Let V be a suvariety of a manifold M. We say that V has extension property, if any bounded holomorphic function on V extends to a holomorphic function on M with the same sup-norm. In the talk we shall explain connections between this problem and operator theory (von Neumann inequality, interpolation problem) as well as with the theory of invariant functions and metrics

We review constructions of Manolescu’s Floer homotopy type, which gives a homotopical refinement of monopole Floer homology. Based on it, we will introduce some homology cobordism/ knot concordance invariant. Using these invariants, we provide relative versions of 10/8 inequalities for 4-manifolds with boundary or surfaces in 4-manifolds. In particular, I’ll explain Manolescu’s relative 10/8 inequality, real 10/8 inequality, and Montague’s 10/8 inequality.

Let X be a semistable p-adic formal scheme. In this talk, we will discuss a prismatic description of semistable local systems on the generic fiber of X. A main new ingredient is a purity result. This is based on a joint work with Heng Du, Tong Liu, Koji Shimizu.

We first review fundamental concepts about Seiberg-Witten theory for closed 4-manifolds. Subsequently, we will introduce a refinement of Seiberg-Witten invariant, called Bauer—Furuta invariant. Using Bauer—Furuta invariant, I will explain how to prove Furuta’s 10/8 inequality and its variant for group actions proven by Bryan and Kato.

The essential dimension of an algebraic object E over a field L is heuristically the number of parameters it takes to define it. This notion was formalized and developed by Buhler and Reichstein in the late 90s, who noticed at the time, that several classical results could be interpreted as theorems about essential dimension. Since the paper of Buhler and Reichstein, most of the progress on essential dimension has had to do with essential dimension of versal G-torsors for an algebraic group G. But recently Farb, Kisin and Wolfson showed that interesting theorems can be proved for certain (usually) non-versal torsors arising from congruence covers of Shimura varieties. I'll explain this work, some extensions of it proved by me and Fakhruddin, and a conjecture on period maps which generalizes the picture.

For hyperbolic systems of conservation laws in one space dimension endowed with a single convex entropy, it is an open question if it is possible to construct solutions via convex integration. Such solutions, if they exist, would be highly non-unique and exhibit little regularity. In particular, they would not have the strong traces necessary for the nonperturbative $L^2$ stability theory of Vasseur. Whether convex integration is possible is a question about large data, and the global geometric structure of genuine nonlinearity for the underlying PDE. In this talk, I will discuss recent work which shows the impossibility, for a large class of 2x2 systems, of doing convex integration via the use of $T_4$ configurations. Our work applies to every well-known 2x2 hyperbolic system of conservation laws which verifies the Liu entropy condition. This talk is based on joint work with László Székelyhidi.

This three-day lecture series aims to explore some topics in mathematical image processing before the era of neural networks, highlighting the techniques and applications that were prevalent at that time. From the classical filter-based models to PDE-based or minimization-based models, a variety of example-driven explanations and underlying mathematical theories are provided. By attending the lecture series, participants will gain a comprehensive understanding of image processing techniques used before the advent of neural networks, exploring the challenges, innovations and applications of classical algorithms. This knowledge will provide a foundation for further exploration in the field of image processing and its evolution into the AI-driven era.

We prove an arithmetic path integral formula for the inverse p-adic absolute values of the p-adic L-functions of elliptic curves over the rational numbers with good ordinary reduction at odd prime p. This is joint work with Jeehoon Park

This three-day lecture series aims to explore some topics in mathematical image processing before the era of neural networks, highlighting the techniques and applications that were prevalent at that time. From the classical filter-based models to PDE-based or minimization-based models, a variety of example-driven explanations and underlying mathematical theories are provided. By attending the lecture series, participants will gain a comprehensive understanding of image processing techniques used before the advent of neural networks, exploring the challenges, innovations and applications of classical algorithms. This knowledge will provide a foundation for further exploration in the field of image processing and its evolution into the AI-driven era.

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3-day lecture series (2 of 3)

This three-day lecture series aims to explore some topics in mathematical image processing before the era of neural networks, highlighting the techniques and applications that were prevalent at that time. From the classical filter-based models to PDE-based or minimization-based models, a variety of example-driven explanations and underlying mathematical theories are provided. By attending the lecture series, participants will gain a comprehensive understanding of image processing techniques used before the advent of neural networks, exploring the challenges, innovations and applications of classical algorithms. This knowledge will provide a foundation for further exploration in the field of image processing and its evolution into the AI-driven era.

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3-day lecture series (1 of 3)

I will discuss the ‘global’ nonlinear asymptotic stability of the traveling front solutions to the Korteweg-de Vries–Burgers equation, and other dispersive-dissipative perturbations of the Burgers equation. Earlier works made strong use of the monotonicity of the profile, for relatively weak dispersion effects. We exploit the modulation of the translation parameter, establishing a new stability criterion that does not require monotonicity. Instead, a certain Schrodinger operator in one dimension must have exactly one negative eigenvalue, so that a rank-one perturbation of the operator can be made positive definite. Counting the number of bound states of the Schrodinger equation, we find a sufficient condition in terms of the ’width’ of a front. We analytically verify that our stability criterion is met for an open set in the parameter regime including all monotone fronts. Our numerical experiments, revealing more stable fronts, suggest a computer-assisted proof. Joint with Blake Barker, Jared Bronski, and Zhao Yang.