In this talk I will lay the foundations of the geometry of p-harmonic forms and L^p-Hodge theory. As an application, I will give strong evidence for (half of) a conjecture of Gromov on the L^p cohomology of negatively curved symmetric spaces.

The infinite Atlas model describes a countable system of competing Brownian particles where the lowest particle gets a unit upward drift and the rest evolve as standard Brownian motions. The stochastic process of gaps between the particles in the infinite Atlas model has a one parameter family {p(a), a > 0} of product form mutually singular stationary distributions. We say that an initial distribution of gaps is in the weak domain of attraction of the stationary measure p(a) if the time averaged laws of the stochastic process of the gaps, when initialized using that distribution, converge to p(a) weakly in the large time limit. We provide general sufficient conditions on the initial gap distribution of the Atlas particles for it to lie in the weak domain of attraction of p(a) for each a. Results on extremality and ergodicity of p(a) will be presented. Finally, I will describe some recent results on fluctuations of the Atlas model from inhomogeneous stationary profiles. This is based on joint work with Sayan Banerjee and Peter Rudzis.

G protein-coupled receptors (GPCRs) are membrane receptors that play a pivotal role in the regulation of reproduction and behavior in humans. Upon binding to specific ligands, they trigger a local cAMP production. Activated receptor are then internalized to different endosomal compartments where they can continue signaling before being recycled or destroyed. Recent studies showed that the different pools of cAMP have different effect on the cell.
In the first part of the talk, I will present a piecewise deterministic Markov process (PDMP) of intracellular signaling. The stochastic part of the model accounts for formation, coagulation, fragmentation and recycling of intracellular vesicles which contain the receptor, whereas the deterministic part of the model represents evolution of chemical reactions due to signaling activity of the receptor. We are interested in the existence of and convergence to a stationary measure. I will present different cases for which we were able to obtain results in this direction.
In the second part of the talk, I will present the numerical workflow (SBML, PEtab and PyPESTO) we use to fit ODEs model of GPCR signaling to longitudinal measure of chemical concentrations (BRET data).

In Newtonian gravity, a self-gravitating gas around a massive object such as a star or a planet is modeled via Vlasov Poisson equation with an external Kepler potential. The presence of this attractive potential allows for bounded trajectories along which the gas neither falls in towards the object or escape to infinity. We focus on this regime and prove first a linear phase mixing result in 3D outside symmetry with exact Kepler potential. Then we also prove a long-time nonlinear phase mixing result in spherical symmetry. The mechanism is phenomenologically similar to Landau damping on a torus but mathematically the situation is quite a lot more complex. This is based on an upcoming joint work with Jonathan Luk at Stanford.

We introduce and analyze a size-structured oocyte population model, with non local nonlinearities on recruitment, growth and mortality rates to take into account interactions between cells. We pay special attention to the form of the recruitment term, and its influence on the asymptotic behavior of the cell population.
This model is well-suited for representing oocyte population dynamics within the fish ovary. The nonlocal nonlinearities enable us to capture the diverse feedback mechanisms acting on the growth of oocytes of varying sizes and on the recruitment of new oocytes.
We firstly investigate the existence and uniqueness of global bounded solutions by transforming the partial differential equation into an equivalent system of integral equations, which can be solved using the Contraction Mapping Principle.
In a second step, we investigate the asymptotic behavior of the model. Under an additional assumption regarding the form of the growth rate, we can, with the use of a classical time-scaling transformation, reduce the study to that of a equation with linear growth speed and nonlinear inflow boundary condition. Using arguments from the theory of abstract semilinear Cauchy problems, we investigate the local stability of stationary solutions of this equation by reducing it to a characteristic equation involving the eigenvalues of the linearized problem around equilibrium states.
When the mortality rate is zero, the study of existence and stability of stationary solutions is simplified. Explicit calculations can be carried out in certain interesting cases.

