## Curtis Porter : CRash CouRse in CR Geometry

- Graduate/Faculty Seminar,Uploaded Videos ( 2252 Views )CR geometry studies real hypersurfaces in complex vector spaces and their generalizations, CR manifolds. In many cases of interest to complex analysis and PDE, CR manifolds can be considered ``curved versions" of homogeneous spaces according to Elie Cartan’s generalization of Klein’s Erlangen program. Which homogeneous space is the ``flat model" of a CR manifold depends on the Levi form, a tensor named after a mathematician who used it to characterize boundaries of pseudoconvex domains. As in the analytic setting, the Levi form plays a central role in the geometry of CR manifolds, which we explore in relation to their homogeneous models.

## Thomas Weighill : Optimal transport methods for visualizing redistricting plans

- Applied Math and Analysis ( 0 Views )Ensembles of redistricting plans can be challenging to analyze and visualize because every plan is an unordered set of shapes, and therefore non-Euclidean in at least two ways. I will describe two methods designed to address this challenge: barycenters for partitioned datasets, and a novel dimension reduction technique based on Gromov-Wasserstein distance. I will cover some of the theory behind these methods and show how they can help us untangle redistricting ensembles to find underlying trends. This is joint work with Ranthony A. Clark and Tom Needham.

## Luya Wang : Deformation inequivalent symplectic structures and Donaldsons four-six question

- Geometry and Topology ( 0 Views )Studying symplectic structures up to deformation equivalences is a fundamental question in symplectic geometry. Donaldson asked: given two homeomorphic closed symplectic four-manifolds, are they diffeomorphic if and only if their stabilized symplectic six-manifolds, obtained by taking products with CP^1 with the standard symplectic form, are deformation equivalent? I will discuss joint work with Amanda Hirschi on showing how deformation inequivalent symplectic forms remain deformation inequivalent when stabilized, under certain algebraic conditions. This gives the first counterexamples to one direction of Donaldson??s ??four-six? question and the related Stabilizing Conjecture by Ruan. In the other direction, I will also discuss more supporting evidence via Gromov-Witten invariants.

## Ran Tao : Fluctuations of half-space KPZ: from 1/2 to 1/3

- Probability ( 0 Views )We study the half-space KPZ equation with a Neumann boundary condition, starting from stationary Brownian initial data. We derive a variance identity that links the fluctuations of the height function to the transversal fluctuations of a half-space polymer model. We then establish optimal fluctuation exponents for the height function in both the subcritical and critical regimes, along with corresponding estimates for the polymer endpoint. Based on a joint work with Yu Gu.

## Peter Dillery : Non-basic rigid packets for discrete L-parameters

- Number Theory ( 0 Views )We formulate a new version of the local Langlands correspondence for discrete L-parameters which involves (Weyl orbits of) packets of representations of all twisted Levi subgroups of a connected reductive group G through which the parameter factors and prove that this version of the correspondence follows if one assumes the pre-existing local Langlands conjectures. Twisted Levi subgroups are crucial objects in the study of supercuspidal representations; this work is a step towards deepening the relationship between the representation theory of p-adic groups and the Langlands correspondence. This is joint work with David Schwein (Bonn).

## Zhenyi Chen : A-infinity Sabloff Duality via the LSFT Algebra

- Geometry and Topology ( 0 Views )The Chekanov-Eliashberg dga is a powerful invariant for Legendrian links. Using augmentations of this dga, one can truncate its differential to produce linearized contact homology. About two decades ago, Sabloff established a duality in this setting, closely linked to the Poincaré duality of Lagrangian fillings. This truncation has since been generalized into a unital A-infinity category, Aug_+. In this talk, I will present new results that extend Sabloff duality from the level of cochain complexes to A-infinity bimodules over Aug_+. The key tool in this extension is Ng's LSFT algebra, which enlarges the Chekanov-Eliashberg dga. If time permits, I will also discuss how the LSFT algebra encodes additional homotopy coherent data, providing further insights into Sabloff duality.

