Difference between revisions of "Colloquia 2012-2013"

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(Abstracts)
(Thu, Oct 20: Bernd Sturmfels (Berkeley))
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A smooth quartic curve in the complex projective plane has 36inequivalent representations as a symmetric determinant of linear forms
 
A smooth quartic curve in the complex projective plane has 36inequivalent representations as a symmetric determinant of linear forms
 
and 63 representations as a sum of three squares. We compute these objects from the 28 bitangents. This expresses Vinnikov quartics
 
and 63 representations as a sum of three squares. We compute these objects from the 28 bitangents. This expresses Vinnikov quartics
as spectrahedra and positive quartics as Gram matrices. We explore the geometry of Gram spectrahedra and the variety of Cayley octads. Interwoven
+
as spectrahedra and positive quartics as Gram matrices. We explore the geometry of Gram spectrahedra and the variety of Cayley octads. Interwoven is a delightful exposition of the 19th century theory of plane quartics.
is a delightful exposition of the 19th century theory of plane quartics.
 
  
 
===Fri, Oct 21: Bernd Sturmfels (Berkeley)===
 
===Fri, Oct 21: Bernd Sturmfels (Berkeley)===

Revision as of 06:38, 13 August 2011

Mathematics Colloquium

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

Fall 2011

date speaker title host(s)
Sep 9 Manfred Einsiedler (ETH-Zurich) TBA Fish
Sep 16 Richard Rimanyi (UNC-Chapel Hill) TBA Maxim
Sep 23 Andrei Caldararu (UW-Madison) The Hodge theorem as a derived self-intersection (local)
Oct 7 Hala Ghousseini (University of Wisconsin-Madison) TBA Lempp
Oct 14 Alex Kontorovich (Yale) On Zaremba's Conjecture Shamgar
oct 19, Wed Bernd Sturmfels (UC Berkeley) Convex Algebraic Geometry distinguished lecturer Shamgar
oct 20, Thu Bernd Sturmfels (UC Berkeley) Quartic Curves and Their Bitangents distinguished lecturer Shamgar
oct 21 Bernd Sturmfels (UC Berkeley) Multiview Geometry distinguished lecturer Shamgar
oct 28 Peter Constantin (University of Chicago) TBA distinguished lecturer
oct 31, Mon Peter Constantin (University of Chicago) TBA distinguished lecturer
Nov 4 Sijue Wu (U Michigan) TBA Qin Li
Nov 11 Henri Berestycki (EHESS and University of Chicago) TBA Wasow lecture
Nov 18 Benjamin Recht (UW-Madison, CS Department) TBA Jordan
Dec 2 Robert Dudley (University of California, Berkeley) From Gliding Ants to Andean Hummingbirds: The Evolution of Animal Flight Performance Jean-Luc
dec 9 Xinwen Zhu (Harvard University) TBA Tonghai

Spring 2012

date speaker title host(s)
Feb 24 Malabika Pramanik (University of British Columbia) TBA Benguria


Abstracts

Fri, Oct 14: Alex Kontorovich (Yale)

On Zaremba's Conjecture

It is folklore that modular multiplication is "random". This concept is useful for many applications, such as generating pseudorandom sequences, or in quasi-Monte Carlo methods for multi-dimensional numerical integration. Zaremba's theorem quantifies the quality of this "randomness" in terms of certain Diophantine properties involving continued fractions. His 40-year old conjecture predicts the ubiquity of moduli for which this Diophantine property is uniform. It is connected to Markoff and Lagrange spectra, as well as to families of "low-lying" divergent geodesics on the modular surface. We prove that a density one set satisfies Zaremba's conjecture, using recent advances such as the circle method and estimates for bilinear forms in the Affine Sieve, as well as a "congruence" analog of the renewal method in the thermodynamical formalism. This is joint work with Jean Bourgain.


Wed, Oct 19: Bernd Sturmfels (Berkeley)

Convex Algebraic Geometry

Convex algebraic geometry is an emerging field at the interface of convex optimization and algebraic geometry. A primary focus lies on the mathematical underpinnings of semidefinite programming. This lecture offers a self-contained introduction. Starting with elementary questions about multifocal ellipses in the plane, we move on to discuss the geometry of spectrahedra, orbitopes, and convex hulls of a real algebraic varieties.

Thu, Oct 20: Bernd Sturmfels (Berkeley)

Quartic Curves and Their Bitangents

A smooth quartic curve in the complex projective plane has 36inequivalent representations as a symmetric determinant of linear forms and 63 representations as a sum of three squares. We compute these objects from the 28 bitangents. This expresses Vinnikov quartics as spectrahedra and positive quartics as Gram matrices. We explore the geometry of Gram spectrahedra and the variety of Cayley octads. Interwoven is a delightful exposition of the 19th century theory of plane quartics.

Fri, Oct 21: Bernd Sturmfels (Berkeley)

Multiview Geometry

The study of two-dimensional images of three-dimensional scenes is a foundational subject for computer vision, known as multiview geometry. We present recent work with Chris Aholt and Rekha Thomas on the polynomials defining images taken by n cameras. Our varieties are threefolds that vary in a family of dimension 11n-15 when the cameras are moving. We use toric geometry and multigraded Hilbert schemes to characterize degenerations of camera positions.