Past

We present a computationally tractable method for simulating arbitrage free implied volatility surfaces. Our approach conciliates static arbitrage constraints with a realistic representation of statistical properties of implied volatility co-movements.
We illustrate our method with two examples. First, we propose a dynamic factor model for the implied volatility surface, and show how our method may be used to remove static arbitrage from model scenarios. As a second example, we propose a nonparametric generative model for implied volatility surfaces based on a Generative Adversarial Network (GAN).

We will construct several algebras of functions definable in R_{an,\exp} which are stable under parametric integration. 

Given one such algebra A, we will study the parametric Mellin and Fourier transforms of the functions in A. These are complex-valued oscillatory functions, thus we leave the realm of o-minimality. However, we will show that some of the geometric tameness of the functions in A passes on to their integral transforms.

Dispersion relations for S-matrices and CFT correlators translate UV consistency into bounds on IR observables. In this talk, I will begin with briefly introducing dispersionrelations in 2D flat space which will guide the analogous discussion in AdS₂/CFT₁. I will introduce a set of functionals acting on the 1D CFT. These will allow us to prove bounds on higher-derivative couplings in weakly coupled non-gravitational EFTs in AdS₂. At the leading order in the bulk-point limit, the bounds agree with the flat-space result. Furthermore we can compute the leading universal effect of finite AdS radius on the bounds.

Direct Multisearch (DMS) is a well-known multiobjective derivative-free optimization class of methods, with competitive computational implementations that are often successfully used for benchmark of new algorithms and in practical applications. As a directional direct search method, its structure is organized in a search step and a poll step, being the latter responsible for its convergence. A first implementation of DMS was released in 2010. Since then, the algorithmic class has continued to be analyzed from the theoretical point of view and new improvements have been proposed for the numerical implementation. Worst-case-complexity bounds have been derived, a search step based on polynomial models has been defined, and parallelization strategies have successfully improved the numerical performance of the code, which has also shown to be competitive for multiobjective derivative-based problems. In this talk we will survey the algorithmic structure of this class of optimization methods, the main theoretical properties associated to it and report numerical experiments that validate its numerical competitiveness.

It has been known for some time that the contact sets between
self-avoiding idealised tubes (i.e. with exactly circular, normal
cross-sections) in various highly compact, tight structures comprise
double lines of contact. I will re-visit those results for two canonical
examples, namely the orthogonal clasp and the ideal trefoil knot. I will
then show experimental and 3D FEM simulation data for deformable elastic
tubes (obtained within the group of Pedro Reis at the EPFL) which
reveals that the ideal contact set lines bound (in a non-rigorous sense)
the actual contact patches that arise in reality.

[1] The shapes of physical trefoil knots, P. Johanns, P. Grandgeorge, C.
Baek, T.G. Sano, J.H. Maddocks, P.M. Reis, Extreme Mechanics Letters 43
(2021), p. 101172, DOI:10.1016/j.eml.2021.101172
[2]  Mechanics of two filaments in tight orthogonal contact, P.
Grandgeorge, C. Baek, H. Singh, P. Johanns, T.G. Sano, A. Flynn, J.H.
Maddocks, and P.M. Reis, Proceedings of the National Academy of Sciences
of the United States of America 118, no. 15 (2021), p. e2021684118
DOI:10.1073/pnas.2021684118

In recent years, a tend of higher category theory is growing from multiple areas of research throughout mathematics, physics and theoretical computer science. Guided by Cobordism Hypothesis, I would like to introduce some basics of `higher representation theory’, i.e. the part of higher category theory where we focus on the fundamental objects: `finite dimensional’ linear n-categories. If time permits, I will also introduce some recent progress in linear higher categories and motivations from condensed matter physics.

It is well known that if a group von Neumann algebra of a (nontrivial) discrete group is a factor, then it is a factor of type II_1. During the talk, I will answer the following question: which types appear as types of injective factors being group von Neumann algebras of discrete quantum groups (or looking from the dual perspective - von Neumann algebras of bounded functions on compact quantum groups)? An important object in our work is the subgroup of real numbers t for which the scaling automorphism tau_t is inner. This is joint work with Piotr Sołtan.

I will present two examples of log-correlated fields in 2 dimensions. It is well known that the log-characteristic polynomial of a uniform unitary matrix converges toward a 1 dimensional log-correlated field, and our first example will be obtained from a dynamical version of this model. The second example will be obtained from a radically different construction, based on the Brownian loop soup that we will introduce. It will lead to a whole family of log-correlated fields. We will focus on the description of the behaviour of these objects, more than on rigorous details.

