Department of Mathematics FAS Harvard University One Oxford Street Cambridge MA 02138 USA Tel: (617) 495-2171 Fax: (617) 495-5132
To post a seminar which takes place at the Mathematics department, please email with date, time, room, title and possibly with an abstract.
Periods of algebraic cycles
on Wednesday, November 21, 2018, at 1:30 - 3:00 pm in CMSA Building, 20 Garden St, G10
The tangent space of the Hodge locus at a point can be described by the so called infinitesimal variation of Hodge structures and the cohomology class of Hodge cycles. For hypersurfaces of dimension $n$ and degree $d$ it turns out that one can describe it without any knowledge of cohomology theories and in a fashion which E. Picard in 1900’s wanted to study integrals/periods. The data of cohomology class is replaced with periods of Hodge cycles, and explicit computations of these periods, will give us a computer implementable description of the tangent space. As an application of this we show that for examples of $n$ and $d$, the locus of hypersurfaces containing two linear cycles whose intersection is of low dimension, is a reduced component of the Hodge locus in the underlying parameter space.

Columbia University
Geometric estimates for complex Monge-Ampere equations
on Tuesday, November 27, 2018, at 4:00 pm in Science Center 507
In the talk, we will present a recent result concerning the uniform gradient and diameter estimates for a family of geometric complex Monge-Ampere equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge-Ampere equations. We will also discuss uniform diameter estimate for a collapsing family of twisted Kahler-Einstein metrics on Kahler manifolds of nonnegative Kodaira dimensions.

Universality and least singular values of random matrix products
on Wednesday, November 28, 2018, at 3:00 - 4:00 pm in CMSA Building, 20 Garden St, G10
We consider the product of m independent iid random matrices as m is fixed and the sizes of the matrices tend to infinity. In the case when the factor matrices are drawn from the complex Ginibre ensemble, Akemann and Burda computed the limiting microscopic correlation functions. In particular, away from the origin, they showed that the limiting correlation functions do not depend on m, the number of factor matrices. We show that this behavior is universal for products of iid random matrices under a moment matching hypothesis. In addition, we establish universality results for the linear statistics for these product models, which show that the limiting variance does not depend on the number of factor matrices either. The proofs of these universality results require a near-optimal lower bound on the least singular value for these product ensembles.

Bryn Mawr College
Upper bounds for constant slope p-adic families of modular forms
on Wednesday, November 28, 2018, at 3:00 pm in Science Center 507
This talk is concerned with the radius of convergence of p-adic families of modular forms --- q-series over a p-adic disc whose specialization to certain integer points is the q-expansion of a classical Hecke eigenform of level p. Numerical experiments by Gouvêa and Mazur in the nineties predicted the general existence of such families but also suggested, in spirit, the radius of convergence in terms of an initial member. Buzzard and Calegari showed, ten years later, that the Gouvêa--Mazur prediction was false. It has since remained open question how to salvage it. Here we will present some recent theoretical results towards such a salvage, backed up by numerical data.

CMSA COLLOQUIUM: Robert Haslhofer
University of Toronto
Recent progress on mean curvature flow
on Wednesday, November 28, 2018, at 4:30 pm in CMSA Building, 20 Garden St, G10
A family of surfaces moves by mean curvature flow if the velocity at each point is given by the mean curvature vector. Mean curvature flow is the most natural evolution in extrinsic geometry and shares many features with Hamilton’s Ricci flow from intrinsic geometry. In the first half of the talk, I will give an overview of the well developed theory in the mean convex case, i.e. when the mean curvature vector everywhere on the surface points inwards. Mean convex mean curvature flow can be continued through all singularities either via surgery or as level set solution, with a precise structure theory for the singular set. In the second half of the talk, I will report on recent progress in the general case without any curvature assumptions. Namely, I will describe our solution of the mean convex neighborhood conjecture and the nonfattening conjecture, as well as a general classification result for all possible blowup limits near spherical or cylindrical singularities. In particular, assuming Ilmanen’s multiplicity one conjecture, we conclude that for embedded two-spheres the mean curvature flow through singularities is well-posed. This is joint work with Kyeongsu Choi and Or Hershkovits.

University of Massachusetts - Boston
Degenerations from Floer cohomology
on Friday, November 30, 2018, at 2:00 - 3:15 pm in Science Center 507
I will explain how, under suitable hypotheses, one can construct a flat degeneration from the symplectic cohomology of log Calabi-Yau varieties to the Stanley-Reisner ring of the dual intersection complex of a compactifying divisor and how this result relates to recent mirror constructions of Gross-Hacking-Keel and Gross-Siebert. Along the way, I will hopefully explain various techniques for understanding symplectic cohomology of affine varieties which apply beyond the log Calabi-Yau setting. Parts of this are based on joint work with Sheel Ganatra.

Princeton University
Structural results in wrapped Floer theory
on Friday, November 30, 2018, at 3:30 pm in Science Center 507
I will discuss results relating different partially wrapped Fukaya categories. These include a Kunneth formula, a 'stop removal' result relating partially wrapped Fukaya categories relative to different stops, and a gluing formula for wrapped Fukaya categories. The techniques also lead to generation results for Weinstein manifolds and for Lefschetz fibrations. The methods are mainly geometric, and the key underlying Floer theoretic fact is an exact triangle in the Fukaya category associated to Lagrangian surgery along a short Reeb chord at infinity. This is joint work with Sheel Ganatra and Vivek Shende.

Boston College
Langlands functoriality, the converse theorem, and integral representations of L-functions
on Wednesday, December 05, 2018, at 3:00 pm in Science Center 507
Langlands functoriality predicts maps between automorphic forms on different groups, dictated by a map of L-groups. One important class of such maps are endoscopic liftings, established by Arthur using the trace formula. In this talk I describe an approach to endoscopic lifting that does not use the trace formula. Instead it relies on the converse theorem of Cogdell and Piatetski-Shapiro and on new integral representations of L-functions of Cai, Friedberg, Ginzburg and Kaplan.