Past

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.

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).

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.

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.

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.

Here is the program.

Monumo is interested in computing physical properties of electric motors (torque, efficiency, back EMF) from their designs (shapes, materials, currents). This involves solving Maxwell's equations (non-linear PDEs). They currently compute the magnetic flux, and then use that to compute the other properties of interest. The main challenge they face is that they want to do this for many, many different designs. There seems to be lots of redundancy here, but exploiting it has proved difficult.

In this talk, we consider the sensitivity to the model uncertainty of an optimization problem. By introducing adapted Wasserstein perturbation, we extend the classical results in a static setting to the dynamic multi-period setting. Under mild conditions, we give an explicit formula for the first order approximation to the value function. An optimization problem with a cost of weak type will also be discussed.

The Zilber-Pink conjecture predicts that there should be only finitely
many algebraic numbers t such that the three elliptic curves with
j-invariants t, -t, 2t are all isogenous to each other.  Using previous
work of Habegger and Pila, it suffices to prove a height bound for such
t.  I will outline the proof of this height bound by viewing periods of
the elliptic curves as values of G-functions.  An innovation in this
work is that both complex and p-adic periods are required.  This is
joint work with Christopher Daw.

Associated to any smooth projective curve C is its degree d Jacobian variety, parametrising isomorphism classes of degree d line bundles on C. Letting the curve vary as well, one is led to the universal Jacobian stack. This stack admits several compactifications over the stack of marked stable curves, depending on the choice of a stability condition. In this talk I will introduce these compactified universal Jacobians, and explain how their moduli spaces can be constructed using Geometric Invariant Theory (GIT). This talk is based on arXiv:2210.11457.

Combining the classical theory of optimal transport with modern operator splitting techniques, I will present a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media,materials science, and biological swarming. Using the JKO scheme, along with the Benamou-Brenier dynamical characterization of the Wasserstein distance, we reduce computing the solution of these evolutionary PDEs to solving a sequence of fully discrete minimization problems, with strictly convex objective function and linear constraint. We compute the minimizer of these fully discrete problems by applying a recent, provably convergent primal dual splitting scheme for three operators. By leveraging the PDE’s underlying variational structure, ourmethod overcomes traditional stability issues arising from the strong nonlinearity and degeneracy, and it is also naturally positivity preserving and entropy decreasing. Furthermore, by transforming the traditional linear equality constraint, as has appeared in previous work, into a linear inequality constraint, our method converges in fewer iterations without sacrificing any accuracy. We prove that minimizers of the fully discrete problem converge to minimizers of the continuum JKO problem as the discretization is refined, and in the process, we recover convergence results for existing numerical methods for computing Wasserstein geodesics. Simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our numerical method will be shown.

Electric 1-form symmetries are generically broken in gauge theories with Chern-Simons terms. In this talk we discuss how infinite subsets of these symmetries become non-invertible topological defects. Time permitting we will also discuss generalizations and applications to the Swampland program in relation to the completeness hypothesis.