Past

We introduce a classification tool for mathematical theories based on Robinson's notion of model companionship; roughly the idea is to attach to a mathematical theory $T$ those signatures $L$ such that $T$ as axiomatized in $L$ admits a model companion. We also introduce a slight strengthening of model companionship (absolute model companionship - AMC) which characterize those model companionable $L$-theories $T$ whose model companion is axiomatized by the $\Pi_2$-sentences for $L$ which are consistent with the universal theory of any $L$-model of $T$.

We use the above to analyze set theory, and we show that the above classification tools can be used to extract (surprising?) information on the continuum problem.

The Gromov boundary of a hyperbolic group is a useful topological invariant, the properties of which can encode all sorts of algebraic information. It has found application to some algorithmic questions, such as finding finite splittings (Dahmani-Groves) and, more recently, computing JSJ-decompositions (Barrett). In this talk we will introduce the boundary of a hyperbolic group. We'll outline how one can approximate the boundary with "large spheres" in the Cayley graph, in order to search for topological features. Finally, we will also discuss how this idea is applied in the aforementioned results. 

Aimed: An introduction to the nonlinear theory of hyperbolic PDEs, as well as its close connections with the other areas of mathematics and wide range of applications in the sciences.

TBA

In 2013, Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees found a remarkable correspondence between SCFTs in 4d with N ≥ 2 and vertex algebras. The chiral algebras of class S, i.e. the vertex algebras associated to theories of Class S, are of particular interest as they exhibit rich algebraic structures arising from the requirement of generalised S-duality. I will explain a mathematical construction of these vertex algebras, due to Arakawa, that is remarkably uniform and requires no knowledge of the underlying SCFT. Time permitting, I will detail a recent generalisation of this construction to the case of the chiral algebras of class S with outer automorphism twist lines.

Non-shearing congruences of null geodesics on four-dimensional Lorentzian manifolds are fundamental objects of mathematical relativity. Their prominence in exact solutions to the Einstein field equations is supported by major results such as the Robinson, Goldberg-Sachs and Kerr theorems. Conceptually, they lie at the crossroad between Lorentzian conformal geometry and Cauchy-Riemann geometry, and are one of the original ingredients of twistor theory.
 
Identified as involutive totally null complex distributions of maximal rank, such congruences generalise to any even dimensions, under the name of Robinson structures. Nurowski and Trautman aptly described them as Lorentzian analogues of Hermitian structures. In this talk, I will give a survey of old and new results in the field.

The non-backtracking matrix and its eigenvalues have many applications in network science and graph mining. For example, in network epidemiology, the reciprocal of the largest eigenvalue of the non-backtracking matrix is a good approximation for the epidemic threshold of certain network dynamics. In this work, we develop techniques that identify which nodes have the largest impact on the leading non-backtracking eigenvalue. We do so by studying the behavior of the spectrum of the non-backtracking matrix after a node is removed from the graph, which can be thought of as immunizing a node against the spread of disease. From this analysis we derive a centrality measure which we call X-degree, which is then used to predict which nodes have a large influence in the epidemic threshold. Finally, we discuss work currently in progress on connections with eigenvector localization and percolation theory.

We tie together two natural but, a priori, different themes. As a starting point consider Erdős and Simonovits's classical edge stability for an $(r + 1)$-chromatic graph $H$. This says that any $n$-vertex $H$-free graph with $(1 − 1/r + o(1)){n \choose 2}$ edges is close to (within $o(n^2)$ edges of) $r$-partite. This is false if $1 − 1/r$ is replaced by any smaller constant. However, instead of insisting on many edges, what if we ask that the $n$-vertex graph has large minimum degree? This is the basic question of minimum degree stability: what constant $c$ guarantees that any $n$-vertex $H$-free graph with minimum degree greater than $cn$ is close to $r$-partite? $c$ depends not just on chromatic number of $H$ but also on its finer structure.

Somewhat surprisingly, answering the minimum degree stability question requires understanding locally colourable graphs -- graphs in which every neighbourhood has small chromatic number -- with large minimum degree. This is a natural local-to-global colouring question: if every neighbourhood is big and has small chromatic number must the whole graph have small chromatic number? The triangle-free case has a rich history. The more general case has some similarities but also striking differences.

We present models of a vapour bubble produced during ureteroscopy and laser lithotripsy treatment of kidney stones. This common treatment for kidney stones involves passing a flexible ureteroscope containing a laser fibre via the ureter and bladder into the kidney, where the fibre is placed in contact with the stone. Laser pulses are fired to fragment the stone into pieces small enough to pass through an outflow channel. Laser energy is also transferred to the surrounding fluid, resulting in vapourisation and the production of a cavitation bubble.