This is joint work with Corey Bregman. We study the homotopy type of embedding spaces of unparameterised links, inspired by work of Brendle and Hatcher. We obtain a simple description of the fundamental group of the embedding space, which I will describe for you. Our main tool is a homotopy equivalent semi-simplicial space of separating spheres. As I will explain, this is a combinatorial object that provides a gateway to studying the homotopy type of embedding spaces of split links via the homotopy type of their individual pieces. 

A graph has a pair $(m,f)$ if it has an induced subgraph on $m$ vertices and $f$ edges. We write $(n,e)\rightarrow (m,f)$  if any graph on $n$ vertices and $e$ edges has a pair $(m,f)$.  Let  $$S(n,m,f)=\{e: ~(n,e)\rightarrow (m,f)\} ~{\rm and}$$     $$\sigma(m,f) =   \limsup_{n\rightarrow \infty}\frac{ |S(n,m,f)|}{\binom{n}{2}}.$$ These notions were first introduced and investigated by Erdős, Füredi, Rothschild, and Sós. They found five pairs $(m,f)$ with  $\sigma(m,f)=1$ and showed that for all other pairs $\sigma(m,f)\leq 2/3$.  We extend these results in two directions.

First, in a joint work with Weber, we show that not only $\sigma(m,f)$ can be zero, but also $S(n,m,f)$  could be empty for some pairs $(m,f)$ and any sufficiently large $n$. We call such pairs $(m,f)$ absolutely avoidable.

Second, we consider a natural analogue $\sigma_r(m,f)$ of $\sigma(m,f)$ in the setting of $r$-uniform hypergraphs.  Weber showed that for any $r\geq 3$ and  $m>r$,  $\sigma_r(m,f)=0$ for most values of $f$.  Surprisingly, it was not immediately clear whether there are nontrivial pairs $(m,f)$,  $(f\neq 0$, $f\neq \binom{m}{r}$,  $r\geq 3$),  for which $\sigma_r(m,f)>0$. In a joint work with Balogh, Clemen, and Weber we show that $\sigma_3(6,10)>0$ and conjecture that in the $3$-uniform case $(6,10)$ is the only such pair.

In this talk we will present some work in progress generalising Lubin-Tate formal group laws to higher dimensions. There have been some other generalisations, but ours is different in that the ring over which the formal group law is defined changes as the dimension increases. We will state some conjectures about these formal group laws, including their relationship to the Drinfeld tower over the p-adic upper half plane, and provide supporting evidence for these conjectures.

A Hele-Shaw Newton's cradle: Circular bubbles in a Hele-Shaw channel. (Daniel Booth)

We present a model for the motion of approximately circular bubbles in a Hele-Shaw cell. The bubble velocity is determined by a balance between the hydrodynamic pressures from the external flow and the drag due to the thin films above and below the bubble. We find that the qualitative behaviour depends on a dimensionless parameter and is found to agree well with experimental observations.  Furthermore, we show how the effects of interaction with cell boundaries and/or other bubbles also depend on the value of this dimensionless parameter For example, in a train of three identical bubbles travelling along the centre line, the middle bubble either catches up with the one in front or is caught by the one behind, forming what we term a Hele-Shaw Newton's cradle.
 

Reciprocity in Fluids (Matthew Cotton)

Reciprocity is a useful, and often underused, way to calculate integrated quantities when a to solution to a related problem is known. In the remaining time, I will overview these ideas and give some example use cases

The Riemann hypothesis (RH) is one of the great open problems in
mathematics. It arose from the study of prime numbers in an analytic
context, and—as often occurs in mathematics—developed analogies in an
algebraic setting, leading to the influential Weil conjectures. RH for
curves over finite fields was proven in the 1940’s by Weil using
algebraic-geometric methods, and later reproven by Stepanov and
Bombieri by elementary means. In this talk, we use RH for curves to
prove the Weil bound for certain (Kloosterman) exponential sums, which
in turn is a fundamental tool in the study of prime numbers.

What is the largest deterministic amount of time T∗ that a suitably normalized martingale X can be kept inside a convex body K in Rd? We show, in a viscosity framework, that T∗ equals the time it takes for the relative boundary of K to reach X(0) as it undergoes a geometric flow that we call (positive) minimum curvature flow. This result has close links to the literature on stochastic and game representations of geometric flows. Moreover, the minimum curvature flow can be viewed as an arrival time version of the Ambrosio–Soner codimension-(d − 1) mean curvature flow of the 1-skeleton of K. We present very preliminary sampling-based numerical approximations to the solution of the corresponding PDE. The numerical part is work in progress.

