Past

The classification of finite simple groups shows that many (those of Lie type) are obtained as (projectivisations of) subgroups of some $GL_{n}(\mathbb{F}_{q})$.

Green first determined the character table of any $GL_{n}(\mathbb{F}_{q})$ in his influential 1955 paper, while others have since given more explicit constructions of certain `cuspidal' representations.

In this talk, I will introduce parabolic induction as a means of obtaining representations of $GL_{n}(\mathbb{F}_{q})$ from those of $GL_{m}(\mathbb{F}_{q})$ where $m<n$.

Finding the irreducible representations of any $GL_{n}(\mathbb{F}_{q})$ then becomes inductive on $n$ for fixed $q$, with the cuspidal representations serving as atoms for this process.

Harish-Chandra's philosophy of cusp forms reduces the problem to the following two steps:

•  Find the cuspidal representations of any $GL_{n}(\mathbb{F}_{q})$
•  Determine the irreducible components of any representation $\sigma_{1}\circ\dots\circ\sigma_{k}$ parabolically induced from cuspidals $\sigma_{i}$

The majority of my talk will then aim to address each of these points.

Natural killer (NK) cells are part of the innate immune system and are capable of killing diseased cells. As a result, NK cells are being used for adoptive cell therapies for cancer patients. The activation of NK cell stimulatory receptors leads to a cascade of intracellular phosphorylation reactions, which activates key signaling species that facilitate the secretion of cytolytic molecules required for cell killing. Strategies that maximize the activation of such intracellular species can increase the likelihood of NK cell killing upon contact with a cancer cell and thereby improve efficacy of NK cell-based therapies. However, NK cell exhaustion, a phenotype characterized by reduced effector functionality, can limit the NK cell’s capacity for cell lysis. Due to the complexity of intracellular signaling, it is difficult to deduce a priori which strategies can enhance species activation.  

To aid in the development of strategies to enhance NK cell activation and limit the NK cell exhaustion, we constructed a mechanistic model of the signaling pathways activated by stimulatory receptors in NK cells. We then extended the model to describe the dynamics of the cytolytic molecules granzyme B (GZMB) and perforin-1 (PRF1). We implemented an information-theoretic approach to perform a global sensitivity analysis and optimal control theory to investigate strategies to enhance intracellular signaling and maximize GZMB and PRF1 secretion. We recently expanded the modeling to investigate the role of NK cell heterogeneity on tumor cell killing. In total, we developed a theoretical framework that provides actionable insight into engineering robust NK cells for clinical applications.

I will discuss a string theory perspective on the Bethe/Gauge correspondence for the XXX superspin chain. I explain how to realize 4d Chern-Simons theory with gauge supergroup using branes, and how the brane configurations for the superspin chain get mapped to 2d N = (2,2) quiver gauge theories proposed by Nekrasov. This is based on my ongoing work with Nafiz Ishtiaque, Faroogh Moosavian and Surya Raghavendran.

Elastic strings are among the simplest one-dimensional continuum bodies and have a rich mechanical and mathematical theory dating back to the derivation of their equations of motion by Euler and Lagrange. In classical treatments, the string is either completely extensible (tensile force produces elongation) or completely inextensible (every segment has a fixed length, regardless of the motion). However, common experience is that a string can be stretched (is extensible), and after a certain amount of tensile force is applied the stretch of the string is maximized (becomes inextensible). In this talk, we discuss a model for these stretch-limited elastic strings, in what way they model elastic behavior, the well-posedness and asymptotic stability of certain simple motions, and (many) open questions.

In this talk I will discuss a number of topics at the interface between C* algebras and Geometric Group Theory, with an emphasis on Kazhdan projections, various versions of amenability and their connection to the geometry of groups. This is based on joint work with P. Nowak and J. Mackay.

Orders in major electronic stock markets are executed through centralised limit order books (LOBs). Large amounts of historical data have led to extensive research modeling LOBs, for the purpose of better understanding their dynamics and building simulators as a framework for controlled experiments, when testing trading algorithms or execution strategies.Most work in the literature models the aggregate view of the limit order book, which focuses on the volume of orders at a given price level, using a point process. In addition to this information, brokers and exchanges also have information on the identity of the agents submitting the order. This leads to a more granular view of limit order book dynamics, which we attempt to model using a heterogeneous model of order flow.

We present a granular representation of the limit order book that allows to account for the origins of different orders. Using client order flow from a major broker, we analyze the properties of variables in this representation. The heterogeneity of order flow is modeled by segmenting clients into different clusters, for which we identify representative prototypes. This segmentation appears to be stable both over time as well as over different stocks. Our findings can be leveraged to build more realistic order flow models that account for the diversity of the market participants.

