Past

I will give an introduction to quasidiagonality of group actions wherein an action on a C^*-algebra is approximated by actions on matrix algebras.  This has implications for crossed product C^*-algebras, especially as pertains to finite dimensional approximation.  I'll sketch the proof that all isometric actions are quasidiagonal, which we can view as a dynamical Petr-Weyl theorem.  Then I will discuss an interplay between quasidiagonal actions and semiprojectivity of C^*-algebras, a property that allows "almost representations" to be perturbed to honest ones.  

We will survey work of Birman-Craggs, Johnson, and Sato on the abelianization of the level 2 congruence group of the mapping class group of a surface, and of the corresponding Torelli group. We will then describe recent work of Lewis providing a common framework for both abelianizations, with applications including a partial answer to a question of Johnson.

Fourth-order elliptic problems arise in a variety of applications from thin plates to phase separation to liquid crystals. A conforming Galerkin discretization requires a finite dimensional subspace of $H^2$, which in turn means that conforming finite element subspaces are $C^1$-continuous. In contrast to standard $H^1$-conforming $C^0$ elements, $C^1$ elements, particularly those of high order, are less understood from a theoretical perspective and are not implemented in many existing finite element codes. In this talk, we address the implementation of the elements. In particular, we present algorithms that compute $C^1$ finite element approximations to fourth-order elliptic problems and which only require elements with at most $C^0$-continuity. We also discuss solvers for the resulting subproblems and illustrate the method on a number of representative test problems.

We present new high-order finite elements discretizing the $L^2$ de Rham complex on triangular and tetrahedral meshes. The finite elements discretize the same spaces as usual, but with different basis functions. They allow for fast linear solvers based on static condensation and space decomposition methods.

The new elements build upon the definition of degrees of freedom given by (Demkowicz et al., De Rham diagram for $hp$ finite element spaces. Comput.~Math.~Appl., 39(7-8):29--38, 2000.), and consist of integral moments on a symmetric reference simplex with respect to a numerically computed polynomial basis that is orthogonal in both the $L^2$- and $H(\mathrm{d})$-inner products ($\mathrm{d} \in \{\mathrm{grad}, \mathrm{curl}, \mathrm{div}\}$).

On the reference symmetric simplex, the resulting stiffness matrix has diagonal interior block, and does not couple together the interior and interface degrees of freedom. Thus, on the reference simplex, the Schur complement resulting from elimination of interior degrees of freedom is simply the interface block itself.

This sparsity is not preserved on arbitrary cells mapped from the reference cell. Nevertheless, the interior-interface coupling is weak because it is only induced by the geometric transformation. We devise a preconditioning strategy by neglecting the interior-interface coupling. We precondition the interface Schur complement with the interface block, and simply apply point-Jacobi to precondition the interior block.

The combination of this approach with a space decomposition method on small subdomains constructed around vertices, edges, and faces allows us to efficiently solve the canonical Riesz maps in $H^1$, $H(\mathrm{curl})$, and $H(\mathrm{div})$, at very high order. We empirically demonstrate iteration counts that are robust with respect to the polynomial degree.

Given a matrix $A$ with integer entries, a subset $S$ of an abelian group and $r\in\mathbb N$, we say that $S$ is $(A,r)$-Rado if any $r$-colouring of $S$ yields a monochromatic solution to the system of equations $Ax=0$. A classical result of Rado characterises all those matrices $A$ such that $\mathbb N$ is $(A,r)$-Rado for all $r \in \mathbb N$. Rödl and Ruciński, and Friedgut, Rödl and Schacht proved a random version of Rado’s theorem where one considers a random subset of $[n]:=\{1,\dots,n\}$.

In this paper, we investigate the analogous random Ramsey problem in the more general setting of abelian groups. Given a sequence $(S_n)_{n\in\mathbb N}$ of finite subsets of abelian groups, let $S_{n,p}$ be a random subset of $S_n$ obtained by including each element of $S_n$ independently with probability $p$. We are interested in determining the probability threshold for $S_{n,p}$ being $(A,r)$-Rado.

Our main result is a general black box for hypergraphs which we use to tackle problems of this type. Using this tool in conjunction with a series of supersaturation results, we determine the probability threshold for a number of different cases. A consequence of the Green-Tao theorem is the van der Waerden theorem for the primes: every finite colouring of the primes contains arbitrarily long monochromatic arithmetic progressions. Using our machinery, we obtain a random version of this result. We also prove a novel supersaturation result for $[n]^d$ and use it to prove an integer lattice generalisation of the random version of Rado's theorem.

