RSS FeedUpcoming Events3-Manifold Seminar: Skein lasagna modules of link homology theories, March 19https://events.berkeley.edu/math/event/243568-3-manifold-seminar-skein-lasagna-modules-of-linkOne can promote every TQFT-like link homology theory in $S^3$ (e.g. Khovanov homology, Heegaard/instanton knot Floer homologies restricted to $S^3$) to an algebraic invariant of a pair $(X,L)$ of compact smooth 4-manifold $X$ with a link $L$ on its boundary, called the skein lasagna module of $(X,L)$ (with respect to the input TQFT). We survey a few properties of this construction, for example how does the invariant change under connected sums, handle attachments, or more general gluings. As an example, we consider the “trivial” TQFT input for framed links, namely the Khovanov-Rozansky $\mathfrak _1$ homology, and show the skein lasagna module is roughly the second relative homology group of $(X,L)$. We also define a refinement of the skein lasagna module, which for the “trivial” TQFT is roughly the framed cobordism set of surfaces in $X\backslash B^4$ rel $L$. https://berkeley.zoom.us/j/93554741436https://events.berkeley.edu/math/event/243568-3-manifold-seminar-skein-lasagna-modules-of-linkProbabilistic Operator Algebra Seminar: Zero Bias in the Free World, March 19https://events.berkeley.edu/math/event/242953-probabilistic-operator-algebra-seminar-zero-biasIn sampling theory, bias plays an important role. Transforms to compensate for bias have wide use, with applications to normal approximation, waiting-time paradoxes, tightness, Skorokhod embedding, concentration of measure, and many other far-flung ideas. In the 1990s, Goldstein and Reinert introduced zero bias , an “infinitesimal” bias transform which connects to Stein’s method. It is an elegant tool for sharp approximation methods, thanks to a (non-linear) relation to independent sums. Recently, it has been shown to have an interesting connection to infinite divisibility. In this talk, I will discuss my current work (joint with Goldstein) to develop the free probability version of the zero bias transform. Using tools from complex analysis and subordination theory, we show interesting analogs of all the classical zero bias properties, and use the free zero bias to give a new and enlightening perspective on free infinite divisibility. Along the way, we discover a collection of surprisingly new results on combining Cauchy transforms that are of independent interest.https://events.berkeley.edu/math/event/242953-probabilistic-operator-algebra-seminar-zero-biasRepresentation theory and tensor categories seminar: R matrices and shuffle algebras, March 19https://events.berkeley.edu/math/event/243464-representation-theory-and-tensor-categoriesFor any quiver Q there is an associated shuffle algebra structure on the ring of symmetric functions, which can be realised as the cohomological Hall algebra of the category of representations of the quiver. Mimicking Green’s coproduct for finitary Hall algebras, one obtains also a kind of localised coproduct, along with an easy construction of an R-matrix for the resulting (localised) Hopf algebra. This provides a very flexible framework for producing new quantum groups, using the formalism of Faddeev, Reshetikhin and Takhtadzhyan, after a choice of favoured modules for the shuffle algebra. I will explain what is known: that using a specific variant of this construction we obtain the new Yangians defined by Maulik and Okounkov (via recent joint work with Botta). I will also present some conjectures regarding the algebra we obtain this way for certain other choices of cohomological Hall algebras and modules over them.https://events.berkeley.edu/math/event/243464-representation-theory-and-tensor-categoriesHarmonic Analysis and Differential Equations Seminar: Modified scattering for the three dimensional Maxwell-Dirac system, March 19https://events.berkeley.edu/math/event/243623-harmonic-analysis-and-differential-equationsAbstract: In this work we prove global well-posedness for the massive Maxwell-Dirac equation in the Lorentz gauge in $\mathbbR ^$, for small and localized initial data, as well as modified scattering for the solutions. In doing so, we heuristically exploit the close connection between massive Maxwell-Dirac and the wave-Klein-Gordon equations, while developing a novel approach which applies directly at the level of the Dirac equations. This is joint work with Sebastian Herr and Martin Spitz.https://events.berkeley.edu/math/event/243623-harmonic-analysis-and-differential-equationsCommutative Algebra and Algebraic Geometry Seminar: Welschinger Signs and the Wronski Map (New conjectured reality), March 19https://events.berkeley.edu/math/event/243546-commutative-algebra-and-algebraic-geometry-seminarA general real rational plane curve \( C \) of degree \( d \) has \( 3(d-2) \) flexes and \( (d-1)(d-2)/2 \) complex double points. Those double points lying in \( \mathbb ^2 \) are either nodes or solitary points. The Welschinger sign of \( C \) is \( (-1)^s \), where \( s \) is the number of solitary points. When all flexes