While in some cases, bubbles have undesirable effects – for example, causing retropulsion of the kidney stone – it is possible to exploit bubbles to make stone fragmentation more efficient. One laser manufacturer employs a method of firing laser pulses in quick succession; the latter pulses pass through the bubble created by the first pulse, which, due to the low absorption rate of vapour in comparison to liquid, increases the laser energy reaching the stone.

As is common in bubble dynamics, we couple the Rayleigh-Plesset equation to an energy conservation equation at the vapour-liquid boundary, and an advection-diffusion equation for the surrounding liquid temperature.1 However, this present work is novel in considering the laser, not only as the cause of nucleation, but as a spatiotemporal source of heat energy during the expansion and collapse of a vapour bubble.
 

Numerical and analytical methods are employed alongside experimental work to understand the effect of laser power, pulse duration and pulse pattern. Mathematically predicting the size, shape and duration of a bubble reduces the necessary experimental work and widens the possible parameter space to inform the design and usage of lasers clinically.

A topological insulator has anomalous boundary spectrum which completely fills up gaps in the bulk spectrum. This ``topologically protected’’ spectral property is a physical manifestation of coarse geometry and index theory ideas. Special examples involve spectral flow and gerbes, related to Hamiltonian anomalies, and they arise experimentally in quantum Hall systems, time-reversal invariant mod-2 insulators, and shallow-water waves.

Gay and Kirby formulated a new way to decompose a (closed, orientable) 4-manifold M, called a trisection.    I’ll describe how to translate from a classical framed link diagram for M to a trisection diagram.   The links so obtained lie on Heegaard surfaces in the 3-sphere,  and have surgeries yielding some number of copies of S^1XS^2.   We can describe families of “elementary" links which have such surgeries, and one can ask whether all links with few components having such surgeries lie in these families.  The answer is almost certainly no.   We nevertheless give a small piece of evidence in favor of a positive answer for a special family of 2-component links.    This is joint work with Rob Kirby.  Gay and Kirby formulated a new way to decompose a (closed, orientable) 4-manifold M, called a trisection.    I’ll describe how to translate from a classical framed link diagram for M to a trisection diagram.   The links so obtained lie on Heegaard surfaces in the 3-sphere,  and have surgeries yielding some number of copies of S^1XS^2.   We can describe families of “elementary" links which have such surgeries, and one can ask whether all links with few components having such surgeries lie in these families.  The answer is almost certainly no.   We nevertheless give a small piece of evidence in favor of a positive answer for a special family of 2-component links.    This is joint work with Rob Kirby.  

Wasserstein distance induces a natural Riemannian structure for the probabilities on the Euclidean space. This insight of classical transport theory is fundamental for tremendous applications in various fields of pure and applied mathematics. We believe that an appropriate probabilistic variant, the adapted Wasserstein distance $AW$, can play a similar role for the class $FP$ of filtered processes, i.e. stochastic processes together with a filtration. In contrast to other topologies for stochastic processes, probabilistic operations such as the Doob-decomposition, optimal stopping and stochastic control are continuous w.r.t. $AW$. We also show that $(FP, AW)$ is a geodesic space, isometric to a classical Wasserstein space, and that martingales form a closed geodesically convex subspace. Finally we consider computational aspects and provide a novel method based on the Sinkhorn algorithm.

The talk is based on articles with Daniel Bartl, Mathias Beiglböck and Stephan Eckstein.

We study the behaviour of anti-self-dual instantons on $\mathbb{R}^3 \times S^1$ (also known as calorons) under codimension-1 collapse, i.e. when the circle factor shrinks to zero length. In this limit, the instanton equation reduces to the well-known Bogomolny equation of magnetic monopoles on $\mathbb{R}^3 $. However, inspired by work of Kraan and van Baal in the mathematical physics literature, we show how $SU(2)$ instantons can be realised as superpositions of monopoles and "rotated monopoles" glued into a singular background abelian configuration consisting of Dirac monopoles of positive and negative charges. I will also discuss generalisations of the construction to calorons with arbitrary structure group and potential applications to the hyperkähler geometry of moduli spaces of calorons. This is joint work with Calum Ross.