## Xiaoqian Xu : Mixing flow and advection-diffusion-reaction equations

- Applied Math and Analysis ( 0 Views )In the study of incompressible fluid, one fundamental phenomenon that arises in a wide variety of applications is dissipation enhancement by so-called mixing flow. In this talk, I will give a brief introduction to the idea of mixing flow and the role it plays in the field of advection-diffusion-reaction equation. I will also discuss about the examples of such flows in this talk.

## Calvin McPhail-Snyder : Towards quantum complex Chern-Simons theory

- Geometry and Topology ( 0 Views )I will discuss recent joint work (with N. Reshetikhin) defining invariants ? of knot (and link and tangle) exteriors with flat ???? connections. The construction is via a geometric version of the Reshetikhin-Turaev construction: it is algebraic and relies on the representation theory of quantum groups. In this talk I will instead focus on the properties of these invariants and explain why I think they are a good candidate for quantum Chern-Simons theory with noncompact gauge group SL??(??). I will also discuss a connection with (and a generalization of) the Volume Conjecture.

## Sergey Cherkis : Gravitational Instantons: the Tesseron Landscape

- Geometry and Topology ( 0 Views )Since their introduction in Euclidean quantum gravity in mid-70??s, hyperkaehler Gravitational Instantons (aka tesserons) found their use in string theory and in supersymmetric quantum field theory. Their classification was recently completed and now their parameter space is being explored. We propose a systematic program of realizing each of these spaces as a moduli space of monopoles: the monopolization program. Monopolization reveals the combinatorial and geometric structure of the parameter space of all these spaces, equips each space with various natural structures (tautological bundles, Dirac-type operators, etc), and connects different types of integrable systems associated to these gravitational instantons.

## Theo McKenzie : Eigenvalue rigidity for random regular graphs

- Probability ( 0 Views )Random regular graphs form a ubiquitous model for chaotic systems. However, the spectral properties of their adjacency matrices have proven difficult to analyze because of the strong dependence between different entries. In this talk, I will describe recent work that shows that despite this, the fluctuation of eigenvalues of the adjacency matrix are of the same order as for Gaussian matrices. This gives an optimal error term for Friedman's theorem that the second eigenvalue of the adjacency matrix of a random regular graph converges to the spectral radius of an infinite regular tree. Crucial is tight analysis of the Green??s function of the adjacency operator and an analysis of the change of the Green's function after a random edge switch. This is based on joint work with Jiaoyang Huang and Horng-Tzer Yau.

## Xiang Cheng : Transformers learn in-context by (functional) gradient descent

- Applied Math and Analysis ( 0 Views )Motivated by the in-context learning phenomenon, we investigate how the Transformer neural network can implement learning algorithms in its forward pass. We show that a linear Transformer naturally learns to implement gradient descent, which enables it to learn linear functions in-context. More generally, we show that a non-linear Transformer can implement functional gradient descent with respect to some RKHS metric, which allows it to learn a broad class of functions in-context. Additionally, we show that the RKHS metric is determined by the choice of attention activation, and that the optimal choice of attention activation depends in a natural way on the class of functions that need to be learned. I will end by discussing some implications of our results for the choice and design of Transformer architectures.

## Laura Wakelin : Finding characterising slopes for all knots

- Geometry and Topology ( 0 Views )A slope p/q is characterising for a knot K if the oriented homeomorphism type of the 3-manifold obtained by performing Dehn surgery of slope p/q on K uniquely determines the knot K. For any knot K, there exists a bound C(K) such that any slope p/q with |q|?C(K) is characterising for K. This bound has previously been constructed for certain classes of knots, including torus knots, hyperbolic knots and composite knots. In this talk, I will give an overview of joint work with Patricia Sorya in which we complete this realisation problem for all remaining knots.

