The problem of determining the size and structure of the interior singular set of area-minimizing surfaces has been studied thoroughly in a number of different frameworks, with many ground-breaking contributions. In the framework of integral currents, when the surface has higher codimension than 1, the presence of singular points with flat tangent cones creates an obstruction to easily understanding the interior singularities. Until recently, little was known in this direction, particularly for surfaces of dimension higher than two, beyond Almgren�??s celebrated dimension estimate on the interior singular set. In this talk I will discuss joint works with Camillo De Lellis and Paul Minter, where we establish (m-2)-rectifiability of the interior singular set of an m-dimensional area-minimizing integral current and classify tangent cones at \mathcal{H}^{m-2}-a.e. interior point.

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.

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.

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.

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.

For the past 25 years, a key player in contact topology has been the Floer-theoretic invariant called Legendrian contact homology. I'll discuss a package of new invariants for Legendrian knots and links that builds on Legendrian contact homology and is derived from rational symplectic field theory. This includes a Poisson bracket on Legendrian contact homology and a symplectic structure on augmentation varieties. Time permitting, I'll also describe an unexpected connection to cluster theory for a family of Legendrian links associated to positive braids. Parts of this are joint work in progress with Roger Casals, Honghao Gao, Linhui Shen, and Daping Weng.

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.

Consider the lattice Z^2, and assign length 1 or 2 to every edge by flipping a series of independent fair coins. This gives a random weighted graph, and looking at distances in this graph gives a random planar metric. This model is expected to have a continuum scaling limit as we decrease the spacing between lattice points. Moreover, most natural models of random planar metrics and random interface growth (the so-called `KPZ universality class') are expected to converge to the same limiting geometry. The goal of this talk is to introduce this limit, known as the directed landscape, and describe at least one model where we can actually prove convergence.

In this talk, I will present an overview of the framework originally proposed by Harder and further developed in collaboration with Raghuram to address rationality problems for special values of certain automorphic L-functions. I will then proceed to state my main results on the rationality of the special values of Langlands-Shahidi L-functions appearing in the constant term of the Eisenstein series associated with the exceptional group of type G_2�?? over a totally imaginary number field. This study marks the first instance where rank-one Eisenstein cohomology is employed to investigate the arithmetic of automorphic L-functions in the presence of multiple L-functions.

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.

The directed landscape is a random directed metric on the plane that is the scaling limit for models in the KPZ universality class. In this metric, typical pairs of points are connected by a unique geodesic. However, certain exceptional pairs are connected by more exotic geodesic networks. The goal of this talk is to describe a full classification for these exceptional pairs. I will also discuss some connections with other models of random geometry.

Despite the impressive performance of rituximab (R) containing regimens like R-CHOP in CD20+ Non-Hodgkin�??s Lymphoma (NHL), 30-60% of R-naïve NHL patients are estimated to be resistant, and approximately 60% of those patients will not respond to subsequent single agent R treatment. Given that antibody dependent cell mediated cytotoxicity (ADCC) and phagocytosis (ADCP) are thought to be the major mechanisms of action of Rituximab, increasing the activation levels of natural killer (NK) and macrophage (MP) cells may be one strategy for overcoming R resistance.

During (and after) the Fields Institute Industrial Problem Solving Workshop in August 2019, academic participants and industry mentors developed and calibrated to literature data a quantitative systems pharmacology (QSP) model of ADCC/ADCP to interrogate which mechanisms of R resistance could be overcome by increased NK or MP activation, and how much effector cell activation would be required to overcome a given degree and mechanism of R resistance.

This work was motivated by a real-world pharmaceutical drug development question, and the academic-industry interactions during and after the workshop resulted in sharknado plots as well as a published QSP model (presented at American Association of Cancer Research Annual Meeting, 2021) that was able to address some of the key questions around overcoming R resistance. The published model was then incorporated into an in-house QSP model supporting the development of a Takeda investigational drug which is being developed to restore R sensitivity in an R-resistant patient population.

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.

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.

In this talk I will describe my recent work with Mahesh Kakde on the Brumer-Stark Conjecture and certain refinements. I will give a broad overview that motivates the conjecture and gives connections to explicit class field theory. I will conclude with a description of recent work (joint w/ Kakde, Jesse Silliman, and Jiuya Wang) in which we complete the proof of the conjecture. Moreover, we deduce a certain special case of the Equivariant Tamagawa Number Conjecture, which has important corollaries. The key aspect of the most recent results, which allows us to handle the prime p=2, is the proof of a version of Ribet's Lemma in the case of characters that are congruent modulo p.

Directed polymers are a statistical mechanics model for random growth. Their partition functions are solutions to a discrete stochastic heat equation. This talk will discuss the logarithmic derivatives of the partition functions, which are solutions to a discrete stochastic Burgers equation. Of interest is the success or failure of the �??one force-one solution principle�?� for this equation. I will reframe this question in the language of polymers, and share some surprising results that follow. Based on joint work with Louis Fan and Timo Seppäläinen.

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.

In this talk, I will discuss the geometric expansion of a local twisted trace formula for some special varieties. This generalizes the local (twisted) trace formula for reductive groups proved by Arthur and Waldspurger. By applying the trace formula, we prove a multiplicity formula for these spherical varieties. And I will also discuss some applications to the multiplicity of the Galois model and the unitary Shalika model. This is a joint work with Raphael Beuzart-Plessis.

Recently, Markov-chain Monte Carlo methods based on non-reversible piecewise deterministic Markov processes (PDMP) are under growing attention, thanks to the increase in performance they usually bring. Beyond their numerical efficacy, the non-reversible and piecewise deterministic characteristics of these processes prompt interesting questions, regarding for instance ergodicity proof and convergence bounds. During this talk, I will particularly focus on the obtained results and open problems left while considering PDMP evolution of particle systems, both in an equilibrium and out-of-equilibrium setting. Hardcore particle systems have embodied a testbed of choice since the first implementations of Markov chain Monte Carlo in the 50�??s. Even today, the entropic barriers they exhibit are still resisting to the state-of-the-art MCMC sampling methods. During this talk, I will review the recent developments regarding sampling such systems and discuss the dynamical bottlenecks that are yet to be solved.

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.