# 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 7, Mo, 3pm, SMI 133 Sastry Pantula (NSCU and DMS/NSF) TBA Joint Math/Stat Colloquium
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

This lecture concerns convex bodies with an interesting algebraic structure. A primary focus lies on the geometry of semidefinite optimization. Starting with elementary questions about ellipses in the plane, we move on to discuss the geometry of spectrahedra, orbitopes, and convex hulls of real varieties.

### Thu, Oct 20: Bernd Sturmfels (Berkeley)

Quartic Curves and Their Bitangents

We present a computational study of plane curves of degree four, with primary focus on writing their defining polynomials as sums of squares and as symmetric determinants. Number theorists will enjoy the appearance of the Weyl group $\displaystyle{ E_7 }$ as the Galois group of the 28 bitangents. Based on joint work with Daniel Plaumann and Cynthia Vinzant, this lecture spans a bridge from 19th century algebra to 21st century optimization.

### Fri, Oct 21: Bernd Sturmfels (Berkeley)

Multiview Geometry

The study of two-dimensional images of three-dimensional scenes is foundational for computer vision. We present work with Chris Aholt and Rekha Thomas on the polynomials characterizing images taken by $\displaystyle{ n }$ cameras. Our varieties are threefolds that vary in a family of dimension $\displaystyle{ 11n-15 }$ when the cameras are moving. We use toric geometry and Hilbert schemes to characterize degenerations of camera positions.