The Mathematics Department is in the Physics Building, which is located at 120 Science Drive, Duke Campus, Durham, NC
When you arrive on Friday, please go to the Math Department Lounge, Physics 101, for refreshments and to meet the rest of us. All talks will be in Physics 128 on the first floor.
The following tentative schedule will be filled out as information is available.
FRIDAY APRIL 17
12 noon Arrivals in Physics 101 with snacks and drinks
3 – 4 Andrew Sageman-Furnas (NC State University)
4 – 4:30 Refreshments in Physics 101
4:30 – 5:30 Simon Brendle (Columbia University)
SATURDAY APRIL 18
8:30 AM Breakfast in Physics 101
9:30 – 10:30 Chi Li (Rutgers)
10:30 – 11 Remainder of breakfast in Physics 101
11 – 12 Tamas Darvas (U-Maryland)
12 – 2 PM LUNCH in nearby restaurants
2 – 3 Ben Lowe (Chicago)
3 – 30 Refreshments in the lobby
3:30 – 4:30 Bruno Staffa (Rice University)
4:30 – 5 Refreshments in the lobby
5 - 6 Dan Cristofaro-Gardiner (U-Maryland)
6 – 7 Reception, Gross Hall, 3rd Floor Atrium: Wine, hors d'oeuvres, brief speeches
7– 9 Banquet, 330 Gross Hall
For speakers and those who have registered and paid for the banquet.
SUNDAY APRIL 19
8:30 AM Breakfast in Physics 101
9:30 –10:30 Sven Hirsch (Columbia University)
10:30 -–11 Remainder of breakfast in the lobby
11 –12 Yi Lai (Irvine)
CONFERENCE ENDS AT 12 NOON
TITLES AND ABSTRACTS (Listed alphabetically by Speaker's Last Name)
Simon Brendle (Columbia University)
Title: Systolic inequalities and the Horowitz–Myers conjecture
Abstract: Let n be an integer with 3 ≤ n ≤ 7, and let g be a Riemannian metric on B^2 x T^{n-2} with scalar curvature at least -n(n-1). We establish an inequality relating the systole of the boundary to the infimum of the mean curvature on the boundary. As a consequence, we obtain a new positive energy theorem where equality holds for the Horowitz–Myers metrics. This is joint work with Pei–Ken Hung.
Dan Cristofaro-Gardiner (U-Maryland)
Title: Symplectic dynamics, Floer-theoretic Weyl laws, and the Simplicity Conjecture
Abstract: The algebraic structure of the group of volume-preserving homeomorphisms of a manifold of dimension at least three has been well-understood since the work of Fathi from the 70s. However, the surface case has long remained a mystery. I will discuss joint work clearing up parts of this mystery, including the resolution of the longstanding Simplicity Conjecture. Some Weyl type laws that have recently been established in low-dimensional symplectic geometry play a key role in the arguments. These can also be used to show that a generic area-preserving diffeomorphism of a surface has a dense set of periodic points, and I will briefly give a sense for how this works.
Tamas Darvas (U-Maryland)
Title: A YTD correspondence for constant scalar curvature metrics
Abstract: Given a compact Kähler manifold, to better understand Mabuchi's K energy we introduce a family of K^beta energies, whose favorable properties are similar to those of the Ding energy from the Fano case. The construction uses Berman's transcendental quantization, and we show that the slope of the K^beta energies along test configurations can be computed using intersection theory. With these ingredients in place we provide a uniform Yau–Tian–Donaldson correspondence that characterizes the existence of a unique constant scalar curvature Kähler metric using test configurations. Combining our techniques with the non-Archimedean approach to K-stability pioneered by Boucksom–Jonsson, we show that the properness of the classical energy can be tested by checking its slope along a distinguished subclass of Li-type models, called log discrepancy models, thus yielding another G-uniform Yau–Tian–Donaldson correspondence. (Joint with Kewei Zhang)
Sven Hirsch (Columbia University)
Title: Causality of Killing vector fields and Killing spinors
Abstract: We analyze the causal type of both Killing vector fields and Killing spinors. As an application we give a proof of Bartnik's stationary vacuum conjecture from 1989 and geometrically characterize Siklos wave spacetimes. This talk is based on joint work with Lan-Hsuan Huang and Yiyue Zhang.
Yi Lai (Irvine)
Title: Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture
Abstract: We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Our proof relies on a complete classification of ancient, asymptotically cylindrical flows. We prove that any such flow is non-collapsed, convex, rotationally symmetric, and belongs to one of three canonical families: ancient ovals, the bowl soliton, or the flying wing translating solitons. This is joint work with Richard Bamler.
Chi Li (Rutgers)
Title: On the volume of Kähler-Einstein Fano varieties
Abstract: I will talk about a joint work with Minghao Miao, which proves that: in any dimension n, the second largest volume of Kähler–Einstein Fano manifolds is obtained exactly by the volume of quadric hypersurface and the product CP^1*CP^(n-1). We prove this by using the theory of K-stability and its new connection with minimal rational curves on Fano manifolds. The singular case will also be discussed in its relation to the volume of Klt singularities.
Ben Lowe (Chicago)
Title: Rigidity and Finiteness of Totally Geodesic Hypersurfaces
Abstract: This talk will be about the following theorem (joint with Filip-Fisher): every closed analytic negatively curved manifold contains only finitely many compact totally geodesic hypersurfaces unless it has constant curvature, in which case it is an arithmetic hyperbolic manifold. I will also give an account of work that preceded it (some joint with Neves) on rigidity statements involving curvature and minimal surfaces, wherein tools and ideas from dynamical systems became important and formed part of the motivation for the work with Filip-Fisher.
Andrew Sageman-Furnas (NC State University)
Title: Constructing isometric tori with the same curvatures
Abstract: A longstanding problem in differential geometry, known as the Bonnet problem, asks: is a compact surface in Euclidean three-space uniquely determined by its metric and mean curvature function? The answer is known to be yes for a topological sphere and yes for a generic surface.
In this talk, we explicitly construct a pair of immersed tori that are related by a mean curvature preserving isometry. These tori are the first examples of compact Bonnet pairs. Moreover, we prove these isometric tori are real analytic. This resolves a second longstanding open problem on whether real analyticity of the metric already determines a unique compact immersion.
We describe the discovery of these analytic tori using discrete differential geometry. It involves exploring immersions of a 5x7 quad decomposition of a torus and a theory of discrete Bonnet pairs.
The smooth/analytic theory is joint work with Alexander Bobenko and Tim Hoffmann, and the discrete theory is joint work with Tim Hoffmann and Max Wardetzky.
Bruno Staffa (Rice University)
Title: Weyl Law for the volume spectrum and applications
Abstract: Let (M^n,g) be an n-dimensional Riemannian manifold and 1 ≤ k ≤ n−1. In the 1980s, Gromov studied the asymptotic behavior of the volumes ω^k_p(M^n,g) of certain (possibly singular) k-dimensional minimal submanifolds N^k_p of (M^n,g) arising from a Morse Theory of the volume functional (the Almgren-Pitts Min-Max Theory). He conjectured that these volumes ω^k_p(M^n,g) behave when p→∞ similarly to the eigenvalues λ_p of the Laplacian in (M^n,g), obeying the so-called Weyl Law for the Volume Spectrum. I will discuss the current progress on that conjecture and some of the ideas in the proof, as well as its applications to density and equidistribution of minimal hypersurfaces and stationary geodesic nets.