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.

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.

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.

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.

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.

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.

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.

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.

The module theory of the Weyl algebra, known as the theory of $D$-modules, has profound applications in various fields. One of the most famous results is the Riemann-Hilbert correspondence, establishing equivalence between holonomic $D$-modules and perverse sheaves on smooth complex varieties. However, when dealing with singular varieties, such correspondence breaks down due to the non-simplicity of Weyl algebras on singular varieties. In our ongoing work, we introduce a new ring of differential operators on certain singular affine varieties, whose definition is analytically derived from harmonic analysis. It should contain the Weyl algebra as a proper subring and shares many properties with the Weyl algebra on smooth varieties. In the talk, after a brief review of the Weyl algebra, I will explain how the new ring of differential operators arises as a consequence of an explicit form of the Poisson summation conjecture and discuss its properties.

Intuitively, renormalization group flow is irreversible, since one would expect the number of massless degrees of freedom of a theory to decrease as one goes to progressively larger distance scales. This has been proven rigorously in two dimensions (the "c-theorem"), but is otherwise an open problem. I will discuss recent efforts to prove an analogous theorem in four dimensions for supersymmetric theories, for which we can make exact statements not usually accessible in their non-supersymmetric counterparts.

It is known that many random matrix models have a dual description in terms of a topological string theory. Moreover, recently, Ooguri, Strominger and Vafa have shown that one can calculate certain properties of N=2 supersymmetric black holes in four dimensions using a topological string theory. In this talk, based on hep-th/0410141, I connect these two developments, and discuss what matrix models can teach us about black holes.

The dynamics of D branes moving in certain string theory backgrounds can be used to learn about supersymmetric field theories in several dimensions. In this talk I will center on the realization of four-dimensional N=2 and N=1 gauge field theories, by using two types of backgrounds. The first (`brane configurations') involves D branes in the presence of NS fivebranes, whereas the second involves D branes sitting at singular points. A wide variety of conformal theories with marginal couplings, and of finite N=1 theories arises from these constructions. I will also emphasize the equivalence of different constructions by T duality.

I will discuss the statistical analysis of flux vacua with special properties such as (no-scale) supersymmetry and discrete symmetries. Unlike generic flux vacua, enumeration of such special vacua can require number-theoretic methods.

The low energy effective action of the type IIB superstring is given by IIB supergravity. First stringy corrections involve terms with six extra derivatives and can be obtained in an elegant way using a superfield formalism. This talk will be concerned with some recent progress in understanding the IIB superfield. In particular, some non-renormalisation properties of D3-brane supergravity solutions will be demonstrated. The last part of the talk will address higher derivative terms involving three-form fluxes.

We discuss aspects of the renormalization group flow of quiver gauge theories in two dimensions, with little or no supersymmetry, in the spirit of earlier work of Zamolodchikov on non-supersymmetric Landau-Ginzburg theory. By using a variety of techniques including analysis of chiral symmetries, identification of operators, and analysis of perturbative gauge theory diagrams, we show that these theories can (at least) be fine-tuned to flow to orbifold conformal field theory in the infrared. This provides a linear sigma model description of string backgrounds with no spacetime supersymmetry, whose applications and limitations we begin to explore.

The talk will begin with a review of some recent progress in non-supersymmetric string theory, motivating a careful study of such backgrounds. Many non-supersymmetric closed string theories have negative modes (``tachyons''). I will describe an approach to understanding the result of tachyon condensation using configurations in string theory of effectively negative but finite tension (such as orientifold planes and their S-duals).

To define a consistent perturbative geometric heterotic compactification one must specify not only a Calabi-Yau manifold M but also a bundle E on M. The bundle E is required to satisfy a subtle constraint known as ``stability,'' which depends upon the Kähler form. This dependence upon the Kähler form is highly nontrivial---the Kähler cone splits into subcones, with a distinct moduli space of bundles in each subcone---and has long been overlooked by physicists. In this talk we shall describe this behavior and its physical manifestation.

Supersymmetry can severely constrain higher derivative terms in the effective action of both Yang-Mills theories and string theories. I will explain how to obtain these constraints.