of \( C \) are real, its parameterization comes from a point on the Grassmannian under the Wronskii map, and every parameterized curve with those flexes is real (this is the Mukhin-Tarasov-Varchenko Theorem). Thus to \( C \) we may associate the local degree of the Wronskii map, which is also \( 1 \) or \( -1 \). My talk will discuss work with Brazelton and McKean towards a possible conjecture that that these two signs associated to \( C \) agree, and the challenges to gathering evidence for this.https://events.berkeley.edu/math/event/243546-commutative-algebra-and-algebraic-geometry-seminarPauline Sperry Lecture: Model theory and cardinal invariants of the continuum, March 19https://events.berkeley.edu/math/event/243622-pauline-sperry-lecture-model-theory-and-cardinalCardinal invariants of the continuum give an interesting way of studying what was, early last century, the conjecturally nonempty region between $\aleph _1$ (the first uncountable cardinal) and the continuum. Even after Cohen’s invention of forcing, they continue to open up many subtle questions about infinity. In this lecture, we will start from the beginning and explore some of this world, including some surprising interactions with model theory.https://events.berkeley.edu/math/event/243622-pauline-sperry-lecture-model-theory-and-cardinalCommutative Algebra and Algebraic Geometry Seminar: Relations between Poincare series for quasi-complete intersection homomorphisms, March 19https://events.berkeley.edu/math/event/243545-commutative-algebra-and-algebraic-geometry-seminarQuasi-complete intersection (q.c.i.) homomorphisms are surjective homomorphisms of local rings for which the Koszul homology on a minimal generating set of the kernel is an exterior algebra. We study base change results for Poincare series along a q.c.i. homomorphism in situations that extend results known for complete intersection (c.i.) homomorphisms. The main new result is joint work with Josh Pollitz, and generalizes a well-known result of Shamash for c.i. homomorphisms which makes use of systems of higher homotopies. Our proof develops base change results involving Poincare series over the Koszul complex.https://events.berkeley.edu/math/event/243545-commutative-algebra-and-algebraic-geometry-seminarApplied Mathematics Seminar, Spring 2024: Fine-grained Theory and Hybrid Algorithms for Randomized Numerical Linear Algebra, March 20https://events.berkeley.edu/math/event/243477-applied-mathematics-seminar-spring-2024Randomized algorithms have gained increased prominence within numerical linear algebra and they play a key role in an ever-expanding range of problems driven by a breadth of scientific applications. In this talk we will explore two aspects of randomized algorithms by (1) providing experiments and accompanying theoretical analysis that demonstrate how their performance depends on matrix structures beyond singular values (such as coherence of singular subspaces), and (2) showing how to leverage those insights to build hybrid algorithms that blend favorable aspects of deterministic and randomized methods. A broad range of randomized algorithms will be considered, relevant motivating applications will be discussed, and numerical experiments will illuminate directions for further research.https://events.berkeley.edu/math/event/243477-applied-mathematics-seminar-spring-2024Combinatorics Seminar: Type C combinatorics of two-pole centralizer algebras, March 20https://events.berkeley.edu/math/event/243574-combinatorics-seminar-type-c-combinatorics-ofHecke algebras arise in one sense as deformations of Weyl groups, but in another sense as quotients of braid algebras. While some types of braid algebras can only loosely be associated to their namesake, some can be concretely imagined as braid diagrams on k strands, possibly in spaces with one or more punctures. The latter produces a connection to quantum group actions on tensor spaces of particular shapes. For example, the finite Hecke algebra of type A has a natural action on a tensor space $V^$, which commutes with the natural action of the quantum group of type A (this is precisely the q-deformation of classical Schur-Weyl duality). Closures of braids then correspond to traces of endomorphisms, giving rise to knot and link invariants via Hecke algebras; actions on tensor space have implications for lattice models in statistical mechanics; and on the connections go.
It is from this perspective of Schur-Weyl duality that our story takes place: classifying representations of the type-C affine Hecke algebra was once relegated to geometric methods. However, what one learns from studying two-pole braids, and the corresponding tensor spaces, unlocks a beautiful combinatorial classification of representation theory of the type-C affine Hecke and Temperley-Lieb algebras.
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Probability seminar: An elementary approach to non-asymptotic random matrix theory for unitarily invariant matrices, March 20https://events.berkeley.edu/live/events/240846-probability-seminar-jorge-garza-vargas

