Past

A central question in infinite group theory is to determine how much global information about a group is encoded in its set of finite quotients. In this talk, we will discuss this problem in the case of algebraically fibered groups, which naturally generalise fundamental groups of compact manifolds that fiber over the circle. The study of such groups exploits the relationships between the geometry of the classifying space, the dynamics of the monodromy map, and the algebra of the group, and as such draws from all of these areas.

A rooted tree $T$ has degree sequence $(d_1,\ldots,d_n)$ if $T$ has vertex set $[n]$ and vertex $i$ has $d_i$ children for each $i$ in $[n]$. 

I will describe a line-breaking construction of random rooted trees with given degree sequences, as well as a way of coupling random trees with different degree sequences that also couples their heights to one another. 

The construction and the coupling have several consequences, and I'll try to explain some of these in the talk.

First, let $T$ be a branching process tree with critical—mean one—offspring distribution, and let $T_n$ have the law of $T$ conditioned to have size $n$. Then the following both hold.
1) $\operatorname{height}(T_n)/\log(n)$ tends to infinity in probability. 
2) If the offspring distribution has infinite variance then $\operatorname{height}(T_n)/n^{1/2}$ tends to $0$ in probability. This result settles a conjecture of Svante Janson.

The next two statements relate to random rooted trees with given degree sequences. 
1) For any $\varepsilon > 0$ there is $C > 0$ such that the following holds. If $T$ is a random tree with degree sequence $(d_1,\ldots,d_n)$ and at least $\varepsilon n$ leaves, then $\mathbb{E}(\operatorname{height}(T)) < C \sqrt{n}$. 
2) Consider any random tree $T$ with a fixed degree sequence such that $T$ has no vertices with exactly one child. Then $\operatorname{height}(T)$ is stochastically less than $\operatorname{height}(B)$, where $B$ is a random binary tree of the same size as $T$ (or size one greater, if $T$ has even size). 

In this talk, I will discuss correlation functions in 6d (2, 0) theories of two 1/2-BPS operators inserted away from a 1/2-BPS surface defect. In the large central charge limit the leading connected contribution corresponds to sums of tree-level Witten diagram in AdS7×S4 in the presence of an AdS3 defect. I will show that these correlators can be uniquely determined by imposing only superconformal symmetry and consistency conditions, eschewing the details of the complicated effective Lagrangian. I will present the explicit result of all such two-point functions, which exhibits remarkable hidden simplicity.

A representation on the space of paths is a map which is compatible with the concatenation operation of paths, such as the path signature and Cartan development (or equivalently, parallel transport), and has been used to define characteristic functions for the law of stochastic processes. In this talk, we consider representations of surfaces which are compatible with the two distinct algebraic operations on surfaces: horizontal and vertical concatenation. To build these representations, we use the notion of higher parallel transport, which was first introduced to develop higher gauge theories. We will not assume any background in geometry or category theory. This is a continuation of the previous talk based on a recent preprint (https://arxiv.org/abs/2311.08366) with Harald Oberhauser.

In the past two decades, starting with the pioneering work of Bourgain, Brezis, and Mironescu, there has been widespread interest in characterizing Sobolev and BV (bounded variation) functions by means of non-local functionals. In my recent work I have studied two such functionals: a BMO-type (bounded mean oscillation) functional, and a functional related to the fractional Sobolev seminorms. I will discuss some of my results concerning the limits of these functionals, the concept of Gamma-convergence, and also open problems. 

Modular curves play a key role in the Langlands programme, being the simplest example of so-called Shimura varieties.  Their less famous cousins, Shimura curves, are also very interesting, and very concrete. 
In this talk I will give a gentle introduction to the arithmetic of Shimura curves, with lots of explicit examples. Time permitting, I will say something about recent work about intersection numbers of geodesics on Shimura curves.

Involutive knot Floer homology, a refinement of knot Floer theory, is a powerful knot invariant which was used to solve several long-standing problems, including the one-is-not-enough result for 4-manifolds with boundary. In this talk, we show that if the involutive knot Floer homology of a knot K admits an invariant splitting, then the induced splitting if the knot Floer homology of P(K), for any pattern P, can be made invariant under its \iota_K involution. As an application, we construct an infinite family of examples of pairs of exotic contractible 4-manifolds which survive one stabilization, and observe that some of them are potential candidates for surviving two stabilizations.
 