We showcase some newly emerging connections between the theory of random walks and operator algebras, obtained by associating concrete algebras of operators to random walks. The C*-algebras we obtain give rise to new and interesting notions of ratio limit space and boundary, which are computed by appealing to various works on the asymptotic behavior of transition probabilities for random walks. Our results are leveraged to shed light on a question of Viselter on symmetry-unique quotients of Toeplitz C*-algebras of subproduct systems arising from random walks.

Techniques that address sequential data have been a central theme in machine learning research in the past years. More recently, such considerations have entered the field of finance-related ML applications in several areas where we face inherently path dependent problems: from (deep) pricing and hedging (of path-dependent options) to generative modelling of synthetic market data, which we refer to as market generation.

We revisit Deep Hedging from the perspective of the role of the data streams used for training and highlight how this perspective motivates the use of highly-accurate generative models for synthetic data generation. From this, we draw conclusions regarding the implications for risk management and model governance of these applications, in contrast to risk management in classical quantitative finance approaches.

Indeed, financial ML applications and their risk management heavily rely on a solid means of measuring and efficiently computing (similarity-)metrics between datasets consisting of sample paths of stochastic processes. Stochastic processes are at their core random variables with values on path space. However, while the distance between two (finite dimensional) distributions was historically well understood, the extension of this notion to the level of stochastic processes remained a challenge until recently. We discuss the effect of different choices of such metrics while revisiting some topics that are central to ML-augmented quantitative finance applications (such as the synthetic generation and the evaluation of similarity of data streams) from a regulatory (and model governance) perspective. Finally, we discuss the effect of considering refined metrics which respect and preserve the information structure (the filtration) of the market and the implications and relevance of such metrics on financial results.

An insightful exercise might be to ask what is the most important idea in linear algebra. Our first answer would not be eigenvalues or linearity, it would be “matrix factorizations.”  We will discuss a blueprint to generate  53 inter-related matrix factorizations (times 2) most of which appear to be new. The underlying mathematics may be traced back to Cartan (1927), Harish-Chandra (1956), and Flensted-Jensen (1978) . We will discuss the interesting history. One anecdote is that Eugene Wigner (1968) discovered factorizations such as the svd in passing in a way that was buried and only eight authors have referenced that work. Ironically Wigner referenced Sigurður Helgason (1962) but Wigner did not recognize his results in Helgason's book. This work also extends upon and completes open problems posed by Mackey²&Tisseur (2003/2005).

Classical results of Random Matrix Theory concern exact formulas from the Hermite, Laguerre, Jacobi, and Circular distributions. Following an insight from Freeman Dyson (1970), Zirnbauer (1996) and Duenez (2004/5) linked some of these classical ensembles to Cartan's theory of Symmetric Spaces. One troubling fact is that symmetric spaces alone do not cover all of the Jacobi ensembles. We present a completed theory based on the generalized Cartan distribution. Furthermore, we show how the matrix factorization obtained by the generalized Cartan decomposition G=K₁AK₂ plays a crucial role in sampling algorithms and the derivation of the joint probability density of A.

Joint work with Sungwoo Jeong.

An insightful exercise might be to ask what is the most important idea in linear algebra. Our first answer would not be eigenvalues or linearity, it would be “matrix factorizations.” We will discuss a blueprint to generate 53 inter-related matrix factorizations (times 2) most of which appear to be new. The underlying mathematics may be traced back to Cartan (1927), Harish-Chandra (1956), and Flensted-Jensen (1978) . We will discuss the interesting history. One anecdote is that Eugene Wigner (1968) discovered factorizations such as the SVD in passing in a way that was buried and only eight authors have referenced that work. Ironically Wigner referenced Sigurður Helgason (1962) but Wigner did not recognize his results in Helgason's book. This work also extends upon and completes open problems posed by Mackey² & Tisseur (2003/2005).

Classical results of Random Matrix Theory concern exact formulas from the Hermite, Laguerre, Jacobi, and Circular distributions. Following an insight from Freeman Dyson (1970), Zirnbauer (1996) and Duenez (2004/5) linked some of these classical ensembles to Cartan's theory of Symmetric Spaces. One troubling fact is that symmetric spaces alone do not cover all of the Jacobi ensembles. We present a completed theory based on the generalized Cartan distribution. Furthermore, we show how the matrix factorization obtained by the generalized Cartan decomposition G=K₁AK₂ plays a crucial role in sampling algorithms and the derivation of the joint probability density of A.

