Seminars from until

To develop a model for non-equilibrium statistical mechanics, the system is typically brought into contact with two thermodynamic reservoirs, known as boundary reservoirs. These reservoirs impose their own particle density at the system's boundary, thereby inducing a current. Over time, a non-equilibrium steady state emerges, characterized by a stationary current value.

Recently, there has been increasing interest in multi-component systems, where various particle species (sometimes referred to as colors) coexist. In such setups, interactions between diferent species are possible alongside the occupation of available sites.

This work focuses on the boundary-driven multi-species stirring process on a one-dimensional lattice. This process extends naturally from the symmetric exclusion process (SEP) when multiple particle species are considered. Its dynamics involve particles exchanging positions with holes or with particles of diferent colors, each occurring at a rate of 1. Additionally, the system interacts with boundary reservoirs that inject, remove, and exchange types of particles.

After defining the process's generator using an appropriate representation of the gl(N) Lie algebra, we establish the existence of an absorbing dual process defined on an extended chain, where the two boundary reservoir are replaced by absorbing extra sites. This dual process shares bulk dynamics with the original but includes extra sites that absorb particles over extended time periods.

This multi-species stirring process can be mapped onto a higher rank open XXX-Heisenberg spin chain, therefore we employ absorbing duality and the matrix product ansatz to derive closed-form expressions for the non-equilibrium steady-state multi-point correlations of the process. This result is reported in [1].

Next, scaling limits of the process are examined, particularly the behavior of the properly scaled empirical density of the process. First, hydrodynamic equations are derived, illustrating typical system behavior (in the spirit of the law of large numbers). Second, fluctuations from this hydrodynamic limit are investigated, revealing a set of Gaussian processes coupled through noise, resembling aspects of the central limit theorem. Finally, large deviation results are reported, describing the probability of rare trajectories deviating from typical behaviors. An additional outcome of this analysis is the identification of a system of hydrodynamic equations featuring a drift due to interaction with an external field. These scaling limit results are reported in [2] and [3].

References

1. F. Casini, R. Frassek, C. Giardinà, Duality for the multispecies stirring process with open boundaries. (2024) J. Phys. A: Math. Theor. 57 295001
2. F. Casini, C. Giardinà, F. Redig, Density Fluctuations for the Multi-Species Stirring Process. (2024) J Theor Probab 37, 33173354.
3. F. Casini, F. Redig, H. van Wiechen, A large deviation principle for the multispecies stirring process. (2024), Arxiv: 2410.20857.

I will describe some requirements on a non semi-simple ribbon category that ensure its admissible skein modules form the state spaces of a 3+1-dimensional TQFT.

We fix an arbitrary symplectic toric manifold M. Its real toric lagrangians are the lagrangian submanifolds of M whose intersection with each torus orbit is clean and an orbit of the subgroup of elements that square to the identity of the torus (basically that subgroup is $\{ 1 , -1\}^n$). In particular, real toric lagrangians are transverse to the principal torus orbits and retain as much symmetry as possible.

This talk will explain why any two real toric lagrangians in M are related by an equivariant symplectomorphism and, therefore, any real toric lagrangian must be the real locus for a real structure preserving the moment map. This is joint work with Yael Karshon.

In this work, we deal with the symmetric exclusion process with k slow bonds equally spaced in the torus with kn sites, where the strength of a slow bond is $\alpha n^{-\beta}$, where $\beta>1$. For k fixed, it was known (T. Franco, P. Gonçalves, A. Neumann, AIHP'13) that the hydrodynamic limit in the diffusive scaling of this process is given by the heat equation with Neumann boundary conditions, meaning that the system does not allow flux through a slow bond in the limit. In this joint work with Tiecheng Xu and Dirk Erhard, we obtain another three superdiffusive scalings for this system. If $k$ is fixed and the (time) scaling is $n^{\theta}$, where $2< \theta<1+\beta$, the system reaches equilibrium instantaneously at each box between two consecutive slow bonds, being constant in time and space. If k is fixed and $\theta=1+\beta$, the density is spatially constant inside each box, and evolves as the discrete heat equation. And if $\theta=1+\beta$ and $k$ goes to infinity, we recover the continuous heat equation (on the torus).

I will offer an introductory exploration into the field of Reinforcement Learning (RL) with a focus on Markov Decision Processes (MDPs). The first session provides a foundational understanding of RL, covering key concepts such as agents, environments, rewards, and actions. It explains the RL problem framework and introduces MDPs, exploring their role as the mathematical framework underpinning RL.