This is joint work with Andrea Freschi and Andrew Treglown (both University of Birmingham).

I will then explain that hybrid quantum-classical protocols are the most promising candidates for achieving early quantum advantage. These have the potential to solve real-world problems---including optimisation or ground-state search---but they suffer from a large number of circuit repetitions required to extract information from the quantum state. I will explain some of our recent results as hybrid quantum algorithms that exploit so-called classical shadows (random unitary protocols) in order to extract and post-process a large amount of information from the quantum computer [PRX 12, 041022 (2022)] and [arXiv:2212.11036]. I will finally identify the most likely areas where quantum computers may deliver a true advantage in the near term.

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.
 

TBC

One way to capture both the elastic and stochastic reaction of purchases to price is through a model where sellers control the intensity of a counting process, representing the number of sales thus far. The intensity describes the probabilistic likelihood of a sale, and is a decreasing function of the price a seller sets. A classical model for ticket pricing, which assumes a single seller and infinite time horizon, is by Gallego and van Ryzin (1994) and it has been widely utilized by airlines, for instance. Extending to more realistic settings where there are multiple sellers, with finite inventories, in competition over a finite time horizon is more complicated both mathematically and computationally. We discuss some dynamic games of this type, from static to two player to the associated mean field game, with some numerical and existence-uniqueness results.

Based on works with Andrew Ledvina and with Emre Parmaksiz.

Bicommutant categories, initially invented for the purposes of Chern-Simons theory and 2d CFT, seem to also appear in other domains of math with examples related to group theory, and dynamical systems.

Learning to process and analyze the raw weight matrices of neural networks is an emerging research area with intriguing potential applications like editing and analyzing Implicit Neural Representations (INRs), weight pruning/quantization, and function editing. However, weight spaces have inherent permutation symmetries – permutations can be applied to the weights of an architecture, yielding new weights that represent the same function. As with other data types like graphs and point clouds, these symmetries make learning in weight spaces challenging.

This talk will overview recent advances in designing architectures that can effectively operate on weight spaces while respecting their underlying symmetries. First, we will discuss our ICML 2023 paper which introduces novel equivariant architectures for learning on multilayer perceptron weight spaces. We first characterize all linear equivariant layers for their symmetries and then construct networks composed of these layers. We then turn to our ICLR 2024 work, which generalizes the approach to diverse network architectures using what we term Graph Metanetworks (GMN). This is done by representing input networks as graphs and processing them with graph neural networks. We show the resulting metanetworks are expressive and equivariant to weight space symmetries of the architecture being processed. Our graph metanetworks are applicable to CNNs, attention layers, normalization layers, and more. Together, these works make promising steps toward versatile and principled architectures for weight-space learning.

Join us for a session with Keith Macksoud, Executive Director at global recruitment consultant Options Group in London and who previously has > 20 years’ experience in Prime Brokerage Sales at Morgan Stanley, Citi, and Deutsche Bank.  Keith will discuss the recruitment process for financial institutions, and how to increase your chances of a successful application. 

Keith will detail his finance background in Prime Brokerage and provide students with an exclusive look behind the scenes of executive search and strategic consulting firm Options Group. We will look at what Options Group does, how executive search firms work and the Firm’s 30-year track record of placing individuals at many of the industries’ largest and most successful global investment banks, investment managers and other financial services-related organisations. 

About Options Group

Options Group is a leading global executive search and strategic consulting firm specializing in financial services including capital markets, global markets, alternative investments, hedge funds, and private banking/wealth management.

https://www.optionsgroup.com/

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.

Malignant transformation of somatic tissues is an evolutionary process, driven by selection for oncogenic mutations. Understanding when these mutations occur, and how fast mutant cell clones expand can improve diagnostic schemes and therapeutic intervention. However, clonal dynamics are not directly accessible in humans, posing a need for inference approaches to reconstruct the division history in normal and malignant cell clones, and to predict their future evolution. Inspired from population genetics theory, we develop mathematical models to detect imprints of clonal selection in the variant allele frequency distribution measured in a single tissue sample of a homeostatic tissue. I will present the theoretical basis of our approach and inference results for the tissue dynamics in physiological and clonal hematopoiesis, obtained from variant allele frequencies measured by snapshot bulk whole genome sequencing of human bone marrow samples.