Mathematicians have been interested in the problem of compactifying the Jacobian variety of curves since the mid XIX century. In this talk we will discuss how all 'reasonable' compactified Jacobians of nodal curves can be classified combinatorically. This suffices to obtain a combinatorial classification of all 'reasonable' compactified universal (over the moduli spaces of stable curves) Jacobians. This is a joint work with Orsola Tommasi.

When the data is large, or comes in a streaming way, randomized iterative methods provide an efficient way to solve a variety of problems, including solving linear systems, finding least square solutions, solving feasibility problems, and others. Randomized Kaczmarz algorithm for solving over-determined linear systems is one of the popular choices due to its efficiency and its simple, geometrically intuitive iterative steps. 
In challenging cases, for example, when the condition number of the system is bad, or some of the equations contain large corruptions, the geometry can be also helpful to augment the solver in the right way. I will discuss our recent work with Michal Derezinski and Jackie Lok on Kaczmarz-based algorithms that use external knowledge about the linear system to (a) accelerate the convergence of iterative solvers, and (b) enable convergence in the highly corrupted regime.

Join us for a discussion about preparing for PhD and PostDoc Interviews. We will be talking to Melanie Rupflin and Mura Yakerson.

This talk concerns some ideas around the question of when a *-homomorphism into a quotient C*-algebra lifts. Lifting of *-homomorphisms arises prominently in the notions of projectivity and semiprojectivity, which in turn are closely related to stability of relations. Blackadar recently defined the notions of l-open and l-closed C*-algebras, making use of the topological space of *-homomorphisms from a C*-algebra A to another C*-algebra B, with the point-norm topology. I will discuss these properties and present new characterizations of them, which lead to solutions of some problems posed by Blackadar. This is joint work with Dolapo Oyetunbi.

We present a new formulation of string and particle amplitudes that emerges from simple one-dimensional models. The key is a new way to parametrize the positive part of Teichmüller space. It also builds on the results of Mirzakhani for computing Weil-Petterson volumes. The formulation works at all orders in the perturbation series, including non-planar contributions. The relationship between strings and particles is made manifest as a "tropical limit". The results are well adapted to studying the scattering of large numbers of particles or amplitudes at high loop order. The talk will in part cover results from arXiv:2309.15913, 2311.09284.

This event is free but requires prior registration. To register, please click here.

Abstract
In the contemporary AI landscape, Large Language Models (LLMs) stand out as game-changers. They redefine not only how we interact with computers via natural language but also how we identify and extract insights from vast, complex datasets. This presentation delves into the nuances of training and customizing LLMs, with a focus on their applications to quantitative finance.


About the speaker
Ioana Boier is a senior principal solutions architect at Nvidia. Her background is in Quantitative Finance and Computer Science. Prior to joining Nvidia, she was the Head of Quantitative Portfolio Solutions at Alphadyne Asset Management, and led research teams at Citadel LLC, BNP Paribas, and IBM T.J. Watson Research. She has a Ph.D. in Computer Science from Purdue University and is the author of over 30 peer-reviewed publications, 15 patents, and the winner of several awards for applied research delivered into products.
View her LinkedIn page

Frontiers in Quantitative Finance is brought to you by the Oxford Mathematical and Computational Finance Group and sponsored by CitiGroup and Mosaic SmartData.
 

One of the fundamental problems in representation theory is determining the representation type of algebras. In this talk, we will introduce the representation type of cyclotomic quiver Hecke algebras, also known as cyclotomic Khovanov-Lauda-Rouquier algebras, especially in affine type A and affine type C. Our main result relies on novel constructions of the maximal dominant weights of integrable highest weight modules over quantum groups. This talk is based on collaborations with Susumu Ariki, Berta Hudak, and Linliang Song.

