Past

Title: When data driven reduced order modelling meets full waveform inversion

Abstract:

This talk is concerned with the following inverse problem for the wave equation: Determine the variable wave speed from data gathered by a collection of sensors, which emit probing signals and measure the generated backscattered waves. Inverse backscattering is an interdisciplinary field driven by applications in geophysical exploration, radar imaging, non-destructive evaluation of materials, etc. There are two types of methods:

(1) Qualitative (imaging) methods, which address the simpler problem of locating reflective structures in a known host medium. 

(2) Quantitative methods, also known as velocity estimation. 

Typically, velocity estimation is  formulated as a PDE constrained optimization, where the data are fit in the least squares sense by the wave computed at the search wave speed. The increase in computing power has lead to growing interest in this approach, but there is a fundamental impediment, which manifests especially for high frequency data: The objective function is not convex and has numerous local minima even in the absence of noise.

The main goal of the talk is to introduce a novel approach to velocity estimation, based on a reduced order model (ROM) of the wave operator. The ROM is called data driven because it is obtained from the measurements made at the sensors. The mapping between these measurements and the ROM is nonlinear, and yet the ROM can be computed efficiently using methods from numerical linear algebra. More importantly, the ROM can be used to define a better objective function for velocity estimation, so that gradient based optimization can succeed even for a poor initial guess.

TBC

The mod p Langlands program is an attempt to relate mod p Galois representations of a local field to mod p representations of the p-adic points of a reductive group. This is inspired by the classical local Langlands (l-adic coefficients) and it is partially a stepping stone towards the p-adic Langlands (p-adic coefficients). I will explain this for GL2/Qp, where one can explicitly describe both sides, and I will relate it to congruences between modular forms. 

A conjecture due to Zilber predicts that the complex exponential field is quasiminimal: that is, that all subsets of the complex numbers that are definable in the language of rings expanded by a symbol for the complex exponential function are countable or cocountable.
Zilber showed that this conjecture would follow from Schanuel's Conjecture and an existential closedness type property asserting that certain systems of exponential-polynomial equations can be solved in the complex numbers; later on, Bays and Kirby were able to remove the dependence on Schanuel's Conjecture, shifting all the focus to the existence of solutions. In this talk, I will discuss recent work about the quasiminimality of a reduct of the complex exponential field, that is, the complex numbers expanded by multivalued power functions. This is joint work with Jonathan Kirby.

Nonlinear filtering is a central mathematical tool in understanding how we process information. Unfortunately, the equations involved are often high dimensional, and therefore, in practical applications, approximate filters are often employed in place of the optimal filter. The error introduced by using these approximations is generally poorly understood. In this talk we will see how, in the case where the underlying process is a continuous-time, finite-state Markov Chain, results on the stability of filters can be strengthened to yield bounds for the error between the optimal filter and a general approximate filter.  We will then consider the 'projection filter', a low dimensional approximation of the filtering equation originally due to D. Brigo and collaborators, and show that its error is indeed well-controlled. The talk is based on joint work with Sam Cohen.

We propose to use stochastic Riemannian coordinate descent on the Stiefel manifold for deep neural network training. The algorithm rotates successively two columns of the matrix, an operation that can be efficiently implemented as a multiplication by a Givens matrix. In the case when the coordinate is selected uniformly at random at each iteration, we prove the convergence of the proposed algorithm under standard assumptions on the loss function, stepsize and minibatch noise. Experiments on benchmark deep neural network training problems are presented to demonstrate the effectiveness of the proposed algorithm.

We will look at the vanishing of group cohomology from the perspective of Kazhdan’s property (T). We will investigate an analogue of this property for any degree, introduced by U. Bader and P. W. Nowak in 2020 and describe a method of proving these properties with computers.

Female PhD students and Postdocs will be giving short talks about their research. This will be followed by a Q&A and a chance to mingle with the speakers over lunch.

Speakers:

• Rhiannon Savage, DPhil Student in Algebra and Geometry
• Shanshan Hua, DPhil Student in Functional Analysis
• Silvia Butti, Postdoc in Theoretical Computer Science
• Anna Berryman, DPhil Student in OCIAM (Oxford Centre for Industrial and Applied Mathematics)
• Carmen Jorge Diaz, DPhil Student in Mathematical Physics

Wreath-like products are a new class of groups, which are close relatives of the classical wreath products. Examples of wreath-like product groups arise from every non-elementary hyperbolic groups by taking suitable quotients. As a consequence, unlike classical wreath products, many wreath-like products have Kazhdan's property (T). 

I will present several rigidity results for von Neumann algebras of wreath-like product groups. We show that any group G in a natural family of wreath-like products with property (T) is W*-superrigid: the group von Neumann algebra L(G) remembers the isomorphism class of G. This provides the first examples of W*-superrigid groups with property (T). For a wider class wreath-like products with property (T), we show that any isomorphism of their group von Neumann algebras arises from an isomorphism of the groups. As an application, we prove that any countable group can be realized as the outer automorphism group of L(G), for an icc property (T) group G. These results are joint with Ionut Chifan, Denis Osin and Bin Sun.  

Time permitting, I will mention an additional application of wreath-like products obtained in joint work with Ionut Chifan and Daniel Drimbe, and showing that any separable II_1 factor is contained in one with property (T). This provides an operator algebraic counterpart of the group theoretic fact that every countable group is contained in one with property (T).