## Benjamin Seeger : Equations on the Wasserstein space and applications

- Probability ( 0 Views )The purpose of this talk is to give an overview of recent work involving differential equations posed on spaces of probability measures and their use in analyzing controlled multi-agent systems. The study of such systems has seen increased interest in recent years, due to their ubiquity in applications coming from macroeconomics, social behavior, and telecommunications. When the number of agents becomes large, the model can be formally replaced by one involving a mean-field description of the population, analogously to similar models in statistical physics. Justifying this continuum limit is often nontrivial and is sensitive to the type of stochastic noise influencing the population, i.e. idiosyncratic or systemic. We will describe settings for which the convergence to mean field stochastic control problems can be resolved through the analysis of a certain Hamilton-Jacobi-Bellman equation posed on Wasserstein spaces. In particular, we develop new stability and regularity results for the equations. These allow for new convergence results for more general problems, for example, zero-sum stochastic differential games of mean-field type. We conclude with a discussion of some further problems for which the techniques for equations on Wasserstein space may be amenable.

## Hongkai Zhao : Mathematical and numerical understanding of neural networks: from representation to learning dynamics

- Applied Math and Analysis ( 0 Views )In this talk I will present both mathematical and numerical analysis as well as experiments to study a few basic computational issues in using neural network to approximate functions: (1) the numerical error that can be achieved given a finite machine precision, (2) the learning dynamics and computation cost to achieve certain accuracy, and (3) structured and balanced approximation. These issues are investigated for both approximation and optimization in asymptotic and non-asymptotic regimes.

## Tatiana Brailovskaya : Matrix superconcentration inequalities

- Probability ( 0 Views )One way to understand the concentration of the norm of a random matrix X with Gaussian entries is to apply a standard concentration inequality, such as the one for Lipschitz functions of i.i.d. standard Gaussian variables, which yields subgaussian tail bounds on the norm of X. However, as was shown by Tracy and Widom in 1990s, when the entries of X are i.i.d. the norm of X exhibits even sharper concentration. The phenomenon of a function of many i.i.d. variables having strictly smaller tails than those predicted by classical concentration inequalities is sometimes referred to as «superconcentration», a term originally dubbed by Chatterjee. I will discuss novel results that can be interpreted as superconcentration inequalities for the norm of X, where X is a Gaussian random matrix with independent entries and an arbitrary variance profile. We can also view our results as a nonhomogeneous extension of Tracy-Widom-type upper tail estimates for the norm of X.

## Noga Alon : Gergen Lecture Seminar 3 Distance problems for Euclidean and other norms Lecture B: Coloring and ordering

- Gergen Lectures ( 0 Views )Distance problems in discrete geometry include fascinating examples of questions that are easy to state and hard to solve. Three of the best known problems of this type, raised in the 40s, are the Erd\H{o}s Unit Distance Problem, his Distinct Distances Problem, and the Hadwiger-Nelson Problem about the chromatic number of the unit distance graph in the plane. I will describe surprisingly tight recent solutions of the analogs of all three problems for typical norms, settling, in a strong form, questions and conjectures of Matou\v{s}ek, of Brass, of Brass, Moser and Pach, and of Chilakamarri. I will also discuss a related work about ordering points according to the sum of their distances from chosen vantage points. The proofs combine Combinatorial, Geometric and Probabilistic methods with tools from Linear Algebra, Topology, and Algebraic Geometry. Based on recent joint works with Matija Buci\'c and Lisa Sauermann, and with Colin Defant, Noah Kravitz and Daniel Zhu.

## Tye Lidman : Cosmetic surgeries and Chern-Simons invariants

- Geometry and Topology ( 0 Views )Dehn surgery is a fundamental construction in topology where one removes a neighborhood of a knot from the three-sphere and reglues to obtain a new three-manifold. The Cosmetic Surgery Conjecture predicts two different surgeries on the same knot always gives different three-manifolds. We discuss how gauge theory, in particular, the Chern-Simons functional, can help approach this problem. This technique allows us to solve the conjecture in essentially all but one case. This is joint work with Ali Daemi and Mike Miller Eismeier.

