Colloquia 2012-2013

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)	Periodic orbits on homogeneous spaces	Fish
Sep 16	Richard Rimanyi (UNC-Chapel Hill)	Global singularity theory	Maxim
Sep 23	Andrei Caldararu (UW-Madison)	The Hodge theorem as a derived self-intersection	(local)
Sep 30
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
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)
Jan 26, Thu	Peter Constantin (University of Chicago)	TBA	distinguished lecturer
Jan 27	Peter Constantin (University of Chicago)	TBA	distinguished lecturer
Feb 24	Malabika Pramanik (University of British Columbia)	TBA	Benguria
March 2	Guang Gong (University of Waterloo)	TBA	Shamgar

Abstracts

Fri, Sept 9: Manfred Einsiedler (ETH-Zurich)

Periodic orbits on homogeneous spaces

We call an orbit xH of a subgroup H<G on a quotient space Gamma \ G periodic if it has finite H-invariant volume. These orbits have intimate connections to a variety of number theoretic problems, e.g. both integer quadratic forms and number fields give rise periodic orbits and these periodic orbits then relate to local-global problems for the quadratic forms or to special values of L-functions. We will discuss whether a sequence of periodic orbits equidistribute in Gamma \ G assuming the orbits become more complicated (which can be measured by a discriminant). If H is a diagonal subgroup (also called torus or Cartan subgroup), this is not always the case but can be true with a bit more averaging. As a theorem of Mozes and Shah show the case where H is generated by unipotents is well understand and is closely related to the work of M. Ratner. We then ask about the rate of approximation, where the situation is much more complex. The talk is based on several papers which are joint work with E.Lindenstrauss, Ph. Michel, and A. Venkatesh resp. with G. Margulis and A. Venkatesh.

Fri, Sept 16: Richard Rimanyi (UNC)

Global singularity theory

The topology of the spaces A and B may force every map from A to B to have certain singularities. For example, a map from the Klein bottle to 3-space must have double points. A map from the projective plane to the plane must have an odd number of cusp points.

To a singularity one may associate a polynomial (its Thom polynomial) which measures how topology forces this particular singularity. In the lecture we will explore the theory of Thom polynomials and their applications in enumerative geometry. Along the way, we will meet a wide spectrum of mathematical concepts from geometric theorems of the ancient Greeks to the cohomology ring of moduli spaces.

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.