Mathematical Sciences

School of Natural Sciences & Mathematics

Geometry Topology Dynamical Systems Seminar

AY 2021-22

Unless otherwise noted, the GTDS Seminar will be held on Mondays @ 3pm at FO 2.404.

For online seminars, zoom link will be shared in this page.

Date

Speaker/Institution

Title/Abstract

Oct 11 at 11am

GR 2.302

Tristan Ozuch

MIT

Noncollapsed degeneration and desingularization of Einstein 4-manifolds

Oct 11

Christos Mantoulidis 

Rice U.

A nonlinear spectrum on closed manifolds

Oct 18 at 2pm

online talk

Tristan Collins

MIT

Mirror symmetry for log del Pezzo surfaces

Zoom Link

Oct 25 at 2pm

GR 4.204

Ronan Conlon

UT Dallas

Asymptotically conical Calabi-Yau manifolds
Nov 1

Aaron Tyrrell

UT Dallas

Renormalized Area for 4-dimensional Minimal Hypersurfaces of a Poincaré-Einstein Space
Nov 8

Jonathan Johnson

OK State

Bi-Orderability Techniques and Double Twist Links

Nov 15

online talk

Henry Segerman

OK State

A survey of veering triangulations

Zoom link

       Nov 22         

Fall Break

 

 

Nov 29

online talk

Hang Huang

Texas A&M

Tensor Ranks and Matrix Multiplication Complexity

Zoom link

 

For questions about the seminar, please contact Baris Coskunuzer

Archive of talks from previous semesters

 

Noncollapsed degeneration and desingularization of
Einstein 4-manifolds

Tristan Ozuch (MIT) October 11, 2021

We study the moduli space of unit-volume Einstein 4-manifolds near its finite-distance boundary, that is, the noncollapsed singularity formation. We prove that any smooth Einstein 4-manifold close to a singular one in a mere Gromov-Hausdorff (GH) sense is the result of a gluing-perturbation procedure that we develop and which handles the presence of multiple trees of singularities at arbitrary scales. This sheds some light on the structure of the moduli space and lets us show that spherical and hyperbolic orbifolds which are Einstein in a synthetic sense cannot be GH-approximated by smooth Einstein 4-manifolds. 

 

A nonlinear spectrum on closed manifolds

Christos Mantoulidis (Rice U.)   October 11, 2021

The p-widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace–Beltrami operator, which was defined by Gromov in the 1980s and corresponds to areas of a certain min-max sequence of hypersurfaces. By a recent theorem of Liokumovich–Marques–Neves, the p-widths obey a Weyl law, just like the eigenvalues do. However, even though eigenvalues are explicitly computable for many manifolds, there had previously not been any >= 2-dimensional manifold for which all the p-widths are known. In recent joint work with Otis Chodosh, we found all p-widths on the round 2-sphere and thus the previously unknown Liokumovich–Marques–Neves Weyl law constant in dimension 2. Our work combines Lusternik–Schnirelmann theory, integrable PDE, and phase transition techniques.

 

Mirror symmetry for log del Pezzo surfaces

Tristan Collins (MIT) October 18, 2021

If X is a del Pezzo surface and D is a smooth anti-canonical divisor, we can regard the complement X\D as a non-compact Calabi-Yau surface.  I will discuss a proof of a strong form of the Strominger-Yau-Zaslow mirror symmetry conjecture for these non-compact surfaces.  It turns out the mirror Calabi-Yau is a rational elliptic surface (in particular, it has an elliptic fibration onto P^1) with a singular fiber which is a chain of nodal spheres.  I will discuss how we can construct special Lagrangian fibrations on these manifolds, as well as moduli of complex and symplectic structures and how hyper-Kahler rotation allows us to construct an identification of these moduli spaces.  This is joint work with A. Jacob and Y.-S. Lin.

 

Asymptotically conical Calabi-Yau manifolds

Ronan Conlon (UTD) October 25, 2021

Asymptotically conical Calabi-Yau manifolds are non-compact Ricci-flat Kähler manifolds that are modelled on a Ricci-flat Kähler cone at infinity. I will present a classification result for such manifolds. This is joint work with Hans-Joachim Hein (Muenster).

 

Renormalized Area for 4-dimensional Minimal Hypersurfaces of a Poincaré-Einstein Space

Aaron Tyrrell (UTD) November 1, 2021

In 1999 Graham and Witten showed that one can define a notion of renormalized area for properly embedded minimal submanifolds of Poincaré-Einstein spaces. For even dimensional submanifolds this quantity is a conformal invariant of ambient metric and the submanifold. In 2008 Alexakis and Mazzeo wrote a paper on this quantity for surfaces in a 3-dimensional PE manifold, getting an explicit formula and studying its functional properties.  We will look at a formula for the renormalized area of a minimal hypersurface of a 5-dimensional Poincaré-Einstein space. 

 

Bi-Orderability Techniques and Double Twist Links

Jonathan Johnson (OK State) November 8, 2021

The bi-orderability of link groups has become a fascinating research topic. In this talk, we will survey some useful tools that have been developed to investigate the bi-orderability of link complements, including results of Linnell-Rhemtulla-Rolfsen, Ito, and Kin-Rolfsen. In particular, we will apply these techniques to the double twist link groups.

 

A survey of veering triangulations

Henry Segerman (OK State) November 15, 2021

Veering triangulations are combinatorial objects introduced by Agol to describe three-manifolds with pseudo-Anosov bundle structures. I’ll describe past work with various coauthors relating veering triangulations to other structures on triangulations: strict angle structures and geometric structures, and in building censuses of veering triangulations. I’ll also talk about ongoing work with Saul Schleimer to extend Agol’s correspondence to manifolds with pseudo-Anosov flows, and work with Jason Manning and Saul Schleimer to produce Cannon-Thurston maps from veering triangulations. These include the first examples of Cannon-Thurston maps that do not come, even virtually, from surface subgroups.

 

Tensor Ranks and Matrix Multiplication Complexity

Hang (Amy) Huang (Texas A&M) November 29, 2021

Tensors are just multi-dimensional arrays. Notions of ranks and border rank abound in the literature. Tensor decompositions also have a lot of application in data analysis, physics, and other areas of science. I will try to give a colloquium-style talk surveying my recent two results about tensor ranks and their application to matrix multiplication complexity. The first result relates different notions of tensor ranks to polynomials of vanishing Hessian. The second one computes the border rank of 3 X 3 permanent. I will also briefly discuss the newest technique we used to achieve our results: border apolarity.