Joint work with Sungwoo Jeong

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Understanding the motion of small bodies at a fluid interface has relevance to a range of natural systems and technological applications. In this talk, we discuss two systems where capillarity and fluid inertia govern the dynamics of millimetric particles at a fluid interface.

In the first part, we present a study of superhydrophobic spheres impacting a quiescent water bath.  Under certain conditions particles may rebound completely from the interface - an outcome we characterize in detail through a synthesis of experiments, modeling, and direct numerical simulation.  In the second half, we introduce a system wherein millimetric disks trapped at a fluid interface are vertically oscillated and spontaneously self-propel.  Such "capillary surfers" interact with each other via their collective wavefield and self-assemble into a myriad of cooperative dynamic states.  Our experimental observations are well captured by a first theoretical model for their dynamics, laying the foundation for future investigations of this highly tunable active system.

The goal of this talk is to present some elementary examples of fusion 2-categories whilst doing as little higher category theory as possible. More precisely, it turns out that up to a canonical completion operation, certain higher fusion categories are entirely described by their maximal subspaces. I will briefly motivate this completion operation in the 1-categorical case, and go on to explain why working with spaces is good enough in this particular case. Then, we will review some fact about $E_n$-algebras, and why they come into the picture. Finally, we will have a look at some small examples arising from finite groups.

Three conjectures on group rings of torsion-free groups are commonly attributed to Kaplansky, namely the unit, zero divisor and idempotent conjectures. For example, the zero divisor conjecture predicts that if $K$ is a field and $G$ is a torsion-free group, then the group ring $K[G]$ has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other conjectures and group properties, and present my recent counterexample to the unit conjecture.

We consider the problem of clustering in two important families of networks: signed and directed, both relatively less well explored compared to their unsigned and undirected counterparts. Both problems share an important common feature: they can be solved by exploiting the spectrum of certain graph Laplacian matrices or derivations thereof. In signed networks, the edge weights between the nodes may take either positive or negative values, encoding a measure of similarity or dissimilarity. We consider a generalized eigenvalue problem involving graph Laplacians, with performance guarantees under the setting of a signed stochastic block model. The second problem concerns directed graphs. Imagine a (social) network in which you spot two subsets of accounts, X and Y, for which the overwhelming majority of messages (or friend requests, endorsements, etc) flow from X to Y, and very few flow from Y to X; would you get suspicious? To this end, we also discuss a spectral clustering algorithm for directed graphs based on a complex-valued representation of the adjacency matrix, which is able to capture the underlying cluster structures, for which the information encoded in the direction of the edges is crucial. We evaluate the proposed algorithm in terms of a cut flow imbalance-based objective function, which, for a pair of given clusters, it captures the propensity of the edges to flow in a given direction. Experiments on a directed stochastic block model and real-world networks showcase the robustness and accuracy of the method, when compared to other state-of-the-art methods. Time permitting, we briefly discuss potential extensions to the sparse setting and regularization, applications to lead-lag detection in time series and ranking from pairwise comparisons.

Dark matter is known to exist, but no-one knows what it is or where it came
from.  We describe a new production mechanism of particle dark matter, which
hinges on a first-order cosmological phase transition.  We then show that
this mechanism can be slightly modified to produce primordial black holes.

While solar mass and supermassive black holes are now known to exist,
primordial black holes have not yet been seen but could solve a number of
problems in cosmology.  Finally, we demonstrate that if an evaporating
primordial black hole is observed, it will provide a unique window onto
Beyond the Standard Model physics.

There is a striking relationship between Willmore surfaces of revolution and elastic curves in hyperbolic half-space. Here the term elastic curve refer to a critical point of the energy given by the integral of the curvature squared. In the talk we will discuss this relationship and use it to study long-time existence and asymptotic behavior for the L2-gradient flow of the Willmore energy, under the condition that the initial datum is a torus of revolution. As in the case of Willmore flow of spheres, we show that if an initial datum has Willmore energy below 8 \pi then the solution of the Willmore flow converges to the Clifford Torus, possibly rescaled and translated. The energy threshold of 8 \pi turns out to be optimal for such a convergence result. 

The lecture is based on joint work with M. Müller (Univ. Freiburg), R. Schätzle (Univ. Tübingen) and A. Spener (Univ. Ulm).

The action of the automorphisms of a formal group on its deformation space is crucial to understanding periodic families in the homotopy groups of spheres and the unsolved Hecke orbit conjecture for unitary Shimura varieties. We can explicitly pin down this squirming action by geometrically modelling it as coming from an action on a moduli space, which we construct using inverse Galois theory and some representation theory (a Hurwitz space). I will show you pretty pictures.