This work is based on a collaboration with Camilo Garcia Trillos, Martin Larsson, and Yufei Zhang.

THIS TALK IS CANCELLED DUE TO ILLNESS -- In this talk I will present classification results for rigid Frobenius algebras in Dijkgraaf–Witten categories ℨ( Vec(G)ᵚ ) over a field of arbitrary characteristic, generalising existing results by Davydov-Simmons. For this purpose, we provide a braided Frobenius monoidal functor from ℨ ( Vect(H)ᵚˡᴴ ) to ℨ( Vec(G)ᵚ ) for any subgroup H of G. I will also discuss about their categories of local modules, which are modular tensor categories  by results of Kirillov–Ostrik in the semisimple case and Laugwitz–Walton in the general case. Joint work with Robert Laugwitz (Nottingham) and Sam Hannah (Cardiff).

Donaldson-Thomas theory is classically defined for moduli spaces of sheaves over a Calabi-Yau threefold. Thanks to recent foundational work of Cao-Leung, Borisov-Joyce and Oh-Thomas, DT theory has been extended to Calabi-Yau 4-folds. We discuss how, in this context, one can define natural K-theoretic refinements of Donaldson-Thomas invariants (counting sheaves on Hilbert schemes) and Pandharipande-Thomas invariants (counting sheaves on moduli spaces of stable pairs) and how — conjecturally — they are related. Finally, we introduce an extension of DT invariants to Calabi-Yau 4-orbifolds, and propose a McKay-type correspondence, which we expect to be suitably interpreted as a wall-crossing phenomenon. Joint work (in progress) with Yalong Cao and Martijn Kool.

In this talk, I will discuss and introduce deep insight from the dynamical system perspective to understand the convergence guarantees of first-order algorithms involving inertial features for convex optimization in a Hilbert space setting.

Such algorithms are widely popular in various areas of data science (data processing, machine learning, inverse problems, etc.).
They can be viewed discrete as time versions of an inertial second-order dynamical system involving different types of dampings (viscous damping,  Hessian-driven geometric damping).

The dynamical system perspective offers not only a powerful way to understand the geometry underlying the dynamic, but also offers a versatile framework to obtain fast, scalable and new algorithms enjoying nice convergence guarantees (including fast rates). In addition, this framework encompasses known algorithms and dynamics such as the Nesterov-type accelerated gradient methods, and the introduction of time scale factors makes it possible to further accelerate these algorithms. The framework is versatile enough to handle non-smooth and non-convex objectives that are ubiquituous in various applications.

Only a decade ago the detection of gravitational waves seemed like a fantasy to most, and merely a handful of 
people in the world believed in the validity and even great potential of using the powerful framework of EFT, and 
more generally -- advances in QFT to study gravity theory for real-world gravitational waves. I will present the 
significant advancement accomplished uniquely via the tower of EFTs with the EFT of spinning gravitating objects, 
and the incorporation of QFT advances, which my work has pioneered since those days. Today, only 6 years after 
the official birth of precision gravity with a rapidly growing influx of gravitational-wave data, and a decade of great 
theoretical progress, the power and insight of using modern QFT for real-world gravity have become incontestable.

Your supervisor is the person you will interact with on a scientific level most of all during your studies here. As a result, it is vital that you establish a good working relationship. But how should you do this? In this session we discuss tips and tricks for getting the most out of your supervisions to maximize your success as a researcher. Note that this session will have no faculty in the audience in order to allow people to speak openly about their experiences. 

Cell-to-cell variability is often a primary source of variability in experimental data. Yet, it is common for mathematical analysis of biological systems to neglect biological variability by assuming that model parameters remain fixed between measurements. In this two-part talk, I present new mathematical and statistical tools to identify cell-to-cell variability from experimental data, based on mathematical models with random parameters. First, I identify variability in the internalisation of material by cells using approximate Bayesian computation and noisy flow cytometry measurements from several million cells. Second, I develop a computationally efficient method for inference and identifiability analysis of random parameter models based on an approximate moment-matched solution constructed through a multivariate Taylor expansion. Overall, I show how analysis of random parameter models can provide more precise parameter estimates and more accurate predictions with minimal additional computational cost compared to traditional modelling approaches.

Here is the program.

Auslander-Reiten theory provides lots of powerful tools to study algebras of finite representation type. One of these is Auslander correspondence, a well-known result establishing a bijection between the class of algebras of finite representation type and their corresponding Auslander algebras. I will present these classical results in a key example: the class of algebras associated to quivers of type A_n. I will talk about well-known results regarding their derived equivalence with another class of algebras, and I will present a more recent result regarding the perfect derived category of the Auslander algebras of type A_n.