Informal Geometric Analysis Seminar (2025.09-2026.06)

The seminar aims to be a relaxed forum for researchers and graduate students interested in geometric analysis, and students of all levels are encouraged to attend and ask questions. Talks will focus on the "big picture" and key ideas and encourage interaction. Talks will be in one of two formats: a) 30-minute talks followed by a ~ 15-30 minute discussion, or b) 50-75 minute talks.

2025 Fall

• October 21

• Speaker: Carlos Esparza (Berkeley)

• Title: Uniqueness of asymptotically conical shrinking gradient Kähler--Ricci solitons

• Abstract: We show that, up to biholomorphism, a given noncompact complex manifold admits at most one shrinking gradient Kähler-Ricci soliton with Ricci curvature vanishing at infinity. Time permitting, we will also discuss how the technique for proving the uniqueness of the soliton vector field can be applied to other settings, such as AC Calabi-Yau manifolds.

• November 4

• Speaker: Ao Sun (Lehigh)

• Title: Geometry of Mean Curvature Flow near Cylindrical Singularities

• Abstract: The cylindrical singularities are prevalent but complicated in geometric flows. We discuss one of the simplest extrinsic flow, the mean curvature flow, and illustrate how the local dynamics of the singularities influence the singular set itself, and the geometry and topology of the flow. This talk is based on joint works with Zhihan Wang (Cornell) and Jinxin Xue (Tsinghua).

• November 11

• Speaker: Yueqiao Wu (JHU)

• Title: K-semistability at infinity

• Abstract: The question of finding and classifying complete Calabi--Yau metrics on smooth affine varieties of Euclidean volume growth goes back to Tian--Yau, who constructed such metrics on X given by the complement of a Kähler--Einstein divisor in a Fano variety. Recent classification results suggest that such metrics on smooth affine varieties come from prescribing the asymptotic geometry using a negative valuation. In this talk, I will revisit Tian--Yau's example, in which case the Kähler--Einstein divisor defines a K-semistable valuation which does not admit a center on X. Generalizing this leads to a valuative criterion for K-semistable valuations at infinity on a given affine variety. Time permitting, I will also explain that these valuations in fact come from Fano type compactifications generalizing the Tian--Yau case. This is based on joint work in progress with Mattias Jonsson.

• November 18

• Speaker: Benjy Firester (MIT)

• Title: Free boundary Monge-Ampere equations and boundary regularity of optimal transport

• Abstract: In this talk, I will present a variational framework to solve a general class of free-boundary Monge–Ampère equations. This approach combines the classical first and second boundary value problems by imposing both the boundary data and the gradient image of the solution. I will explore applications to the Monge–Ampère eigenvalue problem and a reconstruction theorems, and geometric problems including a hemispherical Minkowski problem, Calabi-Yau metrics, and free boundary toric Kähler–Einstein/Kähler-Ricci soliton metrics. Furthermore, I will discuss the connection to the boundary regularity of optimal transport and recent progress inspired by geometric regularity theory techniques.

• December 2

• Speaker: Tamás Darvas (UMD)

• 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)

• December 9

• Speaker: Tang-Kai Lee (Columbia)

• Title: Uniqueness of mean curvature flow evolution

• Abstract: The smooth mean curvature flow often develops singularities, making weak solutions essential for extending the flow beyond singular times, as well as having applications for geometry and topology. Among various weak formulations, the level set flow method is notable for ensuring long-time existence and uniqueness. However, this comes at the cost of potential fattening, which reflects genuine non-uniqueness of the flow after singular times. Even for flows starting from smooth, embedded, closed initial data, such non-uniqueness can occur. Thus, we can't expect genuine uniqueness in general. Addressing this non-uniqueness issue is a difficult problem. With Alec Payne, we establish an intersection principle comparing two intersecting flows. We prove that level set flows satisfy this principle in the absence of non-uniqueness.

2026 Spring

• February 17, 2pm (special time), joint with Hopkins-Maryland geometry seminar

• Speaker: Mattias Jonsson (Michigan)

• Title: On the Yau-Tian-Donaldson conjecture for extremal metrics

• Abstract: Let X be a compact Kähler manifold. Calabi asked whether a given Kähler class on X contains a "canonical" Kähler metric, such as an extremal metric. Roughly speaking, the Yau-Tian-Donaldson conjecture states that if the Kähler class is the first Chern class of an ample line bundle, then the existence of an extremal metric should be governed by an algebro-geometric stability condition. I will present joint work with S. Boucksom, where we prove a version of this conjecture.

• February 19 12.30pm (special time)

• Speaker: Richard Bamler (Berkeley)

• Title: Ancient cylindrical mean curvature flows and the mean convex neighbordood conjecture

• Abstract: We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Specifically, we show that if the tangent flow at a singular point is a multiplicity-one cylinder, then in a neighborhood of that point the flow is mean-convex, its time-slices arise as level sets of a continuous function, and all nearby tangent flows are cylindrical. Moreover, we establish a canonical neighborhood theorem near such points, which characterizes the flow via local models. 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. Our approach relies on two new techniques. The first, called the PDE-ODI principle, converts a broad class of parabolic differential equations into systems of ordinary differential inequalities. This framework bypasses many delicate analytic estimates used in previous work, and yields asymptotic expansions to arbitrarily high order. The second combines a new leading mode condition combined with an "induction over thresholds" argument" to obtain even finer asymptotic estimates (This is joint work with Yi Lai).

• March 24

• Speaker: Charlie Cifarelli (Stony Brook)

• Title: TBA

• Abstract: TBA

• April 21

• Speaker: Mingyang Li (Stony Brook)

• Title: TBA

• Abstract: TBA