler-Einstein metrics on Fano manifolds and its relation to K-stability. I will mainly focus on the analytic part of the theory, discuss how to solve the related complex Monge-Ampere equations and provide concrete examples in both smooth and conical settings. If time permits, I will also say something about the algebraic part of the theory, including the study of K-stability using the Minimal Model Program (joint with Chenyang Xu) and the existence of proper moduli space of smoothable K-polystable Fano varieties (joint with Xiaowei Wang and Chenyang Xu). | ||
+ | |||
+ | ===January 21: Jun Kitagawa (Toronto)=== | ||
+ | |||
+ | ====Regularity theory for generated Jacobian equations: from optimal transport to geometric optics==== | ||
+ | |||
+ | Equations of Monge-Ampere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degeneracy (in the sense of ellipticity). Motivated by an example from geometric optics, I will talk about the class of Generated Jacobian Equations; recently introduced by Trudinger, this class also encompasses, for example, optimal transport, the Minkowski problem, and the classical Monge-Ampere equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field reflector problems (of interest from a physical and practical point of view). This talk is based on joint works with N. Guillen. | ||
===February 12: Mihai Putinar (UC Santa Barbara)=== | ===February 12: Mihai Putinar (UC Santa Barbara)=== |
Revision as of 06:54, 16 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 (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) | Regularity theory for generated Jacobian equations: from optimal transport to geometric optics | 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.
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.
January 15: Chi Li
On Kahler-Einstein metrics and K-stability
The existence of Kahler-Einstein metrics on Kahler manifolds is a basic problem in complex differential geometry. This problem has connections to other fields: complex algebraic geometry, partial differential equations and several complex variables. I will discuss the existence of Kahler-Einstein metrics on Fano manifolds and its relation to K-stability. I will mainly focus on the analytic part of the theory, discuss how to solve the related complex Monge-Ampere equations and provide concrete examples in both smooth and conical settings. If time permits, I will also say something about the algebraic part of the theory, including the study of K-stability using the Minimal Model Program (joint with Chenyang Xu) and the existence of proper moduli space of smoothable K-polystable Fano varieties (joint with Xiaowei Wang and Chenyang Xu).
January 21: Jun Kitagawa (Toronto)
Regularity theory for generated Jacobian equations: from optimal transport to geometric optics
Equations of Monge-Ampere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degeneracy (in the sense of ellipticity). Motivated by an example from geometric optics, I will talk about the class of Generated Jacobian Equations; recently introduced by Trudinger, this class also encompasses, for example, optimal transport, the Minkowski problem, and the classical Monge-Ampere equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field reflector problems (of interest from a physical and practical point of view). This talk is based on joint works with N. Guillen.
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]\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.