The second session delves into core algorithms, including Q-learning and policy gradients. The lecture highlights the connection between MDPs and dynamic programming techniques, emphasizing policy iteration and value iteration. Time allowing, I will finalize with a brief description of some recent research topics and results.

A good introduction to RL is the 2018 book on the subject by Sutton and Barto. We will talk about topics in Chapters 1,2-6 and 13.

A more rigorous introduction to MDPs, including convergence results, can be found in the book by Puterman:

Martin L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming, Wiley, 2005.

I will offer an introductory exploration into the field of Reinforcement Learning (RL) with a focus on Markov Decision Processes (MDPs). The first session provides a foundational understanding of RL, covering key concepts such as agents, environments, rewards, and actions. It explains the RL problem framework and introduces MDPs, exploring their role as the mathematical framework underpinning RL.

The second session delves into core algorithms, including Q-learning and policy gradients. The lecture highlights the connection between MDPs and dynamic programming techniques, emphasizing policy iteration and value iteration. Time allowing, I will finalize with a brief description of some recent research topics and results.

Cells make fate decisions in response to dynamic environments, and multicellular structures emerge from multiscale interplays among cells and genes in space and time. The recent single-cell genomics technology provides an unprecedented opportunity to profile cells for all their genes. While those measurements provide high-dimensional gene expression profiles for all cells, it requires fixing individual cells that lose many important spatiotemporal information. Is it possible to infer temporal relationships among cells from single or multiple snapshots? How to recover spatial interactions among cells, for example, cell-cell communication? In this talk I will present our newly developed computational tools to study cell fate in the context of single cells as a system. In particular, I will show dynamical models and machine-learning methods, with a focus on inference and analysis of transitional properties of cells and cell-cell communication using both high-dimensional single-cell and spatial transcriptomics, as well as multi-omics data for some cases. Through their applications to various complex systems in development, regeneration, and diseases, we show the discovery power of such methods in addition to identifying areas for further method development for spatiotemporal analysis of single-cell data.

We begin with a brief overview of the rapidly developing research area of active matter (a.k.a. active materials). These materials are intrinsically out of equilibrium resulting in novel physical properties whose modeling requires the development of new mathematical tools. We focus on studying the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction. We introduce a minimal two-dimensional free-boundary PDE model that captures the evolution of the cell shape and nonlinear diffusion of myosin.

We first consider a linear diffusion model with two sources of nonlinearity: Keller-Segel cross-diffusion term and the free boundary that models moving/deformable cell membranes. Here we establish asymptotic linear stability and derive the explicit formula for the stability-determining eigenvalue.

Next, we consider the effect of nonlinear myosin diffusion, which results in the change of the bifurcation type from super- to subcritical, and we obtain an asymptotic representation of the bifurcation curve (for small velocities). This allows us to derive an explicit formula for the curvature at the bifurcation point that controls the bifurcation type. In the most recent work in progress with the Heidelberg biophysics group, we study the relation between various types of nonlinear diffusion and bistability.

Finally, we discuss novel mathematical features of this free boundary model with a focus on non-self-adjointness, which plays a key role in the spectral stability analysis. Our mathematics reveals the physical origins of the non-self-adjoint of the operators in this free boundary model.

Joint works with A. Safsten & V. Rybalko (Transactions of AMS 2023, and Phys. Rev. E 2022), with O. Krupchytskyi &T. Laux (Preprint 2024), and with A. Safsten & L. Truskinovsky ( Arxiv preprint 2024). This work has been supported by NSF grants DMS-2404546, DMS-2005262, and DMS-2404546.

Malliavin Calculus, an infinite dimensional calculus on probability spaces, was born with the paper

P. Malliavin - Stochastic calculus of variations and hypoelliptic operators, Proc. Inter. Symp. Stoch. Diff. Eqs. Kyoto 1976, Wiley (1978), 195–263.

and was initially aimed to give a probabilistic counterpart of Hörmander theorem for hypoelliptic operators. Soon it found many developments and other applications within Mathematics (notably in Mathematical Physics), also in Finance.

I will give an introduction to Malliavin Calculus techniques, with a brief reference to some applications.

Malliavin Calculus, an infinite dimensional calculus on probability spaces, was born with the paper

P. Malliavin - Stochastic calculus of variations and hypoelliptic operators, Proc. Inter. Symp. Stoch. Diff. Eqs. Kyoto 1976, Wiley (1978), 195–263.

and was initially aimed to give a probabilistic counterpart of Hörmander theorem for hypoelliptic operators. Soon it found many developments and other applications within Mathematics (notably in Mathematical Physics), also in Finance.

I will give an introduction to Malliavin Calculus techniques, with a brief reference to some applications.

In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.

In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.

In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.