Aug 21, 2024
Wednesday
|
09:15 AM - 09:30 AM
|
|
Welcome to SLMath
|
- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
-
--
- Abstract
- --
- Supplements
-
--
|
09:30 AM - 10:30 AM
|
|
When is an Alexandrov space smoothable?
Catherine Searle (Wichita State University)
|
- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
-
- Abstract
Zoom Link
In this talk, I will discuss the problem of when an Alexandrov space is smoothable. I will review the history of this question and discuss a new result that partially answers it. This is joint work with Pedro Solórzano and Fred Wilhelm.
- Supplements
-
--
|
10:30 AM - 11:00 AM
|
|
Break
|
- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
-
--
|
11:00 AM - 12:00 PM
|
|
Riemannian and Kahler flying wing steady Ricci solitons
Yi Lai (Stanford University)
|
- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
-
- Abstract
Zoom Link
Steady Ricci solitons are fundamental objects in the study of Ricci flow, as they are self-similar solutions and often arise as singularity models. Classical examples of steady solitons are the most symmetric ones, such as the 2D cigar soliton, the O(n)-invariant Bryantsolitons, and Cao’s U(n)-invariant Kahler steady solitons. Recently we constructed a family of flying wing steady solitons in any real dimension n≥3, which confirmed a conjecture by Hamilton in n=3. In dimension 3, we showed all steady gradient solitons are O(2)-symmetric. In the Kahler case, we also construct a family of Kahler flying wing steady gradient solitons with positive curvature for any complex dimension n≥2. This answers a conjecture by H.-D.Cao in the negative. This is partly collaborated with Pak-Yeung Chan and Ronan Conlon.
- Supplements
-
--
|
12:00 PM - 02:00 PM
|
|
Lunch
|
- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
-
--
|
02:00 PM - 03:00 PM
|
|
Eigenvalue problems and free boundary minimal surfaces in spherical caps
Ana Menezes (Princeton University)
|
- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
-
- Abstract
Zoom Link
In a joint work with Vanderson Lima (UFRGS, Brazil), we introduced a family of functionals on the space of Riemannian metrics of a compact surface with boundary, defined via eigenvalues of a Steklov-type problem. In this talk we will prove that each such functional is uniformly bounded from above, and we will characterize maximizing metrics as induced by free boundary minimal immersions in some geodesic ball of a round sphere. Also, we will determine that the maximizer in the case of a disk is a spherical cap of dimension two, and we will prove rotational symmetry of free boundary minimal annuli in geodesic balls of round spheres which are immersed by first eigenfunctions.
- Supplements
-
--
|
03:00 PM - 03:30 PM
|
|
Afternoon Tea
|
- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
-
--
|
03:30 PM - 04:30 PM
|
|
Boundary unique continuation properties
Zihui Zhao (Johns Hopkins University)
|
- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
-
- Abstract
Zoom Link
- Supplements
-
--
|
04:30 PM - 05:30 PM
|
|
Panel Discussion
Sun-Yung Chang (Princeton University), Eleonora Di Nezza (Institut de Mathématiques de Jussieu; École Normale Supérieure), Giada Franz (Massachusetts Institute of Technology), Lan-Hsuan Huang (University of Connecticut), Yi Lai (Stanford University), Catherine Searle (Wichita State University)
|
- Location
- SLMath: Eisenbud Auditorium
- Video
-
--
- Abstract
- --
- Supplements
-
--
|
06:15 PM - 08:00 PM
|
|
Dinner
|
- Location
- --
- Video
-
--
- Abstract
- --
- Supplements
-
--
|
|
Aug 22, 2024
Thursday
|
09:30 AM - 10:30 AM
|
|
Non-Kähler Calabi-Yau metrics
Anna Fino (Università di Torino; Florida International University)
|
- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
-
- Abstract
Zoom Link
Pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kähler manifolds. In the talk I will review general results on pluriclosed metrics in relation to the pluriclosed flow and the generalized Ricci flow. Next I will discuss some recent results on compact complex manifolds admitting a pluriclosed metric with vanishing Bismut-Ricci form.
- Supplements
-
--
|
10:30 AM - 10:40 AM
|
|
Group Photo
|
- Location
- SLMath: Front Courtyard
- Video
-
--
- Abstract
- --
- Supplements
-
--
|
10:40 AM - 11:00 AM
|
|
Break
|
- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
-
--
|
11:00 AM - 12:00 PM
|
|
Rigidity of mass-preserving 1-Lipschitz maps from integral current spaces into Euclidean space
Raquel Perales (Cimat)
|
- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
-
- Abstract
Zoom Link
We will prove that given an n-dimensional integral current space and a 1-Lipschitz map, from this space onto the n-dimensional Euclidean ball, that preserves the mass of the current and is injective on the boundary, then the map has to be an isometry. We deduce as a consequence the stability of the positive mass theorem for graphical manifolds as originally formulated by Huang--Lee--Sormani. (Joint work with G. Del Nin).
