Geometry/Topology Seminar

from Thursday
June 01, 2017 to Sunday
December 31, 2017
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Tuesday
September 12, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Christina Sormani, CUNY
Rectifiablility of Gromov-Hausdorff Limits

The Gromov-Hausdorff limits of sequences of Riemannian manifolds do not in general have any regularity. Ranaan Schul and Stefan Wenger produced a sequence of such manifolds which are noncollapsing (and even have a uniform contractibility function) whose limit space is not even countably $H^m$ rectifiable. If one has a uniform lower bound on sectional curvature then the limit is an Alexandrov space which is rectifiable by a theorem of Burago-Gromov-Perelman, and if one one has a uniform lower bound on Ricci curvature, Cheeger-Colding proved rectifiability and more. By definition the intrinsic flat limits of Riemannian manifolds are always countably $H^m$ rectifiable, and so one may prove a GH limit is rectifiable by proving the GH and intrinsic flat limits agree. This was first done in a joint paper with Wenger applying the Gromov filling volume, and more recently in joint work with Portegies applying a new notion we call the sliced filling volume. I will also present work of postdocs in this area including: Matveev-Portegies, Perales and Li-Perales. This work is completely disjoint from my work on the convergence of manifolds with nonnegative scalar curvature as the GH and intrinsic flat limits do not agree and GH limits need not exist in that setting.


Tuesday
September 26, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Mark Lawrence, Nazarbeyev University
Totally real tori in $S^1 × C$ and their polynomial hulls.

If $K ⊂ C^n$ is a compact subset, there is in general no hope of finding analytic structure in the polynomial hull $\hat{K}$ \K. Even for sets which are quite smooth, there are difficulties. In this talk, some theorems about analytic structure in the polynomial hull of a totally real torus in $S^1 × C$ will be discussed. Connections with knot theory (torus knots, quasipositivity) and holomorphic motions a la Slodkowski will be explained.


Tuesday
October 10, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Luca di Cerbo, Stony Brook University
Minimal Volumes, Hyperbolic Geometry and Lattices in PU(2, 1)

The study of (minimal) volumes in hyperbolic geometry has attracted quite a bit of attention in the mathematical community. In the first part of the talk, I will give an overview of this fascinating field and review some classical results. In the second part of the talk, I will describe new results concerning complex hyperbolic surfaces. This is based on a couple of joint papers with M. Stover.


Tuesday
October 17, 2017

4:00 PM - 5:00 PM
Math Tower P-131
Jiyuan Han, UW-Madison
On closedness of ALE SFK metrics on minimal ALE Kahler surfaces

For certain cases with some topological assumption that gives the boundedness of Sobolev constant, we construct the space of ALE SFK metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with Jeff Viaclovsky.


Tuesday
October 24, 2017

4:00 PM - 5:30 PM
SCGP 102
Mark Haskins, University of Bath
Codimension One Collapse and Special Holonomy Metrics

In this talk we describe recent developments related to codimension one collapse of exceptional holonomy metrics. i.e. where a family of special holonomy metrics on a space of dimension n converges in some limit to a metric on a space of dimension n-1. Interesting examples occur for hyperkaehler 4-manifolds, G_2 holonomy manifolds and Spin_7 holonomy manifolds. The talk will focus on the G_2 holonomy case, but will also draw on the better understood hyperkaehler case for inspiration and for useful analogies. These mathematical developments are closely related to important limits in physics, e.g. in the context of G_2 holonomy metrics it is related to the identification of the weak coupling limit of M theory compactified on a G_2 holonomy space being Type IIA String Theory on a 6-dimensional space. This is work joint with Lorenzo Foscolo and Johannes Nordstrom.


Tuesday
October 31, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Nishanth Gudapati, Yale University
An energy functional for axially symmetric Maxwell perturbations of Kerr-de Sitter black holes

The Kerr-de Sitter black hole family is a solution of Einstein's equations of general relativity with a positive cosmological constant. After reviewing some background on these spacetimes, we shall discuss the proof that there exists a phase space of canonical variables for the (unconstrained) axially symmetric Maxwell's equations propagating on Kerr-de Sitter, such that their motion is restricted to the level sets of a positive-definite Hamiltonian, despite the ergo-region. If time permits, we shall discuss the equivalent results for the corresponding fully coupled (and constrained) Einstein-Maxwell initial value problem.


Tuesday
November 07, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Mehdi Lejmi, CUNY
On the Chern-Yamabe problem

On an almost-Hermitian manifold, the Chern connection is the unique Hermitian connection with J-anti-invariant torsion. In this talk, we compare the Chern scalar curvature with the Riemannian one. Moreover, we study an analog of Yamabe problem by looking for an almost Hermitian metric with constant Chern scalar curvature in a conformal class, extending results of Angella, Calamai and Spotti to the non-integrable case.

We also study the Chern-Yamabe flow and get existence of solutions when the Chern scalar curvature is small enough. This is joint work with Markus Upmeier and Ali Maalaoui.


Tuesday
November 14, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Eveline Legendre, Universite de Toulouse
Localization formula applied to Sasaki geometry

We apply an extension of the Duistermaat--Heckman Theorem to study the volume, the total scalar curvature and the Einstein-Hilbert functionals defined on the Sasaki cone, and prove that they are all proper. This implies that the (transversal) Futaki invariant always admits a zero in a Sasaki cone.


Tuesday
November 21, 2017

4:00 PM - 5:30 PM
SCGP 102
Misha Gromov, IHES & NYU
TBA


Tuesday
November 28, 2017

4:00 PM - 5:00 PM
Math Tower P-131
Antonio Aché, University of Notre Dame
Sharp Sobolev-trace inequalities of order four

We establish sharp Sobolev inequalities of order four on Euclidean d-balls for d greater than or equal to four. When d=4, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean 2-balls. Our method relies on the use of scattering theory on hyperbolic d-balls. As an application, we characterize the extremals of the main term in the log-determinant formula corresponding to the conformal Laplacian coupled with the boundary Robin operator on Euclidean 4-balls. This is joint work with Alice Chang.


Tuesday
December 05, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Boris Botvinnik, University of Oregon
Topology of the spaces of metrics of positive scalar or positive Ricci curvature

I will first review Hitchin's index-difference map from the space of positve-scalar-curvature metrics to real K-theory, and discuss my joint result with J. Ebert and O. Randal-Williams, that shows that the index-difference map induces nontrivial homomorphisms of appropriate homotopy groups.

I will then describe some recent related results on the space of metrics of positive Ricci curvature. In particular, I will discuss my joint work with J. Ebert and D. Wraith, which shows that the space of such metrics on connected sums of certain products of spheres has non-trivial rational homotopy groups in specific dimensions.


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Instructions for subscribing to Stony Brook Math Department Calendars