Stochastic modeling and analysis can help answer pressing medical questions. In this talk, I will attempt to justify this claim by describing recent work on two problems in medicine. The first problem concerns ovarian tissue cryopreservation, which is a proven tool to preserve ovarian follicles prior to gonadotoxic treatments. Can this procedure be applied to healthy women to delay or eliminate menopause? How can it be optimized? The second problem concerns medication nonadherence. What should you do if you miss a dose of medication? How can physicians design dosing regimens that are robust to missed/late doses? I will describe (a) how stochastics theory offers insights into these questions and (b) the mathematical questions that emerge from this investigation. The first problem is based on joint work with Joshua Johnson (University of Colorado School of Medicine), John Emerson (Yale University), and Kutluk Oktay (Yale School of Medicine).

The classical branching rules describe the spectrum of an irreducible complex representation of a compact Lie group to its subgroup. The local Gan–Gross–Prasad conjecture generalizes the branching problem to classical groups over local fields of characteristic zero. After the pioneering work of Waldspurger, there has been significant progress on the conjecture using various approaches. In my talk, I will introduce a relatively uniform approach to prove the conjecture, including joint work with Z. Luo and joint work with R. Chen and J. Zou.

Hilbert modular surfaces are 2-dimensional analogues of modular curves, parametrizing polarized abelian surfaces with endomorphism and level structure. Modular curves are stratified by genus, and canonical equations for modular curves are obtained from the graded ring of modular forms. Similar to how curves are stratified by genus, surfaces are organized by their numerical invariants; the Enriques-Kodaira classification organizes smooth surfaces by Kodaira dimension, Hodge numbers, and Chern numbers. In this talk, we explain how to compute these invariants and equations for certain Hilbert modular surfaces. This is joint work with Eran Assaf, Angie Babei, Ben Breen, Sara Chari, Edgar Costa, Juanita Duque-Rosero, Alex Horawa, Jean Kieffer, Avi Kulkarni, Grant Molnar, Abhijit S. Mudigonda, Michael Musty, Sam Schiavone, Shikhin Sethi, and Samuel Tripp.

The random distribution Holomorphic multiplicative chaos (HMC) with Gaussian inputs is recently introduced independently by Najnudel, Paquette, and Simm as a limiting object on the unit complex circle of characteristic polynomial of circular beta ensembles, and by Soundararajan and Zaman as an analogue of random multiplicative functions. In this talk, we will explore this rich connection between HMC and random matrix theory, number theory, and Gaussian multiplicative chaos. We will also discuss the regularity of this distribution, alongside the fractional moments and tightness of its Fourier coefficients (also referred to as secular coefficients). Furthermore, we introduce non-Gaussian HMC, and discuss the Gaussian universality and two phase transitions phenomenon in the fractional moments of its secular coefficients. A transition from global to local effect is observed, alongside an analysis of the critical local-global case. As a result, we unveil the regularity of some non-Gaussian HMC and tightness of their secular coefficients. Based on joint work with Zhenyuan Zhang.

Physics-Informed Neural Networks (PINNs) have received much attention recently due to their potential for high-performance computations for complex physical systems, including data-based computing, systems with unknown parameters, and others. The idea of PINNs is to approximate the equations and boundary and initial conditions through a loss function for a neural network. PINNs combine the efficiency of data-based prediction with the accuracy and insights provided by the physical models. However, applications of these methods to predict the long-term evolution of systems with little friction, such as many systems encountered in space exploration, oceanography/climate, and many other fields, need extra care as the errors tend to accumulate, and the results may quickly become unreliable. We provide a solution to the problem of data-based computation of Hamiltonian systems utilizing symmetry methods. Many Hamiltonian systems with symmetry can be written as a Lie-Poisson system, where the underlying symmetry defines the Poisson bracket. For data-based computing of such systems, we design the Lie-Poisson neural networks (LPNets). We consider the Poisson bracket structure primary and require it to be satisfied exactly, whereas the Hamiltonian, only known from physics, can be satisfied approximately. By design, the method preserves all special integrals of the bracket (Casimirs) to machine precision. LPNets yield an efficient and promising computational method for many particular cases, such as rigid body or satellite motion (the case of SO(3) group), Kirchhoff's equations for an underwater vehicle (SE(3) group), and others. Joint work with Chris Eldred (Sandia National Lab), Francois Gay-Balmaz (CNRS and ENS, France), and Sophia Huraka (U Alberta). The work was partially supported by an NSERC Discovery grant.

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.