The Monster group M is the largest sporadic simple group. It is also the group of automorphisms of 196, 884-dimensional Fischer-Norton-Griess algebra V_M. In 2009, A. A. Ivanov offered an axiomatic approach to studying the structure of V_M by introducing the notions of Majorana algebra and Majorana representation. Later, the theory developed, and Majorana representations of several groups were constructed. Our talk is dedicated to the existence of standard Majorana representations of 3-transposition groups for the Fischer list. The main result is that the groups from the Fischer list which admit a standard Majorana representation can be embedded into the Monster group.

Positive logic is a generalisation of full first-order logic, where negation is not built in, but can be added as desired. In joint work with Jan Dobrowolski we succesfully generalised the recent development on Kim-independence in NSOP1 theories to the positive setting. One of the important theorems in this development is the independence theorem, whose statement is very similar to the well-known statement for simple theories, and allows us to amalgamate independent types. In this talk we will have a closer look at the proof of this theorem, and what needs to be changed to make the proof work in positive logic compared to full first-order logic.

To every action of a discrete group on a compact Hausdorff space one can canonically associate a C*-algebra, called the crossed product. The crossed product construction is an extremely popular one, and there are numerous results in the literature that describe the structure of this C* algebra in terms of the dynamical system. In this talk, we will focus on one of the central notions in the realm of the classification of simple, nuclear C*-algebras, namely Jiang-Su stability. We will review the existing results and report on the most recent progress in this direction, going beyond the case of free actions both for amenable and nonamenable groups. 

Parts of this talk are joint works with Geffen, Kranz, and Naryshkin, and with Geffen, Gesing, Kopsacheilis, and Naryshkin. 

Diffusion models, which transform noise into new data instances by reversing a Markov diffusion process, have become a cornerstone in modern generative models. A key component of these models is to learn the score function through score matching. While the practical power of diffusion models has now been widely recognized, the theoretical developments remain far from mature. Notably, it remains unclear whether gradient-based algorithms can learn the score function with a provable accuracy. In this talk, we develop a suite of non-asymptotic theory towards understanding the data generation process of diffusion models and the accuracy of score estimation. Our analysis covers both the optimization and the generalization aspects of the learning procedure, which also builds a novel connection to supervised learning and neural tangent kernels.

This is based on joint work with Yinbin Han and Meisam Razaviyayn (USC).

A recent conjecture proposed by Harris and Venkatesh relates the action of derived Hecke operators on the space of weight one modular forms to certain Stark units. In this talk, I will explain how this can be rephrased as a conjecture about "tame" analogues of triple product periods for a triple of mod p eigenforms of weights (2,1,1). I will then present an elliptic counterpart to this conjecture relating a tame triple product period to a regulator for global points of elliptic curves in rank 2. This conjecture can be proved in some special cases for CM weight 1 forms, with techniques resonating with the so-called Jochnowitz congruences. This is joint work in preparation with Henri Darmon. 

We give a strongly polynomial algorithm for minimum cost generalized flow, and as a consequence, for all linear programs with at most two nonzero entries per row, or at most two nonzero entries per column. While strongly polynomial algorithms for the primal and dual feasibility problems have been known for a long time, various combinatorial approaches used for those problems did not seem to carry over to the minimum-cost variant.

Our approach is to show that the ‘subspace layered least squares’ interior point method, an earlier joint work with Allamigeon, Dadush, Loho and Natura requires only a strongly polynomial number of iterations for minimum cost generalized flow. We achieve this by bounding the straight line complexity, introduced in the same paper. The talk will give an overview of the interior point method as well as the combinatorial straight-line complexity analysis for the particular setting. This is joint work with Daniel Dadush, Zhuan Khye Koh, Bento Natura, and Neil Olver.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

Let q be a prime power and let C be a smooth curve defined over F_q. The number of points of C over the finite extensions of F_q are determined by the Zeta function of C, which can be written in the form P_C(t)/((1-t)(1-qt)), where P_C(t) is a polynomial of degree 2g and g is the genus of C; this is often called the L-polynomial of C. We use a Chebotarev-like statement (over function fields instead of Z) due to Katz in order to study the distribution, as C varies, of the coefficients of P_C(t) in a non-archimedean setting.