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Printer-friendly versionThursday, November 20, 2025
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| 4:00pm |
11/20/2025 - 4:00pm 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.
Location:
KT201
11/20/2025 - 4:30pm 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. Location:
KT 801
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