Classical random matrix theory focuses on the study of highly structured models (e.g. Wigner and Wishart matrices) which are presented as a sequence of random matrices defined for every dimension, whose asymptotic (i.e. as the dimension goes to infinity) spectral properties must be understood in detail. However, modern problems in data and computer science require only a coarser understanding of the random matrices in question, but necessitate nonasymptotic results in settings where the models are less structured and do not necessarily belong to a prescribed sequence of matrices.

In this work we provide new tools for analyzing the spectral distribution of self-adjoint noncommutative polynomials evaluated in arbitrary independent unitarily invariant Hermitian random matrices of a fixed dimension. With these tools we are able to recover some of Parraud’s results which quantify the distance of the spectral distribution of random matrices with the aforementioned structure from the spectral distribution of their free (in the sense of free probability) limit.


This is joint work with Chi-Fang Chen and Joel Tropp.

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Topics in Dynamical Systems: Lyapunov Exponents for Random Dynamical Systems, March 21https://events.berkeley.edu/math/event/243565-topics-in-dynamical-systems-lyapunov-exponents-forMany dynamical systems encountered in celestial and statistical mechanics are expected to be chaotic. Mathematically speaking, a dynamical system is chaotic if it has a positive Lyapunov exponent. It is often a daunting task to show that a dynamical system has a positive Lyapunov exponent. As it turns out, the chaos is far more tractable when the system is subjected to random noise. As an example, we consider stochastic differential equations, and discuss formulas of Furstenberg-Khasminskii, and Bedrossian-Blumental-Punshon Smith for their top Lyapunov exponent.https://events.berkeley.edu/math/event/243565-topics-in-dynamical-systems-lyapunov-exponents-forMathematics Department Colloquium: Monotiling, March 21https://events.berkeley.edu/math/event/239065-mathematics-department-colloquium-tbdThe discoveries of the Hat and Spectre — single shapes that can be used to form tilings of the plane, but only can form non-periodic ones — lay to rest the longstanding question of the existence of an “aperiodic monotile” but it remains an open question: How complex can the behavior of a single shape of tile be? Can we even tell whether or not a given shape will tile the plane — is the “monotiling problem” even decidable? We’ll survey the status of several related decision and existence problems, across a range of settings, such as hyperbolic space, subshifts on groups, or tilings by a single monotile.https://events.berkeley.edu/math/event/239065-mathematics-department-colloquium-tbdNonlinear Algebra Seminar: The Two Lives of the Grassmannian, March 25https://events.berkeley.edu/math/event/243505-nonlinear-algebra-seminar-the-two-lives-of-theThe Grassmannian parametrizes linear subspaces of a real vector space. It is both a projective variety (via Plücker coordinates) and an affine variety (via orthogonal projections). We examine these two representations, through the lenses of linear algebra, commutative algebra, and statistics.https://events.berkeley.edu/math/event/243505-nonlinear-algebra-seminar-the-two-lives-of-theSpectral Theory Seminar: Classical/Quantum correspondence in Lindblad evolution, March 27https://events.berkeley.edu/math/event/243544-spectral-theory-seminar-classicalquantumAbstract. I will introduce and motivate the Lindblad master equation which models interaction of a quantum system with a larger “open” system. I will then show that (under some assumptions) the evolution of a quantum observable remains close to the classical Fokker–Planck evolution (in the Hilbert– Schmidt norm) for times vastly exceeding the Ehrenfest time (the limit of such agreement without coupling to the open system). This is based on joint work with Jeff Galkowski.https://events.berkeley.edu/math/event/243544-spectral-theory-seminar-classicalquantumMathematics 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-indigenousProbability seminar: Hao Wu, April 10https://events.berkeley.edu/live/events/240849-probability-seminar-hao-wu

TBA

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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 24https://events.berkeley.edu/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