UC San Diego Geometry Seminar 2020-2021

Unless otherwise noted, all seminars are on Wed 11:00 - 12:00 in Zoom setting. To obtain the Zoom ID please contact Luca Spolaor at lspolaor@ucsd.edu
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Winter 2021 Schedule

January 13

Riccardo Tione, EPFL
Title: Anisotropic energies: examples, rectifiability and regularity
Abstract: Anisotropic energies are functionals defined by integrating over a generalized surface (such as a current or a varifold) an integrand depending on the tangent plane to the surface. In the case of a constant positive integrand, one obtains the area functional, and hence one can see anisotropic energies as a generalization of it. A long standing question in geometric measure theory is to establish regularity properties of critical points to such functionals. In this talk, I will discuss some recent developments on this theory, addressing in particular the question of rectifiability of stationary points and regularity of stationary Lipschitz graphs. The talk is based on joint work with Antonio De Rosa.

January 20

Yiming Zhao, MIT
Title: Mass transport problem on the unit sphere via Gauss map
Abstract: In this talk, I will discuss when two probability measures on the unit sphere can be transported to one another using the Gauss map of a convex body. Here, a convex body is a compact convex subset of the Euclidean n-space with non-empty interior. Notice that the boundary of a convex body might not be smooth-in general, it can even contain a fractal structure. This problem can be viewed as the problem of reconstructing a convex body using partial data regarding its Gauss map. When smoothness is assumed, it reduces to a Monge-Ampere type equation on the sphere. However, in this talk, we will work with generic convex bodies and talk about how variational argument can work in this setting. This is joint work with Karoly Boroczky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang.

January 27

Yi Lai, UC Berkeley
Title: A family of 3d steady gradient solitons that are flying wings
Abstract We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension n \geq 4, we find a family of Z2 x O(n - 1)-symmetric but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operator. We show that these solitons are non-collapsed.

February 3


Jonathan Zhu, Princeton, 4:00-5:00pm
Title: Explicit Łojasiewicz inequalities for mean curvature flow shrinkers
Abstract: Łojasiewicz inequalities are a popular tool for studying the stability of geometric structures. For mean curvature flow, Schulze used Simon’s reduction to the classical Łojasiewicz inequality to study compact tangent flows. For round cylinders, Colding and Minicozzi instead used a direct method to prove Łojasiewicz inequalities. We’ll discuss similarly explicit Łojasiewicz inequalities and applications for other shrinking cylinders and Clifford shrinkers.

February 10

Tamas Darvas, Univ. Maryland
 Title: The closure of test configurations and algebraic singularity types
Abstract: Given a Kähler manifold X with an ample line bundle L, we consider the metric space of L^1 geodesic rays associated to the first Chern class of L. We characterize rays that can be approximated by ample test configurations. At the same time, we also characterize the closure of algebraic singularity types among all singularity types of quasi-plurisubharmonic functions, pointing out the very close relationship between these two seemingly unrelated problems. By Bonavero's holomorphic Morse inequalities, the arithmetic and non-pluripolar volumes of algebraic singularity types coincide. We show that in general the arithmetic volume dominates the non-pluripolar one, and equality holds exactly on the closure of algebraic singularity types. Joint work with Mingchen Xia.

February 17


Mat Langford, 4:00-5:00pm
Title: Ancient solutions out of polytopes.
Abstract: I will show how to construct a very large family of new examples of convex ancient and translating solutions to mean curvature flow in all dimensions. At $t=-\infty$, these examples resemble a family of standard Grim hyperplanes of certain prescribed orientations. The existence of such examples has been suggested by Hamilton and Huisken—Sinestrari. Our examples include solutions with symmetry group $D\times \mathbb Z_2$, where $D$ is the symmetry group of any given regular polytope, and, surprisingly, many examples which admit only a single reflection symmetry. We also exhibit a family of eternal solutions which do not evolve by translation, settling a conjecture of Brian White in the negative. Time permitting, I will present further structure and partial classification results for this class of solutions, as well as some open questions and conjectures. Joint with T. Bourni and G. Tinaglia

February 24

Robin Neumayer, Northwestern
Title: $d_p$ Convergence and $\epsilon$-regularity theorems for entropy and scalar curvature lower bounds
Abstract: In this talk, we consider Riemannian manifolds with almost non-negative scalar curvature and Perelman entropy. We establish an $\epsilon$-regularity theorem showing that such a space must be close to Euclidean space in a suitable sense. Interestingly, such a result is false with respect to the Gromov-Hausdorff and Intrinsic Flat distances, and more generally the metric space structure is not controlled under entropy and scalar lower bounds. Instead, we introduce the notion of the $d_p$ distance between (in particular) Riemannian manifolds, which measures the distance between $W^{1,p} Sobolev spaces, and it is with respect to this distance that the $\epsilon$ regularity theorem holds. We will discuss various applications to manifolds with scalar curvature and entropy lower bounds, including a compactness and limit structure theorem for sequences, a uniform $L^\infty$ Sobolev embedding, and a priori $L^p$ scalar curvature bounds for $p<1$ This is joint work with Man-Chun Lee and Aaron Naber.

March 3


Saikee Yeung, Purdue Univ.
Title: Limit of Weierstrass measure on stable curves
Abstract: Let M be a Riemann surface. A usual Weierstrass point is a point on M at which there is a holomorphic one form vanishing to order at least the dimension of the space of holomorphic one forms. This notion can be generalized to Ln for any ample line bundle L on M. It is observed by Olsen in 70’s that the distribution of Weierstrass point is dense on M in the usual complex topology as n → ∞. A surprising result of Mumford and Neeman in 80’s states that the weighted Dirac measure of Weierstrass points of Ln approaches the pull back of the flat metric from its Jacobian variety as n → ∞. On the other hand, it is observed in the work of Ballico, Furio, Gatto, Lax, Little and others in 80-90’s that the picture on a stable rational curve is different in the sense that a corresponding Weierstrass measure is not dense in complex topology. The goal of the talk is to explain a result with Ngaifung Ng in this direction. We obtained the analogue of the result of Mumford and Neeman on Weierstrass measure for stable curves which are either irreducible or of compact type. This is achieved by relating the problem to and studying the asymptotic behavior of the Bergman kernel on a Riemann surface Mt as t approaches the boundary of the moduli space of curves in its Deligne-Mumford compactification.

March 10


Valentino Tosatti, Northwestern
Title: Smooth asymptotics for collapsing Ricci-flat metrics
Abstract: I will discuss the problem of understanding the collapsing behavior of Ricci-flat Kahler metrics on a Calabi-Yau manifold that admits a holomorphic fibration structure, when the Kahler class degenerates to the pullback of a Kahler class from the base. I will present new work with Hans-Joachim Hein where we obtain a priori estimates of all orders for the Ricci-flat metrics away from the singular fibers, as a corollary of a complete asymptotic expansion.

Spring 2019 Schedule

Fall 2019 Schedule

Questions: lni@math.ucsd.edu lspolaor@ucsd.edu