Printer-friendly versionAbstracts
Week of November 16, 2025
| Group Actions and Dynamics | Asymptotically large free semigroups in Zariski dense discrete subgroups of Lie groups |
3:50pm -
KT 203
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Abstract: An important quantity in the study of discrete groups of isometries of Riemannian manifolds, Gromov hyperbolic spaces, and other interesting geometric objects is the critical exponent. For a discrete subgroup of isometries of the quaternionic hyperbolic space or octonionic projective plane, Kevin Corlette established in 1990 that the critical exponent detects whether a discrete subgroup is a lattice or has infinite covolume. Precisely, either the critical exponent equals the volume entropy, in which case the discrete subgroup is a lattice, or the critical exponent is less than the volume entropy by some definite amount, in which case the discrete subgroup has infinite covolume. In 2003, Leuzinger extended this gap theorem for the critical exponent to any discrete subgroup of a Lie group having Kazhdan’s property (T) (for instance, a discrete subgroup of SL(n,R), where n is at least 3). In this talk, I will present a result which shows that no such gap phenomenon holds for discrete semigroups of Lie groups. More precisely, for any Zariski dense discrete subgroup of a Lie group, there exist free, finitely generated, Zariski dense subsemigroups whose critical exponents are arbitrarily close to that of the ambient discrete subgroup. As an application, we show that the critical exponent is lower semicontinuous in the Chabauty topology whenever the Chabauty limit of a sequence of Zariski dense discrete subgroups is itself a Zariski dense discrete subgroup. |
| Geometry, Symmetry and Physics | The application of the factorization method in the study of the Gaiotto conjecture | 4:30pm - |
The Gaiotto conjecture says that the category of representations of the quantum supergroup is equivalent to the category of certain sheaves on the affine Grassmannian. It is the quantum version of a particular case of the relative Langlands conjecture by Ben-Zvi-Sakellaridis-Venkatesh. In the first half of this talk, we will review the background and explain the statement of the Gaiotto conjecture. In the second half, we will sketch the proof with a focus on the factorization method. If time permits, we will also explain the application of the method in related questions. The talk is based on work and ongoing work joint with Michael Finkelberg and Roman Travkin. Seminar talk is supported in part by the Mrs. Hepsa Ely Silliman Memorial Fund. |
| Geometry & Topology | Stretch Factors on Infinite-Type Surfaces |
4:30pm -
KT 207
|
Given a finite-type surface S, it is still unknown exactly which numbers occur as stretch factors of pseudo-Anosov maps on S. It appears there is more flexibility in producing a homeomorphism with a given stretch factor if we allow for the surface to be infinite-type. I’ll discuss current work with Marissa Loving and Chenxi Wu showing every weak Perron number is the stretch factor of some end-periodic homeomorphism on some infinite-type surface. |
| Colloquium | New Approach to Matrix Perturbation: Beyond the Worst-Case Analysis |
4:00pm -
KT 101
|
Matrix-perturbation bounds quantify how the spectral characteristics of a baseline matrix A change under additive noise E. Classical results, including Weyl’s inequality for eigenvalues and the Davis–Kahan theorem for eigenvectors and eigenspaces, have long played a foundational role in mathematics. These bounds are known to be sharp in worst-case analysis. In the 10 years, we have been working to develop a perturbation framework that leverages the interaction between E and the eigenvectors of A. This perspective yields quantitative improvements over classical bounds, particularly when E is random, a common scenario in applications. This talk surveys these developments and main ideas, focusing on recent results concerning eigenspace perturbation. If time allows, we will discuss extensions to other spectral functionals and applications in different areas. |
| Analysis | On the optimal Sobolev threshold for evolution equations with rough nonlinearities |
4:00pm -
KT201
|
Consider a general evolution equation of the form
$\partial_tu-A(D)u=F(u,\overline{u},\nabla u, \nabla \overline{u})$
where $A(D)$ is a Fourier multiplier of either dispersive or parabolic type, and the nonlinear term $F$ has limited regularity (e.g. it is Hölder continuous up to a certain order). In this talk, I will describe a robust set of techniques which can be used in many cases to predict the highest possible Sobolev exponent $s=s(q,d)$ for which the above evolution can be well-posed in $W_x^{s,q}(\mathbb{R}^d)$. I will discuss how these principles can be rigorously implemented in the model cases of the nonlinear Schr"odinger and nonlinear heat equations. More precisely, we are able to show that the nonlinear heat equation
$\partial_tu-\Delta u=|u|^{p-1}u$
is well-posed in $W_x^{s,q}(\mathbb{R}^d)$ when $\max\{0,s_c\}<s<2+p+\frac{1}{q}$ and is strongly ill-posed when $s\geq 2+p+\frac{1}{q}$ and $p-1\not\in 2\mathbb{N}$ in the sense of non-existence of solutions even for smooth, small and compactly supported data. When $q=2$, we establish the same ill-posedness result for the nonlinear Schrödinger equation and the corresponding well-posedness result when $p\geq \frac{3}{2}$. Identifying the optimal Sobolev threshold for even a single non-algebraic $p>1$ has been a longstanding folklore open problem in the literature. As an amusing corollary of the fact that our ill-posedness threshold is dimension independent, we may conclude by taking $d$ sufficiently large relative to $p$ that there are nonlinear Schrödinger equations which are ill-posed in every Sobolev space $H_x^s(\mathbb{R}^d)$. This is based on a joint work with Mitchell Taylor.
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| Quantum Topology and Field Theory | Towards a Dolbeault AGT correspondence |
4:30pm -
KT 801
|
Abstract: The AGT correspondence and its extensions posit geometric constructions of vertex algebras and their modules from cohomology of variants of moduli of sheaves on surfaces. Physically, the correspondence has found an explanation through the holomorphic-topological twist of the six dimensional N=(2,0) superconformal field theories. In this talk, I’ll propose a variant of the AGT correspondence coming from the so-called minimal twist of these theories. Instead of vertex algebras, the natural algebras appearing will be holomorphic factorization algebras in three complex dimensions. From this data, I will explain how one extracts an associative algebra and a module which conjecturally agrees with a quantization of moduli of Higgs sheaves on surfaces. In examples, the pair conjecturally admits a Hodge-deRham deformation to the Heisenberg algebra and its action on cohomology of Hilbert schemes of surfaces, constructed in work of Grojnowski-Nakajima. |
| 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! |
| Geometry, Symmetry and Physics | The application of the factorization method in the study of the Gaiotto conjecture, part 2 | 3:00pm - |
This is the continuation of the Nov 17 talk. |