Math

School of Mathematics

Joint equidistribution of arithmetic orbits, joinings, and rigidity of higher rank diagonalizable actions I

Elon Lindenstrauss
Hebrew University of Jerusalem
March 2, 2015
An important theme in homogenous dynamics is that two parameter diagonalizable actions have much more rigidity than one parameter actions. One manifestation of this rigidity is rigidity of joinings of such actions. Joinings are an important concept in the study of dynamical systems, and allow detection of "hidden common denominator" of two (or more) dynamical systems. While one parameter diagonalizable actions on homogenous spaces have lots of joinings, for two parameter actions any measurable joining can be provided to be in fact of a very precise algebraic kind.

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 $\mathrm{SL}(2,\mathbb Z) \backslash \mathrm{SL}(n,\mathbb R)/ \mathrm{SO}(n)$.

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.

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 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 $C$.

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.

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 log-concavity conjecture and the tropical Laplacian

June Huh
Princeton University; Veblen Fellow, School of Mathematics
February 17, 2015
The log-concavity conjecture predicts that the coefficients of the chromatic (characteristic) polynomial of a matroid form a log-concave sequence. The known proof for realizable matroids uses algebraic geometry in an essential way, and the conjecture is open in its full generality. I will give a survey of known results and introduce a stronger conjecture that a certain Laplacian matrix associated to a matroid has exactly one negative eigenvalue.

Symplectic homology via Gromov-Witten theory

Luis Diogo
Columbia University
February 13, 2015
Symplectic homology is a very useful tool in symplectic topology, but it can be hard to compute explicitly. We will describe a procedure for computing symplectic homology using counts of pseudo-holomorphic spheres. These counts can sometimes be performed using Gromov-Witten theory. This method is applicable to a class of manifolds that are obtained by removing, from a closed symplectic manifold, a symplectic hypersurface of codimension 2. This is joint work with Samuel Lisi.

Extending the Prym map

Samuel Grushevsky
Stony Brook University
February 10, 2015
The Torelli map associates to a genus g curve its Jacobian - a $g$-dimensional principally polarized abelian variety. It turns out, by the works of Mumford and Namikawa in the 1970s (resp. Alexeev and Brunyate in 2010s), that the Torelli map extends to a morphism from the Deligne-Mumford moduli of stable curves to the Voronoi (resp. perfect cone) toroidal compactification of the moduli of abelian varieties. The Prym map associates to an etale double cover of a genus g curve its Prym - a principally polarized $(g-1)$-dimensional abelian variety.