Abstracts

Week of November 30, 2025

December 1, 2025
Geometric Analysis and Application Diameter bound and Geometric estimates along mean curvature flow 3:45pm -
KT 906

Abstract: In this talk, we will study the mean curvature flow in R^3.  We show that the intrinsic diameter of the surface along mean curvature flow is uniformly bounded as one approaches the first singular time, which confirms a conjecture of Haslhofer. In addition, we establish several sharp quantitative estimates of mean curvature flow. This is a joint work with Yiqi Huang (MIT).
Group Actions and Dynamics Representations of quadratic forms 4:00pm -
KT203

Let $q,Q$ be two integral quadratic forms in $m < n$ variables. One can ask when $q$ can be represented by $Q$ - that is, whether there exists an $n \times m$-integer matrix $T$ such that $Q \circ T = q$. Naturally, a necessary condition is that such a representation exists locally, meaning over the real numbers and modulo $N$ for every positive integer $N$. In the absence of local obstructions, does a (global) representation of $q$ by $Q$ exist?

In this talk, we will discuss known results in this direction focusing on recent work with Manfred Einsiedler, Elon Lindenstrauss, and Amir Mohammadi in which such a local-global principle when n-m >2. Our proof establishes much more general effective equidistribution results for periodic orbits of semisimple groups on compact adelic quotients. We will discuss the connection between these topics as well as some new ideas going into the proof of our dynamical results.
Geometry, Symmetry and Physics Bounded t-structures on semiorthogonal components 4:30pm -
KT 801

I will discuss a new method for constructing t-structures on semiorthogonal components of triangulated categories, which leads in particular to the first examples of bounded t-structures on phantom categories. This is joint work with Alexander Kuznetsov and Shengxuan Liu.
December 2, 2025
Geometry & Topology TBA 4:30pm -
KT 207

TBA
Geometry, Symmetry and Physics Pseudo-holomorphic curves with a fixed complex structure in positive symplectic manifolds 4:30pm -
KT 801

In this talk, I will show that fixed-domain Gromov–Witten invariants of a positive symplectic manifold (e.g., a smooth Fano variety) are signed counts of J-holomorphic curves in X satisfying prescribed incidence conditions. This provides a symplectic analogue of a conjecture of Lian and Pandharipande, recently disproved in the algebraic setting by Beheshti, Lehmann, Lian, Riedl, Starr, and Tanimoto. The proof relies on constructing the fixed-domain Gromov–Witten pseudocycle without the use of inhomogeneous or domain-dependent perturbations, answering an old question posed by Ruan and Tian.
December 4, 2025
Analysis Mixing Estimates for Sobolev and BV Vector Fields 4:00pm -
Zoom

I will discuss recent work on quantitative estimates for the passive scalar transport equation when the advecting velocity field is below the Lipschitz regularity class. In this setting, the classical Cauchy-Lipschitz theory fails to give control on the rate at which the solution may grow, usually measured by Sobolev norms. Despite this, well-posedness of the transport equation has been known when the vector field lies in W^{1,1} since 1989, and for vector fields in BV since 2004, under the additional assumption that the divergence remains bounded. It was conjectured by Bressan in 2003 that the same quantitative “mixing estimates” given by the Cauchy-Lipschitz theory should hold also in this setting. Since then, much progress has been made in recovering these estimates when the vector field lies in W^{1,p} with p>1, however the endpoint case p=1, and in particular BV, has remained a challenging open problem. In this talk I will discuss the first quantitative mixing and stability estimates for a passive scalar transported  by W^{1,1} and BV autonomous vector fields with zero divergence. The proof involves quantitative weak harmonic estimates in BV, and a double pigeonhole principle to quantify the weak-* compactness of the solution space.
Quantum Topology and Field Theory TBA 4:30pm -
KT 801

TBA
December 5, 2025
Friday Morning Seminar Friday Morning Seminar 10:00am -
KT 801

We have impromptu and (sometimes) scheduled talks, on topics in probability, combinatorics, geometry, and dynamics.

Everyone is welcome!