- Supplements
-
--
|
12:00 PM - 02:00 PM
|
|
Lunch
|
- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
-
--
|
02:00 PM - 03:00 PM
|
|
Almost convexity of the Mabuchi functional in singular settings
Eleonora Di Nezza (Institut de Mathématiques de Jussieu; École Normale Supérieure)
|
- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
-
- Abstract
Zoom Link
The Mabuchi functional M was introduced by Mabuchi in the 80's in relation to the existence of canonical metrics on a compact Kähler manifold. The critical points of M are indeed constant scalar curvature Kähler (cscK) metrics. Recently, Chen and Cheng proved that the existence of a (smooth) cscK metric is equivalent to the properness of such functional. In order to look for singular metrics, it is then natural to study the properties of the Mabuchi functional in singular settings. In this talk we prove that this functional is (almost) convex in the very general "big case".
- Supplements
-
--
|
03:00 PM - 03:30 PM
|
|
Afternoon Tea
|
- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
-
--
|
03:30 PM - 04:30 PM
|
|
Quartically pinched submanifolds for the mean curvature flow in the sphere.
Artemis Vogiatzi (School of Mathematical Sciences, Queen Mary, University of London)
|
- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
-
--
- Abstract
Zoom Link
By using a sharp quartic curvature pinching for the mean curvature flow in $\mathbb{S}^{n+m}$, $m\ge2$, we improve the quadratic curvature conditions. Using a blow up argument, we prove a codimension and a cylindrical estimate, where in regions of high curvature, the submanifold becomes approximately codimension one, quantitatively, and is weakly convex and moves by translation or is a self shrinker. With a decay estimate, the rescaling converges smoothly to a totally geodesic limit in infinite time, without using any iteration procedures or integral analysis. Our approach relies on the preservation of the quartic pinching condition along the flow and gradient estimates that control the mean curvature in regions of high curvature.
- Supplements
-
--
|
|
Aug 23, 2024
Friday
|
09:30 AM - 10:30 AM
|
|
Einstein metrics in Gromov Thurston manifolds
Ursula Hamenstaedt (Rheinische Friedrich-Wilhelms-Universität Bonn)
|
- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
-
--
- Abstract
Zoom Link
We show that for every $n\geq 4$ and every $\epsilon >0$ there exists a closed manifold $M$ which admits a metric of curvature contained in the interval $[-1-\epsilon,-1+\epsilon]$, which admits a negatively curved Einstein metric but no metric of constant curvature.
- Supplements
-
--
|
10:30 AM - 11:00 AM
|
|
Break
|
- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
-
--
|
11:00 AM - 12:00 PM
|
|
Renormalized Curvature Integrals on Poincare-Einstein manifolds
Yueh Ju Lin (Wichita State University)
|
- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
-
- Abstract
Zoom Link
Poincare-Einstein (PE) manifolds such as Poincare ball model are complete Einstein manifolds with a well-defined conformal boundary. There is a rich interplay between the conformal geometry of the boundary of a PE manifold and the Riemannian geometry of its interior. A first step in studying the moduli space of PE manifolds is to develop a good understanding of its global invariants. In even dimensions, renormalized curvature integrals give many such invariants. In this talk, I will discuss a general procedure for computing renormalized curvature integrals on PE manifolds. In particular, this explains the connection between the Gauss-Bonnet-type formulas of Albin and Chang-Qing-Yang for the renormalized volume, and explicitly identify a scalar conformal invariant in the latter formula. Our procedure also produces similar formulas for compact Einstein manifolds. This talk is based on joint works with Jeffrey Case, Ayush Khaitan, Aaron Tyrrell, and Wei Yuan.
- Supplements
-
--
|
12:00 PM - 02:00 PM
|
|
Lunch
|
- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
-
--
|
02:00 PM - 03:00 PM
|
|
Non-convexity of level sets of k-Hessian equations in convex ring
Ling Xiao (University of Connecticut)
|
- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
-
--
- Abstract
Zoom Link
In this talk we construct explicit examples that show the sublevel sets of the solution of a k-Hessian equation defined on a convex ring do not have to be convex.
- Supplements
-
--
|
03:00 PM - 03:30 PM
|
|
Afternoon Tea
|
- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
-
--
|
03:30 PM - 04:30 PM
|
|
Relating the index and the topology of (free boundary) minimal surfaces
Giada Franz (Massachusetts Institute of Technology)
|
- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
-
- Abstract
Zoom Link
In this talk, I will discuss existing results and open problems about estimating the Morse index of a (free boundary) minimal surface from below by a function of its topology.
I will mostly focus on results proving that the Morse index of a free boundary minimal surface in a three-dimensional Riemannian manifold grows linearly with the product of its area and its topology. This is joint work with Santiago Cordero-Misteli, and it is inspired by Antoine Song's result in the closed case.
- Supplements
-
--
|
|