Proper base change for zero cycles

Moritz Kerz
University of Regensburg; Member, School of Mathematics
February 17, 2015
We study the restriction map to the closed fiber for the Chow group of zero-cycles over a complete discrete valuation ring. It turns out that, for proper families of varieties and for certain finite coefficients, the restriction map is an isomorphism. One can also ask whether for other motivic cohomology groups with finite coefficients one gets a restriction isomorphism.

The cohomology groups of Hilbert schemes and compactified Jacobians of planar curves

Luca Migliorini
University of Bologna; Member, School of Mathematics
February 18, 2015
I will first discuss a relation between the cohomology groups (with rational coefficients) of the compactified Jacobian and those of the Hilbert schemes of a projective irreducible curve $C$ with planar singularities, which extends the classical Macdonald formula, relating the cohomology groups of the symmetric product of a nonsingular curve to those of its Jacobian. The result follows from a "Support theorem" for the relative Hilbert scheme family associated with a versal deformation of the curve $C$.

Eigencurve over the boundary of the weight space

Liang Xiao
University of Connecticut
February 19, 2015
Eigencurve was introduced by Coleman and Mazur to parametrize modular forms varying $p$-adically. It is a rigid analytic curve such that each point corresponds to an overconvegent eigenform. In this talk, we discuss a conjecture on the geometry of the eigencurve: over the boundary annuli of the weight space, the eigencurve breaks up into infinite disjoint union of connected components and the weight map is finite and flat on each component. This was first verified by Buzzard and Kilford by an explicit computation in the case of $p = 2$ and tame level 1.

The symplectic displacement energy

Peter Spaeth
GE Global Research
February 20, 2015
To begin we will recall Banyaga's Hofer-like metric on the group of symplectic diffeomorphisms, and explain its conjugation invariance up to a factor. From there we will prove the positivity of the symplectic displacement energy of open subsets in compact symplectic manifolds, and then present examples of subsets with finite symplectic displacement energy but infinite Hofer displacement energy. The talk is based on a joint project with Augustin Banyaga and David Hurtubise.

Lower bounds for clique vs. independent set

Mika Göös
University of Toronto
February 23, 2015
We prove an $\omega(\log n)$ lower bound on the conondeterministic communication complexity of the Clique vs. Independent Set problem introduced by Yannakakis (STOC 1988, JCSS 1991). As a corollary, this implies superpolynomial lower bounds for the Alon--Saks--Seymour conjecture in graph theory. Our approach is to first exhibit a query complexity separation for the decision tree analogue of the UP vs. coNP question---namely, unambiguous DNF width vs. CNF width---and then "lift" this separation over to communication complexity using a result from prior work.

Arthur's trace formula and distribution of Hecke eigenvalues for $\mathrm{GL}(n)$

Jasmin Matz
Member, School of Mathematics
February 23, 2015
A classical problem in the theory of automorphic forms is to count the number of Laplace eigenfunctions on the quotient of the upper half plane by a lattice $L$. For $L$ a congruence subgroup in $\mathrm{SL}(2,\mathbb Z)$ the Weyl law was proven by Selberg giving an asymptotic count for these eigenfunctions. Further, Sarnak studied the distribution of the Hecke eigenvalues of these eigenfunctions. In higher rank, Lindenstrauss-Venkatesh proved the Weyl law for Hecke-Maass forms on $\mathrm{SL}(2,\mathbb Z) \backslash \mathrm{SL}(n,\mathbb R)/ \mathrm{SO}(n)$.

Computing inverses

Louis Rowen
Bar Ilan University
February 24, 2015
We compare methods of computing inverses of matrices over division rings. The most direct way is via Cohn's theory of full matrices, which was improved by Malcolmson, Schofield, and Westreich. But it is simpler to work with finite dimensional representations and generic matrices. In this talk, mostly expository, we describe the relevant algebraic techniques, including a description of full matrices, Sylvester rank functions, coproducts, and generic division algebras and its underlying theory of polynomial identities.

Projectivity of the moduli space of KSBA stable pairs and applications

Zsolt Patakfalvi
Princeton University
February 24, 2015
KSBA (Kollár-Shepherd-Barron-Alexeev) stable pairs are higher dimensional generalizations of (weighted) stable pointed curves. I will present a joint work in progress with Sándor Kovács on proving the projectivity of this moduli space, by showing that certain Hodge-type bundles are ample on it. I will also mention applications to the subadditivity of logarithmic Kodaira dimension, and to the ampleness of the CM line bundle.