Title: New bounds for equiangular lines and spherical two-distance sets

Date: 01/31/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The set of points in a metric space is called an s-distance set if pairwise distances between these points admit only s distinct values. Two-distance spherical sets with the set of scalar products {alpha, -alpha}, alpha in [0,1), are called equiangular. The problem of determining the maximal size of s-distance sets in various spaces has a long history in mathematics. We determine a new method of bounding the size of an s-distance set in two-point homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical two-distance set in R^n is n(n+1)/2 with possible exceptions for some n = (2k+1)^2-3, k a positive integer. We also prove the universal upper bound ~ 2 n a^2/3 for equiangular sets with alpha = 1/a and, employing this bound, prove a new upper bound on the size of equiangular sets in an arbitrary dimension. Finally, we classify all equiangular sets reaching this new bound.

Speaker: Keerthi Madapusi Pera, University of Chicago

Title: Periods, L-functions and abelian varieties

Date: 02/01/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Periods are a special class of complex numbers, arising as integrals of differential forms on algebraic varieties. L-functions are analytic objects that generalize the Riemann zeta function. Both are objects admitting deceptively simple definitions, but carry deep arithmetic information.
In this talk, I'll explain a relationship between periods of abelian varieties with complex multiplication, and certain Artin L-functions, originally conjectured by P. Colmez, and sketch a proof of it that arose out of joint work with Andreatta, Goren and Ben Howard. Among other applications, this relationship has led to a proof by J. Tsimerman of the Andre-Oort conjecture for Siegel modular varieties.

Speaker: Leonid Chekhov, Steklov Mathematical Institute

Title: Quantum cluster algebras from geometry

Date: 02/02/2017

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

We identify the Teichmuller space $T_{g,s,n}$ of (decorated) Riemann
surfaces $\Sigma_{g,s,n}$ of genus $g$, with $s>0$ holes and $n>0$
bordered cusps located on boundaries of holes uniformized by Poincare with
the character variety of $SL(2,R)$-monodromy problem. The effective
combinatorial description uses the fat graph technique; observables are
geodesic functions of closed curves and $\lambda$-lengths of paths
starting and terminating at bordered cusps decorated by horocycles. Such
geometry stems from special 'chewing gum' moves corresponding to colliding
holes (or sides of the same hole) in a Riemann surface with holes. We
derive Poisson and quantum structures on sets of observables relating them
to quantum cluster algebras of Berenstein and Zelevinsky. A seed of the
corresponding quantum cluster algebra corresponds to the partition of
$\Sigma_{g,s,n}$ into ideal triangles, $\lambda$-lengths of their sides
are cluster variables constituting a seed of the algebra; their number
$6g-6+3s+2n$ (and, correspondingly, the seed dimension) coincides with the
dimension of $SL(2,R)$-character variety given by
$[SL(2,R)]^{2g+s+n-2}/\prod_{i=1}^n B_i$,
where $B_i$ are Borel subgroups associated with bordered cusps. I also discuss the
very recent results enabling constructing monodromy matrices of SL(2)-connections out of
the corresponding cluster variables.
The talk is based on the joint papers with with M.Mazzocco and V.Roubtsov

Title: Cutting plane theorems for Integer Optimization and computer-assisted proofs

Date: 02/03/2017

Time: 10:00 AM - 11:00 AM

Place: 1502 Engineering Building

Optimization problems with integer variables form a class of mathematical
models that are widely used in Operations Research and Mathematical Analytics.
They provide a great modeling power, but it comes at a high price: Integer
optimization problems are typically very hard to solve, both in theory and practice.
The state-of-the-art solvers for integer optimization problems use cutting-plane
algorithms. Inspired by the breakthroughs of the polyhedral method for
combinatorial optimization in the 1980s, generations of researchers have studied the
facet structure of convex hulls to develop strong cutting planes. However, the
proofs of cutting planes theorems were hand-written, and were dominated by
tedious and error-prone case analysis.
We ask how much of this process can be automated: In particular, can we use
algorithms to discover and prove theorems about cutting planes? I will present our
recent work towards this objective. We hope that the success of this project would
lead to a tool for developing the next-generation cutting planes that answers the
needs prompted by ever-larger applications and models.