Geometry Seminar

                                                                                                                      Vanderbilt University
                                                                                                                                Fall  2018

   Organizers: Rares Rasdeaconu, Larry Rolen

   Fridays, 3:10-4:00pm in SC 1310 (unless otherwise noted)

                Tuesday, October 9th, 3:10-4:00pm, Room SC1310 

                Speaker: Ian Wagner, Emory University 

                Title: Harmonic Hecke eigenlines and
Mazur's problem

We construct two families of harmonic Maass Hecke eigenforms.  Using these families we construct p-adic
                harmonic Maass forms in the sense of Serre.  The p-adic properties of these forms answer a question of Mazur about the
                existence of an "eigencurve-type" object in the world of harmonic Maass forms.
(Contact Person: Larry Rolen)

                Friday, Oct 19th - No meeting (Fall Break)

                Friday, October 26th

Rudy Rodsphon, Northeastern University

                Title:  On a conjecture of Connes and Moscovici  

In the early eighties, Connes developed his Noncommutative Geometry program, mostly to extend index theory
                to situations where usual tools of differential topology are not applicable. A typical situation is foliations whose holonomy
                does not necessarily preserve any transverse measure, or equivalently the orbit space of the action of the full group of
                diffeomorphisms of a manifold. In the end of the nineties, Connes and Moscovici worked out an equivariant index problem
                in these contexts, and left a conjecture about the calculation of this index in terms of characteristic classes. A large portion of
                the talk will be expository, and will survey the history of this problem. Time permitting, we will explain our recent solution to
                this conjecture. No prior knowledge of the subject will be assumed. This is a joint work with Denis Perrot. (Contact Person:
                Gennadi Kasparov

                Friday, November 9th

Scott Wilson, CUNY Queens College

                Title:  Extensions of some results in complex and Kahler geometry to the non-integrable setting

In this talk I will survey recent joint work with Joana Cirici which extends Dolbeault cohomology to all
                almost complex manifolds, and generalizes many of the foundational results for compact Kahler manifolds to the
                non-integrable setting. Among these are the so-called Hodge, Serre and Lefschetz dualities, as well as certain topological
                bounds on solutions to geometric equations. All of this work stems from a careful study of the exterior derivative on the
                differential forms of an almost complex manifold, and this talk will begin with an elementary discussion of that situation.
                Preprints are available at: arXiv:1809.1416 and arXiv:1809.1414. (Contact Person: Rares Rasdeaconu


                Friday, Nov 16 & Nov 23 - No meeting (Thanksgiving Break)

                Friday, November 30th

                Speaker:   Frank Thorne
, University of South Carolina

                Title:  Gauss' Circle Problem and Arithmetic Statistics

Suppose you have a circle centered at the origin, with radius r. How many integer lattice points (x, y) are contained
                within it? Gauss proved that the answer is pi*r^2 + O(r), and I will explore Gauss' argument a little bit: Can the error terms be
                improved? Can we demand congruence conditions on x and y? And to what other shapes can the argument be generalized?
                It turns out that related questions are at the core of many recent papers in "arithmetic statistics", including those mentioned
                in Manjul Bhargava's 2014 Fields Medal citation. I will give a brief overview of this, with an emphasis on how variations on
                Gauss' method lead to a variety of arithmetic theorems.
(Contact Person: Larry Rolen)

                Friday, December 7th

                Speaker:  Gueo Grancharov
Florida International University
                Title:   Non-Kaehler structures on foliations

After briefly explaining the difference between Kaehler and non-Kaehler structures, I'll focus on one of
                the main sources of compact non-Kaheler manifolds - various types of foliations. These include twistor spaces, principal
                bundles and complex suspensions. I'll explain what type of special Hermitian metrics they carry and report on some
                topologically new examples of manifolds admitting solutions of Strominger-Hull system.
(Contact Person: Rares Rasdeaconu)


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