The Calogero-Painlevé systems were introduced in 2001 by K. Takasaki as a natural generalization of the classical Painlevé equations to the case of the several Painlevé “particles” coupled via the Calogero type interactions. In 2014, I. Rumanov discovered a remarkable fact that a particular case of the Calogero– Painlevé II equation describes the Tracy-Widom distribution function for the general $\beta$-ensembles with the even values of parameter $\beta$. in 2017 work of M. Bertola, M. Cafasso , and V. Rubtsov, it was proven that all Calogero-Painlevé systems are Lax integrable, and hence their solutions admit a Riemann-Hilbert representation. This important observation has opened the door to rigorous asymptotic analysis of the Calogero-Painlevé equations which in turn yields the possibility of rigorous evaluation of the asymptotic behavior of the Tracy-Widom distributions for the values of $\beta$ beyond the classical $\beta =1, 2, 4$. In the talk these recent developments will be outlined with a special focus on the Calogero-Painlevé system corresponding to $\beta = 6$. This is a joint work with Andrei Prokhorov.

COURSE_PROPOSAL (12).pdf

The space of unordered tuples. The notion of differentiability and the theory of metric Sobolev in the context of multi-valued functions. Multivalued maximum principle and Holder regularity. Estimates on the Hausdorff dimension of the singular set of Dir-minimizing functions. If time permits: mass minimizing currents and their link with Dir-minimizers. 

To a group theorist ribbon groups look like knot groups, except  that we know everything about knot groups and next to nothing about ribbon groups.

I will talk about an old paper of mine with Peter Teichner where several questions on ribbon groups naturally arise.

The Satake isomorphism is a fundamental result in p-adic groups, and the affine Grassmannian is the natural setting where this geometrizes to the Geometric Satake Correspondence. In fact, it suffices to work with affine Grassmannian slices, which retain all of the information.

Recently, Braverman, Finkelberg, and Nakajima showed that affine Grassmannian slices arise as Coulomb branches of certain quiver gauge theories. Remarkably, their construction works in Kac-Moody type as well. Their work opens the door to studying affine Grassmannians and Geometric Satake Correspondence for Kac-Moody groups. Unfortunately, it is difficult at present to do any explicit geometry with the Coulomb branch definition. For example, a basic feature is that affine Grassmannian slices embed into one another. However, this is not apparent from the Coulomb branch definition. In this talk, I will explain why these embeddings are necessarily subtle. Nonetheless, I will show a way to construct the embeddings using fundamental monopole operators.

This is joint work with Alex Weekes.

"Colouring" a tournament means partitioning its vertex set into acylic subsets; and the "domination number" is the size of the smallest set of vertices with no common in-neighbour. In some ways these are like the corresponding concepts for graphs, but in some ways they are very different. We give a survey of some recent results and open questions on these topics.

Joint with Tung Nguyen and Alex Scott.

Supersymmetric localization allows one to reduce the computation of the partition function of a supersymmetric theory to a finite-dimensional integral, but the space over which one integrates is often singular. In this talk I will explain how one can use shifted symplectic geometry to get rigorous definitions of partition functions and state spaces in theories with extended supersymmetry. For instance, this gives a field-theoretic origin of DT invariants of CY4 manifolds. This is a report on joint work with Brian Williams.

Diffusion Limited Aggregation is a very simple mathematical model which describes a wide range of natural phenomena. Despite its simplicity, there is very little progress in understanding its large-scale structure. Since its introduction by Witten and Sander over 40 years ago, there was only one mathematical result. In 1987 Kesten obtained an upper bound on the growth rate. In this talk I will discuss DLA and some related models and the recent progress in understanding DLA. In particular, a new simpler proof of Kesten result which generalizes to other aggregation models.

Quasiconvexity is the fundamental existence condition for variational problems, yet it is poorly understood. Two outstanding problems remain: 

• 1) does rank-one convexity, a simple necessary condition, imply quasiconvexity in two dimensions? 

• 2) can one prove existence theorems for quasiconvex energies in the context of nonlinear Elasticity? 

In this talk we show that both problems have a positive answer in a special class of isotropic energies. Our proof combines complex analysis with the theory of gradient Young measures. On the way to the main result, we establish quasiconvexity inequalities for the Burkholder function which yield, in particular, many sharp higher integrability results. 
The talk is based on joint work with Kari Astala, Daniel Faraco, Aleksis Koski and Jan Kristensen.

I will construct the differential geometric, gauge-theoretic, and duality covariant model of classical four-dimensional Abelian gauge theory on an orientable four-manifold of arbitrary topology. I will do so by implementing the Dirac-Schwinger-Zwanziger (DSZ) integrality condition in classical Abelian gauge theories with general duality structure and interpreting the associated sheaf cohomology groups geometrically. As a result, I will obtain that four-dimensional Abelian gauge theories are theories of connections on Siegel bundles, namely principal bundles whose structure group is the generically non-abelian disconnected group of automorphisms of an integral affine symplectic torus. This differential-geometric model includes the electric and magnetic gauge potentials on an equal footing and describes the equations of motion through a first-order polarized self-duality condition for the curvature of a connection. This condition is reminiscent of the theory of four-dimensional Euclidean instantons, even though we consider a two-derivative theory in Lorentzian signature. Finally, I will elaborate on various applications of this differential-geometric model, including a mathematically rigorous description of electromagnetic duality in Abelian gauge theory and the reduction of the polarized self-duality condition to a Riemannian three-manifold, which gives as a result a new type of Bogomolny equation.

The affinity between statistical physics and machine learning has a long history. Theoretical physics often proceeds in terms of solvable synthetic models; I will describe the related line of work on solvable models of simple feed-forward neural networks. I will then discuss how this approach allows us to analyze uncertainty quantification in neural networks, a topic that gained urgency in the dawn of widely deployed artificial intelligence. I will conclude with what I perceive as important specific open questions in the field.