The Mathematics Colloquium in the Department of Mathematics & Computer Science at Rutgers-Newark takes place on Wednesdays 4-5pm, either in person at 204 Smith Hall, 101 Warren St., or via Zoom. All are welcome!

For more information or to be added to our mailing list, please email Kyle Hayden (kyle.hayden@rutgers.edu).

Schedule — Fall 2025

September 17
Minghao Miao (Nanjing University / Rutgers New Brunswick)
The Volume of K-Semistable Fano Manifolds

In 2015, K. Fujita showed that for any n-dimensional K-semistable Fano manifold, the anti-canonical volume is always less than or equal to that of complex projective space (CP^n). In this talk, I will discuss my recent joint work with Chi Li on characterizing the second-largest volume. We prove that for any n-dimensional K-semistable Fano manifold X that is not isomorphic to CPⁿ, the volume is at most 2n^n, with the equality holds if and only if X is a smooth quadric hypersurface or CP^1 × CP^{n-1}. This result applies, in particular, to all Fano manifolds admitting Kähler–Einstein metrics. Our proof is based on a new connection between K-stability and minimal rational curves.

October 1
Alex He (Oklahoma State University)
Practical algorithms in 3-manifold topology

For any fixed compact 3-manifold M, there are infinitely many ways to triangulate M. So given two 3-manifold triangulations, how can we algorithmically decide whether or not they triangulate the same 3-manifold? This is the 3-manifold homeomorphism problem, which is (to say the least) very hard. Nevertheless, we might hope that one day, we can eventually develop a practical algorithm for the homeomorphism problem (that is, an algorithm that is both simple enough to implement in software, and efficient enough that we can actually run the software). This talk will survey some work that has been done to build towards this long-term goal. Specifically, I will discuss some practical algorithms that have been developed to solve some simpler, but still fundamental, problems in low-dimensional topology, such as the problem of computing the prime factorisation of a knot.

October 8
Grace Garden (IMJ-PRG)
Character varieties and essential surfaces

In the seminal work of Culler and Shalen (1983) a method is outlined to detect essential surfaces in a three-manifold by studying their SL(2,C)-character variety. The method underscores connections between the theory of incompressible surfaces in three-manifolds, splittings of fundamental groups, group actions on trees, and the geometry of representation varieties, and led to many developments in low-dimensional topology. In this talk, I will provide an overview of this theory and give intuition through examples. Where possible, I will also discuss recent work extending the theory to algebraically closed fields of arbitrary characteristic. This is joint work with Stephan Tillmann.

October 15
Yikai Teng (Rutgers Newark)
Khovanov homology and exotic planes

Since the 1980s, mathematicians have discovered uncountably many “exotic” embeddings of R² in R⁴, i.e., embeddings that are topologically but not smoothly isotopic to the standard xy-plane. However, until today, there have been no direct, computable invariants that could detect such exotic behavior (with prior results relying on indirect arguments). In this talk, we define the end Khovanov homology, which is the first known combinatorial invariant of properly embedded surfaces in R⁴ up to ambient diffeomorphism. Moreover, we apply this invariant to detect new exotic planes, including the first known example of an exotic plane that is a Lagrangian submanifold of the standard symplectic R⁴.

October 29
Ao Sun (Lehigh University)
Singular behavior of mean curvature flow

Mean curvature flow describes how a surface evolves to reduce its area as quickly as possible. A central challenge in understanding this flow is the formation of singularities. In this talk, I will discuss recent progress on the singular behavior of mean curvature flow, with a focus on joint work with Zhihan Wang (Cornell) and Jinxin Xue (Tsinghua) on cylindrical singularities.

October 31
Zhitong Su (Hunan Normal University)
A decomposition lemma in convex integration via classical algebraic geometry

We consider a problem of improving the regularity of flexible solutions of a nonlinear PDE, which can be viewed as a kind of linearization of the codimension one local isometric embedding equation in Nash-Kuiper Theorem.
Our approach is based on a decomposition lemma that separates part of the error term arising from convex integration into an elliptic system. The argument involves applications of Adams’ theorem on vector fields on spheres, and classical projective duality. Consequently, our improvement on the Hölder exponent of the solutions depends on the Radon-Hurwitz number, exhibiting an eightfold periodicity that reflects Bott periodicity. This is joint work with Weijun Zhang.

November 5
Huai-Dong Cao (Lehigh)
Hamilton-Ivey-type curvature pinching estimates of Ricci solitons

A magic feature of the Ricci flow in three dimensions is the well-known Hamilton–Ivey curvature pinching estimate. Roughly speaking, it asserts that when the curvature blows up along the 3D Ricci flow, the positive part blows up at a faster rate than (absolute value of) the negative part. As a consequence, all 3D shrinking or steady gradient Ricci solitons (or more generally, ancient solutions) arising as limits of parabolic blowups of the flow must have nonnegative sectional curvature. This is extremely powerful in the analysis of 3D Ricci flow singularity models, as it enables the effective use of the Li–Yau–Hamilton differential Harnack inequality and the geometry of non-negatively curved three-manifolds.

In recent years, various generalizations of Hamilton–Ivey curvature pinching have been developed for general shrinking and steady Ricci solitons, and more broadly, for ancient solutions, in both dimension three and higher dimensions. In this talk, I will discuss some new progress on Hamilton–Ivey-type curvature pinching for gradient Ricci solitons, based on my joint work with Junming Xie.

November 12
Charlie Reid (Yale)
Higher Teichmüller theory, and not-so-simple closed curves

A hyperbolic structure on a surface is captured by a representation of the fundamental group into PSL(2,R). Higher rank Teichmüller theory aims to go beyond hyperbolic geometry by studying representations into bigger lie groups, for instance PSL(n,R). I will discuss a “higher” version of one piece of hyperbolic geometry–Thurston’s compactification of Teichmüller space. Boundary points of this compactification are measured laminations, measure-theoretic objects generalizing simple closed curves. I will discuss compactifications of certain higher Teichmüller spaces where we will see closed curves with more intricate restrictions on self-intersection appearing in the boundary.

Bonus: November 14 (at New Brunswick)
Tamás Darvas (University of Maryland)
A YTD correspondence for constant scalar curvature metrics

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)

November 19 (part 1) & December 3 (part 2)
Erez Lapid (Weizmann Institute of Science)
Some new results and conjectures about representations of GL_n over a non-archimedean local field

The seminal work of Joseph Bernstein and Andrei Zelevinsky in the 1970s on representation theory of GL_n over a non-archimedean local field culminated in Zelevinsky’s classification of the irreducible ones in terms of mysterious objects and seemingly innocuous combinatorial objects. The latter have rich geometric structure and show up in other contexts of representation theory such as Lustzig’s canonical bases.

In my talks I will present recent progress on understanding the composition series, and in particular the reducibility, of parabolic induction of representations of general linear groups. The study leads to new geometric constructions in the context of Lusztig’s nilpotent varieties and new relations to classical combinatorial constructions such as the RSK correspondence. I will strive to keep the talks self-contained and logically independent of each other.

Based on joint works with Alberto Minguez, Rami Aizenbud and Max Gurevich.