RSS FeedUpcoming EventsInformal String-Math Seminar: Algebra of the Infrared with Twisted Masses, April 1https://events.berkeley.edu/math/event/244004-informal-string-math-seminar-algebra-of-theThe “Algebra of the Infrared” refers to a collection of homotopical algebra structures (discovered by Gaiotto-Moore-Witten) that one associates to a massive two-dimensional N=(2,2) quantum field theory (subject to certain constraints). This provides a powerful framework for working out the category of boundary conditions of such QFTs. Specializing to the example of massive Landau-Ginzburg models, one is lead to a novel “web-like” construction of the Fukaya-Seidel category. In this talk, after reviewing these developments, I will discuss work-in-progress with Greg Moore which seeks to generalize the web framework to N=(2,2) theories with non-trivial twisted masses. In the Landau-Ginzburg context this amounts to studying LG models defined by a holomorphic one-form dW which is closed but not necessarily exact. Among the results we will announce are a Koszul duality theorem for boundary algebras in such theories, and a categorification of the wall-crossing formula for the \(CP^1\) model with twisted masses.https://events.berkeley.edu/math/event/244004-informal-string-math-seminar-algebra-of-theArithmetic Geometry and Number Theory RTG Seminar: Regular de Rham Galois representations in the completed cohomology of modular curves, April 1https://events.berkeley.edu/math/event/244032-arithmetic-geometry-and-number-theory-rtg-seminarLet $p$ be a prime. I want to explain how to use the geometry of modular curves at infinite level and the Hodge–Tate period map to study de Rham $p$-adic Galois representations appearing in the $p$-adically completed cohomology of modular curves. We will show that these Galois representations up to twists come from modular forms and, if time permits, we give a geometric description of the locally analytic representations of $\operatorname _2(\mathbf Q_p)$ associated to them. These results were previously known by totally different methods.https://events.berkeley.edu/math/event/244032-arithmetic-geometry-and-number-theory-rtg-seminar3-Manifold Seminar: Lee skein lasagna modules and lasagna s-invariants; gluing and extending down, April 2https://events.berkeley.edu/math/event/244089-3-manifold-seminar-lee-skein-lasagna-modules-andWe begin by recalling the definition of (ordinary/refined) skein lasagna modules and finishing up the previous discussion on $gl_1$ homology. Then we turn to the case when the input TQFT is the Lee deformation of the $gl_2$ homology. We determine the structure of Lee skein lasagna modules except the quantum filtration, and extract numerical invariants from the quantum filtration function in the spirit of Rasmussen’s s-invariant. We show that these invariants give lower bounds to the smooth genus function of a 4-manifold. In the rest of the talk, we digress to discuss some gluing formulas, and remark that the skein lasagna module can be extended down to lower dimensions (in the sense of Lurie).https://events.berkeley.edu/math/event/244089-3-manifold-seminar-lee-skein-lasagna-modules-andRepresentation theory and tensor categories seminar: Are categories of modular representations of finite groups (locally) regular?, April 2https://events.berkeley.edu/math/event/244058-representation-theory-and-tensor-categoriesFor a symmetric tensor triangular category T we suggest a notion of “local regularity” which agrees with the classical concept for a derived category D(A) for a commutative ring A. We study this property for stable categories of a finite group scheme (in positive characteristic) and discover that it is related to many other, such as dualizability and various finiteness conditions. I’ll review some fundamental principles of tensor triangular geometry and the stratification of the stable category of a finite group G over a field of positive characteristic. Then we specialize to the fibers of the stable category at homogeneous prime ideals p in the cohomology ring and show that they are regular.
Joint work with D. Benson, S. Iyengar, H. Krause.
