School of Mathematics

Unlinked fixed points of Hamiltonian diffeomorphisms and a dynamical construction of spectral invariants

Sobhan Seyfaddini
Massachusetts Institute of Technology
April 17, 2015
Hamiltonian spectral invariants have had many interesting and important applications in symplectic geometry. Inspired by Le Calvez's theory of transverse foliations for dynamical systems of surfaces, we introduce a new dynamical invariant, denoted by $N$, for Hamiltonians on surfaces (except the sphere). We prove that, on the set of autonomous Hamiltonians, this invariant coincides with the classical spectral invariant. This is joint work with Vincent Humilière and Frédéric Le Roux.

Syzygies, gonality and symmetric products of curves

Robert Lazarsfeld
Stony Brook University
April 14, 2015
In the mid 1980s, Mark Green and I conjectured that one could read off the gonality of an algebraic curve $C$ from the syzygies among the equations defining any one sufficiently positive embedding of $C$. Ein and I recently noticed that a small variant of the ideas used by Voisin in her work on canonical curves leads to a quick proof of this gonality conjecture. The proof involves the geometry of certain vector bundles on the symmetric product of $C$.

Factorization of birational maps on steroids

Dan Abramovich
Brown University
April 14, 2015
Searching literature you will find the following statement (I'm paraphrasing): "If $X_1,X_2$ are nonsingular schemes proper over a complete DVR $R$ with residue characteristic 0, and $\phi: X_1 \to X_2$ is birational, then $\phi$ can be factored as a sequence of blowups and blowdown between nonsingular schemes proper over $R$, with nonsingular blowup centers." along with a demonstration: "The method of [Włodarczyk] or [AKMW] works word-for-word." In revenge you will find elsewhere (I'm paraphrasing): "Since a proof of weak factorization of birational maps over a complete D

Embedding the derived category of a curve into a Fano variety

Alexander Kuznetsov
Steklov Mathematical Institute, Moscow
April 14, 2015
According to the conjecture of Bondal, the derived category of coherent sheaves on any smooth projective variety can be embedded as a semiorthogonal component into the derived category of a Fano variety of higher dimension. I will explain how this embedding can be constructed for general curves of arbitrary genus. This is a joint work with Anton Fonarev.

Quadratic families of elliptic curves and unirationality of degree 1 conic bundles

János Kollár
Princeton University
April 13, 2015
We consider elliptic curves whose coefficients are degree 2 polynomials in a variable $t$. We prove that for infinitely many values of $t$ the resulting elliptic curve has rank at least 1. All such curves together form an algebraic surface which is birational to a conic bundle with 7 singular fibers. The main step of the proof is to show that such conic bundles are unirational. (joint work with M. Mella)

A new approach to the sensitivity conjecture

Michael Saks
Rutgers University
April 13, 2015
The sensitivity conjecture is a major outstanding foundational problems about boolean functions is the sensitivity conjecture. In one of its many forms, it asserts that the degree of a boolean function (i.e. the minimum degree of a real polynomial that interpolates the function) is bounded above by some fixed power of its sensitivity (which is the maximum vertex degree of the graph defined on the inputs where two inputs are adjacent if they differ in exactly one coordinate and their function values are different).

The André-Oort conjecture follows from the Colmez conjecture

Jacob Tsimerman
University of Toronto
April 9, 2015
The André-Oort conjecture says that any subvariety of a Shimura variety with a Zariski dense set of CM points must itself be a Shimura subvariety. In recent years, this has been the subject of much work. We explain how this conjecture for the moduli space of principally polarized abelian varieties of some dimension $g$ follows from current knowledge and a conjecture of Colmez regarding the Faltings heights of CM abelian varieties--in fact its enough to assume an averaged version of the Colmez conjecture.

Interleaved products in special linear groups: mixing and communication complexity

Emanuele Viola
Northeastern University
April 7, 2015
Let $S$ and $T$ be two dense subsets of $G^n$, where $G$ is the special linear group $\mathrm{SL}(2,q)$ for a prime power $q$. If you sample uniformly a tuple $(s_1,\ldots,s_n)$ from $S$ and a tuple $(t_1,\ldots,t_n)$ from $T$ then their interleaved product $s_1.t_1.s_2.t_2 \ldots s_n.t_n$ is equal to any fixed $g$ in $G$ with probability $(1/|G|)(1 + |G|^{-\Omega(n)})$.

Counting and dynamics in $\mathrm{SL}_2$

Michael Magee
Member, School of Mathematics
April 6, 2015
In this talk I'll discuss a lattice point count for a thin semigroup inside $\mathrm{SL}_2(\mathbb Z)$. It is important for applications I'll describe that one can perform this count uniformly throughout congruence classes. The approach to counting is dynamical---with input from both the real place and finite primes. At the real place one brings ideas of Dolgopyat concerning oscillatory functions into play​. At finite places the necessary expansion property follows from work of Bourgain and Gamburd (at one prime) or Bourgain, Gamburd and Sarnak (at squarefree moduli).

Fredholm theory for higher order elliptic boundary value problems in non-smooth domains

Irina Mitrea
Temple University; von Neumann Fellow, School of Mathematics
April 6, 2015
One of the most effective methods for solving boundary value problems for basic elliptic equations of mathematical physics in a given domain is the method of layer potentials. Its essence is to reduce the entire problem to an integral equation on the boundary of the domain which is then solved using Fredholm theory. Until now, this approach has been primarily used in connection with second order operators for which a sophisticated, far-reaching theory exists.