## Theodore Drivas : The Feynman-Lagerstrom criterion for boundary layers

- Class Department of Mathematic ( 99 Views )We study the boundary layer theory for slightly viscous stationary flows forced by an imposed slip velocity at the boundary. According to the theory of Prandtl (1904) and Batchelor (1956), any Euler solution arising in this limit and consisting of a single "eddy" must have constant vorticity. Feynman and Lagerstrom (1956) gave a procedure to select the value of this vorticity by demanding a necessary condition for the existence of a periodic Prandtl boundary layer description. We will show that this condition is also sufficient for the existence of a periodic boundary layer described by the Prandtl equations. This is joint work with S. Iyer and T. Nguyen.

## Samuel Isaacson : Spatial Jump Process Models for Estimating Antibody-Antigen Interactions

- Mathematical Biology ( 103 Views )Surface Plasmon Resonance (SPR) assays are a standard approach for quantifying kinetic parameters in antibody-antigen binding reactions. Classical SPR approaches ignore the bivalent structure of antibodies, and use simplified ODE models to estimate effective reaction rates for such interactions. In this work we develop a new SPR protocol, coupling a model that explicitly accounts for the bivalent nature of such interactions and the limited spatial distance over which such interactions can occur, to a SPR assay that provides more features in the generated data. Our approach allows the estimation of bivalent binding kinetics and the spatial extent over which antibodies and antigens can interact, while also providing substantially more robust fits to experimental data compared to classical bivalent ODE models. I will present our new modeling and parameter estimation approach, and demonstrate how it is being used to study interactions between antibodies and spike protein. I will also explain how we make the overall parameter estimation problem computationally feasible via the construction of a surrogate approximation to the (computationally-expensive) particle model. The latter enables fitting of model parameters via standard optimization approaches.

## Robin Zhang : Harris-Venkatesh plus Stark

- Number Theory ( 64 Views )The class number formula describes the behavior of the Dedekind zeta function at s = 0. The Stark conjecture extends the class number formula, describing the behavior of Artin L-functions at s = 0 in terms of units. The Harris??Venkatesh conjecture, originally motivated by the conjectures of Venkatesh and Prasanna??Venkatesh on derived Hecke algebras, can be viewed as an analogue to the Stark conjecture modulo p. In this talk, I will draw an introductory picture, formulate a unified conjecture combining Harris??Venkatesh and Stark for modular forms of weight 1, and describe the proof of this in the imaginary dihedral case. Time permitting, I will also describe some new questions and in-progress work modulo pn.

## Ayman Said : Small scale creation of the Lagrangian flow in 2d perfect fluids

- Class Department of Mathematic ( 95 Views )In this talk I will present a recent result showing that for all solutions of the 2d Euler equations with initial vorticity with finite Sobolev smoothness then an initial data dependent norm of the associated Lagrangian flow blows up in infinite time like $t^{\frac{1}{3}}$. This initial data dependent norm quantifies the exact $L^2$ decay of the Fourier transform of the solution. This adapted norm turns out to be the exact quantity that controls a low to high frequency cascade which I will then show to be the quantitative phenomenon behind a microlocal generalized Lyapunov function constructed by Shnirelman.

## Eylem Zeliha Yildiz : Braids in planar open books and fillable surgeries.

- Geometry and Topology ( 129 Views )We'll give a useful description of braids in $\underset{n}{\#}(S^1\times S^2)$ using surgery diagrams, which will allow us to address families of knots in lens spaces that admit fillable positive contact surgery. We also demonstrate that smooth $16$ surgery to the knot $P(-2,3,7)$ bounds a rational ball, which admits a Stein handlebody. This answers a question left open by Thomas Mark and Bülent Tosun.