Tue, 14 May 2024
11:00
L5

A graph discretized approximation of diffusions with drift and killing on a complete Riemannian manifold

Hiroshi Kawabi
(Keio University)
Abstract

In this talk, we present a graph discretized approximation scheme for diffusions with drift and killing on a complete Riemannian manifold M. More precisely, for a given Schrödinger operator with drift on M having the form A = Δ b + V , we introduce a family of discrete time random walks in the  ow generated by the drift b with killing on a sequence of proximity graphs, which are constructed by partitions cutting M into small pieces. As a main result, we prove that the drifted Schrodinger semigroup {e—tA}t≥0 is approximated by discrete semigroups generated by the family of random walks with a suitable scale change. This result gives a  nite dimensional summation approximation of a Feynman-Kac type functional integral over M. Furthermore, when M is compact, we also obtain a quantitative error estimate of the convergence.
This talk is based on a joint work with Satoshi Ishiwata (Yamagata University), and the full paper can be found on https://doi.org/10.1007/s00208-024-02809-9.

Tue, 21 May 2024

14:00 - 15:00
L5

TBC

Elijah Bodish
(MIT)
Abstract

to follow

Thu, 14 Mar 2024
16:00
L5

Free Interface Problems and Stabilizing Effects of Transversal Magnetic Fields

Professor Zhouping Xin
(The Chinese University of Hong Kong)
Abstract

Dynamical interface motions are important flow patterns and fundamental free boundary problems in fluid mechanics, and have attracted huge attention in the mathematical community. Such waves for purely inviscid fluids are subject to various instabilities such as Kelvin-Helmholtz and Rayleigh-Taylor instabilities unless other stabilizing effects such as surface tension, Taylor-sign conditions or dissipations are imposed. However, in the presence of magnetic fields, it has been known that tangential magnetic fields may have stabilizing effects for free surface waves such as plasma-vacuum or plasma-plasma interfaces (at least locally in time), yet whether transversal magnetic fields (which occurs often for interfacial waves for astrophysical plasmas) can stabilize typical free interfacial waves remain to be some open problems. In this talk, I will show the stabilizing effects of the transversal magnetic fields for some interfacial waves for both compressible and incompressible multi-dimensional magnetohydrodynamics (MHD).

First, I will present the local (in time) well-posedness in Sobolev space of multi- dimensional compressible MHD contact discontinuities, which are the most typical interfacial waves for astrophysical plasma and prototypical fundamental waves for systems of hyperbolic conservations. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversally, which leads to a two-phase free boundary problem that may have nonlinear Rayleigh- Taylor instability and whose front symbols have no ellipticity. We overcome such difficulties by exploiting full the transversality of the magnetic fields and designing a nonlinear approximate problem, which yields the local well-posed without loss of derivatives and without any other conditions such as Rayleigh-Taylor sign conditions or surface tension. Second, I will discuss some results on the global well-posedness of free interface problems for the incompressible inviscid resistive MHD with transversal magnetic fields. Both plasma-vacuum and plasma-plasma interfaces are studied. The global in time well-posedness of both interface problems in a horizontally periodic slab impressed by a uniform non-horizontal magnetic field near an equilibrium are established, which reveals the strong stabilizing effect of the transversal field as the global well- posedness of the free boundary incompressible Euler equations (without the irrotational assumptions) around an equilibrium is unknown. This talk is based on joint work with Professor Yanjin Wang. 

Fri, 16 Feb 2024

15:00 - 16:00
L5

Morse Theory for Tubular Neighborhoods

Antoine Commaret
(INRIA Sophia-Antipolis)
Abstract
Given a set $X$ inside a Riemaniann manifold $M$ and a smooth function $f : X -> \mathbb{R}$, Morse Theory studies the evolution of the topology of the closed sublevel sets filtration $X_c = X \cap f^{-1}(-\infty, c]$ when $c \in \mathbb{R}$ varies using properties on $f$ and $X$ when the function is sufficiently generic. Such functions are called Morse Functions . In that case, the sets $X_c$ have the homotopy type of a CW-complex with cells added at every critical point. In particular, the persistent homology diagram associated to the sublevel sets filtration of a Morse Function is easily understood. 
 
In this talk, we will give a broad overview of the classical Morse Theory, i.e when $X$ is itself a manifold, before discussing how this regularity assumption can be relaxed. When $M$ is a Euclidean space, we will describe how to define a notion of Morse Functions, first on sets with positive reach (a result from Joseph Fu, 1988), and then for any tubular neighborhood of a set at a regular value of its distance function, i.e when $X = \{ x \in M, d_Y(x) \leq \varepsilon \}$ where $Y \subset M$ is a compact set and $\varepsilon > 0$ is a regular value of $d_Y$ the distance to $Y$ function.
 