https://events.berkeley.edu/math/event/244058-representation-theory-and-tensor-categories
Topology Seminar: The combinatorial and gauge-theoretic foam evaluation functors are not the same, April 3https://events.berkeley.edu/math/event/244031-topology-seminar-the-combinatorial-andKronheimer and Mrowka used gauge theory to define a functor $J^\sharp $ from a category of webs in $R^3$ to the category of finite-dimensional vector spaces over the field of two elements. They also suggested a possible combinatorial replacement $J^\flat $ for $J^\sharp $, which Khovanov and Robert proved is well-defined on a subcategory of planar webs. We exhibit a counterexample that shows the restriction of the functor $J^\sharp $ to the subcategory of planar webs is not the same as $J^\flat $.https://events.berkeley.edu/math/event/244031-topology-seminar-the-combinatorial-andMathematics Department Colloquium: The Bonnet problem: Is a surface characterized by its metric and curvatures?, April 4https://events.berkeley.edu/math/event/239064-mathematics-department-colloquium-the-bonnetA longstanding problem in differential geometry, known as the Bonnet problem, is whether a compact surface is uniquelly determined by its metric and mean curvature function. It is known that this is the case for generic surfaces, and also for topological spheres. We explicitly construct a pair of immersed tori in three dimensional Euclidean space that are related by a mean curvature preserving isometry. These tori are the first examples of compact Bonnet pairs. Moreover, we prove these isometric tori are real analytic. This resolves a second longstanding open problem on whether real analyticity of the metric already determines a unique compact immersion. Discrete differential geometry is used to find crucial geometric properties of surfaces. This is a joint work with Tim Hoffmann and Andrew Sageman-Furnas.https://events.berkeley.edu/math/event/239064-mathematics-department-colloquium-the-bonnetRehumanizing Math: Black, Latinx, Indigenous: Transition to working group, April 5https://events.berkeley.edu/math/event/243552-rehumanizing-math-black-latinx-indigenousThis week we will have an organizational meeting to transition into a working group on best practices in mentoring, teaching, research, as well as community building activities. Details announced by faculty email.https://events.berkeley.edu/math/event/243552-rehumanizing-math-black-latinx-indigenousProbabilistic Operator Algebra Seminar: Free probability of type B prime, April 9https://events.berkeley.edu/math/event/243735-probabilistic-operator-algebra-seminar-freeFree probability of type B was invented by Biane–Goodman–Nica and then was generalized by Belinschi–Shlyakhtenko and Fevrier–Nica to infinitesimal free probability. The latter found its applications to eigenvalues of perturbed random matrices in the work of Shlyakhtenko and Cebron–Dahlquist–Gabriel. This talk offers a new framework, called “free probability of type B’ “, which appears in the large size limit of independent unitarily invariant random matrices with perturbations. Our framework is related to boolean, free, (anti)-monotone, cyclic-(anti)monotone and conditionally free independences. We then apply the new framework to the principal minor of unitarily invariant random matrices , which leads to the definition of a multivariate inverse Markov-Krein transform.https://events.berkeley.edu/math/event/243735-probabilistic-operator-algebra-seminar-freeProbability seminar: Hao Wu, April 10/live/events/240849-probability-seminar-hao-wu

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Mathematics Department Colloquium: Geometry emerging from spectra, April 11https://events.berkeley.edu/math/event/239063-mathematics-department-colloquium-tbdWe give a gentle introduction to the spectral approach to geometry, where we replace spaces by commutative algebras, and capture the metric by combining it with the vibration spectrum of a suitable operator on the geometric space. We will give many examples and also show how to reconstruct geometry from this spectral data. This will allow us to generalize to noncommutative spaces, which we illustrate by some of the key examples. We will conclude by establishing some convergence results on the emerging geometric spaces when an increasing part of the spectrum is available.https://events.berkeley.edu/math/event/239063-mathematics-department-colloquium-tbdMathematics Department Colloquium: TBD, April 18https://events.berkeley.edu/math/event/239665-mathematics-department-colloquium-tbdhttps://events.berkeley.edu/math/event/239665-mathematics-department-colloquium-tbdProbabilistic Operator Algebra Seminar: An Application of Free Probability in the Study of Noncommutative Constraint Satisfaction Problems, April 23https://events.berkeley.edu/math/event/243051-probabilistic-operator-algebra-seminar-anIn this talk I explore an application of free probability in our recent work on operator – or noncommutative – variants of constraint satisfaction problems (CSPs). These higher-dimensional variants are a core topic of investigation in quantum information, where they arise as nonlocal games and entangled multiprover interactive proof systems (MIP*) . The idea of higher-dimensional relaxations of CSPs is also important in the classical computer science literature. For example, since the celebrated work of Goemans and Williamson on the Max-Cut CSPs, higher dimensional vector relaxations have been central in the design of approximation algorithms for classical CSPs. We introduce a framework for designing approximation algorithms for noncommutative CSPs. In the study of classical CSPs, $k$-ary decision variables are often represented by $k$-th roots of unity, which generalize to the noncommutative setting as order-$k$ unitary operators. In our framework, using representation theory, we develop a way of constructing unitary solutions from SDP relaxations, extending the pioneering work of Tsirelson on XOR games. Then we introduce a novel rounding scheme to transform these unitary solutions to order-$k$ unitaries. Our main technical innovation here is a theorem guaranteeing that, for any set of unitary operators, there exists a set of order-$k$ unitaries that closely mimics it. As an integral part of the rounding scheme, we prove a random matrix theory result that characterizes the distribution of the relative angles between eigenvalues of random unitaries using tools from free probability. Based on joint work with Eric Culf and Taro Sprig arXiv:2312.16765.https://events.berkeley.edu/math/event/243051-probabilistic-operator-algebra-seminar-anProbability seminar: Wai Tong (Louis) Fan, April 24/live/events/240850-probability-seminar-wai-tong-louis-fan

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Mathematics Department Colloquium: TBD, May 2https://events.berkeley.edu/math/event/239143-mathematics-department-colloquium-tbdhttps://events.berkeley.edu/math/event/239143-mathematics-department-colloquium-tbd