Following the risk-taking model of Seel and Strack, n players decide when to stop privately observed Brownian motions with drift and absorption at zero. They are then ranked according to their level of stopping and paid a rank-dependent reward. We study the optimal reward design where a principal is interested in the average performance and the performance at a given rank. While the former can be related to reward inequality in the Lorenz sense, the latter can have a surprising shape. Next, I will present the mean-field version of this problem. A particular feature of this game is to be tractable without necessarily being smooth, which turns out to offer a cautionary tale. We show that the mean field equilibrium induces n-player ε-Nash equilibria for any continuous reward function— but not for discontinuous ones. We also analyze the quality of the mean field design (for maximizing the median performance) when used as a proxy for the optimizer in the n-player game. Surprisingly, the quality deteriorates dramatically as n grows. We explain this with an asymptotic singularity in the induced n-player equilibrium distributions. (Joint work with M. Nutz)

Most of the talk will be about the Farrell-Jones conjecture from the point of view of an outsider. I'll try to explain what the conjecture is about, why one wants to know it, and how to prove it in some cases. The motivation for the talk is my recent work with Fujiwara and Wigglesworth where we prove this conjecture for (virtually torsion-free hyperbolic)-by-cyclic groups. If there is time I will outline the proof of this result.

A celebrated result in geometry is the Kobayashi-Hitchin correspondence, which states that a holomorphic vector bundle on a compact Kähler manifold admits a Hermite-Einstein metric if and only if the bundle is slope polystable. Recently, Dervan and Sektnan have conjectured an analogue of this correspondence for fibrations whose fibres are compact Kähler manifolds admitting Kähler metrics of constant scalar curvature. Their conjecture is that such a fibration is polystable in a suitable sense, if and only if it admits an optimal symplectic connection. In this talk, I will provide an introduction to this theory, and describe my recent work on the conjecture. Namely, I show that existence of an optimal symplectic connection implies polystability with respect to a large class of fibration degenerations. The techniques used involve analysing geodesics in the space of relatively Kähler metrics of fibrewise constant scalar curvature, and convexity of the log-norm functional in this setting. This is work for my PhD thesis, supervised by Frances Kirwan and Ruadhaí Dervan.

A proposal for the worldsheet string theory that is dual to free N=4 SYM in 4d will be explained. It is described by a free field sigma model on the twistor space of AdS5 x S5, and is a direct generalisation of the corresponding model for tensionless string theory on AdS3 x S3. I will explain how our proposal fits into the general framework of AdS/CFT, and review the various checks that have been performed.
 

In this talk I will present a gravitational interpretation for the superconformal index of N = 4 SYM theory in the large N limit. I will start by reviewing the so-called Bethe Ansatz formulation of the field theory index and its large N expansion (which includes both perturbative and non-perturbative corrections in 1/N). In the gravity side, according the rules of AdS/CFT correspondence, the index—interpreted as the supersymmetric partition function of N = 4 SYM—should be equivalent to the gravitational partition function on AdS_5 x S^5. The latter is classically evaluated as a sum over Euclidean gravity solutions with appropriate boundary conditions. In this context, I will show that (in the case of equal angular momenta) the contribution to the index of each Bethe Ansatz solution that admits a proper large N limit is captured by a complex black hole solution in the gravity side, which reproduces both its perturbative and non-perturbative behavior. More specifically, the number of putative black hole solutions turns out to be much larger than the number of Bethe Ansatz solutions. A resolution of this issue is found by requiring the gravity solutions to be “stable” under the non-perturbative corrections. This ensures that all the extra gravity solutions are ruled out and leads to a precise match between field theory and gravity.

In recent years, topological and geometric data analysis (TGDA) has emerged as a new and promising field for processing, analyzing and understanding complex data. Indeed, geometry and topology form natural platforms for data analysis, with geometry describing the ''shape'' behind data; and topology characterizing / summarizing both the domain where data are sampled from, as well as functions and maps associated with them. In this talk, I will show how topological (and geometric ideas) can be used to analyze graph data, which occurs ubiquitously across science and engineering. Graphs could be geometric in nature, such as road networks in GIS, or relational and abstract. I will particularly focus on the reconstruction of hidden geometric graphs from noisy data, as well as graph matching and classification. I will discuss the motivating applications, algorithm development, and theoretical guarantees for these methods. Through these topics, I aim to illustrate the important role that topological and geometric ideas can play in data analysis.