The goal of this talk is to give you a glimpse of the Langlands program, a central topic at the intersection of algebraic number theory, algebraic geometry and representation theory. I will focus on a celebrated instance of the Langlands correspondence, namely the modularity of elliptic curves. In the first part of the talk, I will give an explicit example, discuss the different meanings of modularity for rational elliptic curves, and mention applications. In the second part of the talk, I will discuss what is known about the modularity of elliptic curves over more general number fields.

Title: Elliptic curves and modularity

Abstract: The goal of this talk is to give you a glimpse of the Langlands program, a central topic at the intersection of algebraic number theory, algebraic geometry and representation theory. I will focus on a celebrated instance of the Langlands correspondence, namely the modularity of elliptic curves. In the first part of the talk, I will give an explicit example, discuss the different meanings of modularity for rational elliptic curves, and mention applications. In the second part of the talk, I will discuss what is known about the modularity of elliptic curves over more general number fields.

Pseudo metrics between persistence modules can be defined starting from Noise Systems [1].  Such metrics are used to compare the modules directly or to extract stable vectorisations. While the stability property directly follows from the axioms of Noise Systems, finding algorithms or closed formulas to compute the distances or associated vectorizations  is often a difficult problem, especially in the multi-parameter setting. In this seminar I will show how extra properties of Noise Systems can be used to define algorithms. In particular I will describe how to compute stable vectorisations with respect to Wasserstein distances [2]. Lastly I will discuss ongoing work (with D. Lundin and R. Corbet) for the computation of a geometric distance (the Volume Noise distance) and associated invariants on interval modules.

[1] M. Scolamiero, W. Chachólski, A. Lundman, R. Ramanujam, S. Oberg. Multidimensional Persistence and Noise, (2016) Foundations of Computational Mathematics, Vol 17, Issue 6, pages 1367-1406. doi:10.1007/s10208-016-9323-y.

[2] J. Agerberg, A. Guidolin, I. Ren and M. Scolamiero. Algebraic Wasserstein distances and stable homological invariants of data. (2023) arXiv: 2301.06484.

Sequential biomedical data is ubiquitous, from time-resolved data about patient encounters in the clinical realm to DNA sequences in the biological domain.  The talk will review our latest work in representation learning from longitudinal data, with a particular focus on finding optimal representations for complex and sparse healthcare data. We show how these representations are useful for comparing patient journeys and finding patients with similar health outcomes. We will also venture into the field of genome engineering, where we build models that work on DNA sequences for predicting editing outcomes for base and prime editors. 

I will discuss logarithmic corrections to various CFT partition functions in the context of the AdS4/CFT3 correspondence for theories arising on the worldvolume of M2-branes. I will use four-dimensional gauged supergravity and heat kernel methods and present general expressions for the logarithmic corrections to the gravitational on-shell action or black hole entropy for a number of different supergravity backgrounds. I will outline several subtleties and puzzles in these calculations and contrast them with a similar analysis of logarithmic corrections performed directly in the eleven-dimensional uplift of a given four-dimensional supergravity background. This analysis suggests that four-dimensional supergravity consistent truncations are not the proper setting for studying logarithmic corrections in AdS/CFT. These results have important implications for the existence of scale-separated AdS vacua in string theory and for effective field theory in AdS more generally.

Roughly speaking, class field theory for a number field K describes the abelianization of its absolute Galois group in terms of the idele class group of K. Geometric class field theory is what we get when K is instead the function field of a smooth projective geometrically connected curve X over a finite field. In this talk, I give a precise statement of geometric class field theory in the unramified case and describe how one can prove it by showing the Picard stack of X is the “free dualizable commutative group stack on X”. A key part is to show that the usual “divisor class group exact sequence“ can be done in families to give the adelic uniformization of the Picard stack by the moduli space of Cartier divisors on X. 

I will recall the Zilber-Pink conjecture for Shimura varieties and give my perspective on current progress towards a proof.

In this talk, we present an algorithm to compute p-adic heights on hyperelliptic curves with good reduction. Our algorithm improves a previous algorithm of Balakrishnan and Besser by being considerably simpler and faster and allowing even degree models. We discuss two applications of our work: to apply the quadratic Chabauty method for rational and integral points on hyperelliptic curves and to test the p-adic Birch and Swinnerton-Dyer conjecture in examples numerically. This is joint work with Steffen Müller.