 
If needed, here are three references :
 
Morse Theory , John Milnor, 1963
 
Curvature Measures and Generalized Morse Theory, Joseph Fu, 1988
Morse Theory for Tubular Neighborhoods, Antoine Commaret, 2024, Arxiv preprint https://arxiv.org/abs/2401.04034
Tue, 23 Jan 2024
11:00
L5

Wilson-Ito diffusions

Massimiliano Gubinelli
(Mathematical Institute)
Abstract

In a recent preprint, together with Bailleul and Chevyrev we introduced a class of random fields which try to model the basic properties of quantum fields. I will try to explain the basic ideas and some of the many open problems.

To read the preprint, please click here.

Fri, 02 Feb 2024

15:00 - 16:00
L5

Algebraic and Geometric Models for Space Communications

Prof. Justin Curry
(University at Albany)
Further Information

Justin Curry is a tenured Associate Professor in the Department of Mathematics and Statistics at the University at Albany SUNY.

His research is primarily in the development of theoretical foundations for Topological Data Analysis via sheaf theory and category theory.

Abstract

In this talk I will describe a new model for time-varying graphs (TVGs) based on persistent topology and cosheaves. In its simplest form, this model presents TVGs as matrices with entries in the semi-ring of subsets of time; applying the classic Kleene star construction yields novel summary statistics for space networks (such as STARLINK) called "lifetime curves." In its more complex form, this model leads to a natural featurization and discrimination of certain Earth-Moon-Mars communication scenarios using zig-zag persistent homology. Finally, and if time allows, I will describe recent work with David Spivak and NASA, which provides a complete description of delay tolerant networking (DTN) in terms of an enriched double category.

Fri, 26 Jan 2024

15:00 - 16:00
L5

Expanding statistics in phylogenetic tree space

Gillian Grindstaff
(Mathematical Institute)
Abstract
For a fixed set of n leaves, the moduli space of weighted phylogenetic trees is a fan in the n-pointed metric cone. As introduced in 2001 by Billera, Holmes, and Vogtmann, the BHV space of phylogenetic trees endows this moduli space with a piecewise Euclidean, CAT(0), geodesic metric. This has be used to define a growing number of statistics on point clouds of phylogenetic trees, including those obtained from different data sets, different gene sequence alignments, or different inference methods. However, the combinatorial complexity of BHV space, which can be most easily represented as a highly singular cube complex, impedes traditional optimization and Euclidean statistics: the number of cubes grows exponentially in the number of leaves. Accordingly, many important geometric objects in this space are also difficult to compute, as they are similarly large and combinatorially complex. In this talk, I’ll discuss specialized regions of tree space and their subspace embeddings, including affine hyperplanes, partial leaf sets, and balls of fixed radius in BHV tree space. Characterizing and computing these spaces can allow us to extend geometric statistics to areas such as supertree contruction, compatibility testing, and phylosymbiosis.


 

Tue, 27 Feb 2024
11:00
L5

Deep Transfer Learning for Adaptive Model Predictive Control

Harrison Waldon
(Oxford Man Institute)
Abstract

This paper presents the (Adaptive) Iterative Linear Quadratic Regulator Deep Galerkin Method (AIR-DGM), a novel approach for solving optimal control (OC) problems in dynamic and uncertain environments. Traditional OC methods face challenges in scalability and adaptability due to the curse-of-dimensionality and reliance on accurate models. Model Predictive Control (MPC) addresses these issues but is limited to open-loop controls. With (A)ILQR-DGM, we combine deep learning with OC to compute closed-loop control policies that adapt to changing dynamics. Our methodology is split into two phases; offline and online. In the offline phase, ILQR-DGM computes globally optimal control by minimizing a variational formulation of the Hamilton-Jacobi-Bellman (HJB) equation. To improve performance over DGM (Sirignano & Spiliopoulos, 2018), ILQR-DGM uses the ILQR method (Todorov & Li, 2005) to initialize the value function and policy networks. In the online phase, AIR-DGM solves continuously updated OC problems based on noisy observations of the environment. We provide results based on HJB stability theory to show that AIR-DGM leverages Transfer Learning (TL) to adapt the optimal policy. We test (A)ILQR-DGM in various setups and demonstrate its superior performance over traditional methods, especially in scenarios with misspecified priors and changing dynamics.

Mon, 05 Feb 2024

16:30 - 17:30
L5

Characterising rectifiable metric spaces using tangent spaces

David Bate
(Warwick)
Abstract

This talk will present a new characterisation of rectifiable subsets of a complete metric space in terms of local approximation, with respect to the Gromov-Hausdorff distance, by finite dimensional Banach spaces. Time permitting, we will discuss recent joint work with Hyde and Schul that provides quantitative analogues of this statement.
 

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