The alchemists wanted to create gold, Hilbert wanted an algorithm to solve Diophantine equations, researchers want to make deep learning robust in AI, MATLAB wants (but fails) to detect when it provides wrong solutions to linear programs etc. Why does one not succeed in so many of these fundamental cases? The reason is typically methodological barriers. The history of  science is full of methodological barriers — reasons for why we never succeed in reaching certain goals. In many cases, this is due to the foundations of mathematics. We will present a new program on methodological barriers and foundations of mathematics,  where — in this talk — we will focus on two basic problems: (1) The instability problem in deep learning: Why do researchers fail to produce stable neural networks in basic classification and computer vision problems that can easily be handled by humans — when one can prove that there exist stable and accurate neural networks? Moreover, AI algorithms can typically not detect when they are wrong, which becomes a serious issue when striving to create trustworthy AI. The problem is more general, as for example MATLAB's linprog routine is incapable of certifying correct solutions of basic linear programs. Thus, we’ll address the following question: (2) Why are algorithms (in AI and computations in general) incapable of determining when they are wrong? These questions are deeply connected to the extended Smale’s 9th and 18th problems on the list of mathematical problems for the 21st century. 

Holographic correlation functions are under good analytic control when none of the single trace operators live in long multiplets. This is famously the case for SCFTs with sixteen supercharges but it is also possible to construct examples with eight supercharges by exploiting space filling branes in AdS. In particular, one can study 4d N=2 theories which are related to each other by an S-fold in much the same way that N=3 theories are related to N=4 Super Yang-Mills. I will describe how modern methods provide a window into their correlation functions with an emphasis on anomalous dimensions. To compare the different S-folds we will need to go to one loop, and to go to one loop we will need to account for operator mixing. This provides an example of resolving degeneracy by resolving degeneracy.

This event will be hybrid and will take place in L1 and on Teams. A link will be available 30 minutes before the session begins.

`Conferences and collaboration’ is a Fridays@4 group discussion. The goal is to have an open and honest conversion about the hurdles posed by these things, led by a panel of graduate students and postdocs. Conferences can be both exciting and stressful - they involve meeting new people and learning new mathematics, but can be intimidating new professional experiences. Many of us also will either have never been to one in person, or at least not been to one in the past two years. Optimistically looking towards the world opening up again, we thought it would be a good time to ask questions such as:
-Which talks should I go to?
-How to cope with incomprehensible talks. Is it imposter syndrome or is the speaker just bad?
-Should I/how should I go about introducing myself to more senior people in the field?
-How do you start collaborations? Does it happen at conferences or elsewhere?
-How do you approach workload in collaborations?
-What happens if a collaboration isn’t working out?
-FOMO if you like working by yourself. Over the hour we’ll have a conversation about these hurdles and most importantly, talk about how we can make conferences and collaborations better for everyone early in their careers.

Affine Hecke algebras and their representations play an important role in the representation theory of p-adic groups since they classify smooth representations generated by Iwahori-fixed vectors. The Deligne-Langlands correspondence, which was proved by Kazhdan and Lusztig, parametrises these representations by geometric data on the Langlands dual group. This talk is supposed to be a gentle introduction to this topic. I will also briefly talk about how this correspondence can be lifted to the derived level.

 In addition, another class of cell-cell communication is by long, thin cellular protrusions that are ~100 microns (many cell-lengths) in length and ~100 nanometers (below traditional microscope resolution) in width. These protrusions have been recently discovered in many organisms, including nanotubes humans and airinemes in zebrafish. But, before establishing communication, these protrusions must find their target cell. Here we demonstrate airinemes in zebrafish are consistent with a finite persistent random walk model. We study this model by stochastic simulation, and by numerically solving the survival probability equation using Strang splitting. The probability of contacting the target cell is maximized for a balance between ballistic search (straight) and diffusive (highly curved, random) search. We find that the curvature of airinemes in zebrafish, extracted from live cell microscopy, is approximately the same value as the optimum in the simple persistent random walk model. We also explore the ability of the target cell to infer direction of the airineme’s source, finding the experimentally observed parameters to be at a Pareto optimum balancing directional sensing with contact initiation.

The infamous Ramanujan tau-function is the starting point for many mysterious conjectures and difficult open problems within the realm of modular forms. In this talk, I will discuss some of our recent results pertaining to odd values of the Ramanujan tau-function. We use a combination of tools which include the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Gyory, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves. This is joint work with Mike Bennett (UBC), Adela Gherga (Warwick) and Samir Siksek (Warwick).

In 2008 Pila and Zannier used a Theorem coming from Logic, proven by Pila and Wilkie, to give a new proof of the Manin-Mumford Conjecture, creating a new, powerful way to prove Theorems in Diophantine Geometry. The Pila-Wilkie Theorem gives an upper bound on the number of rational points on analytic varieties which are not algebraic; this bound usually contradicts a Galois-theoretic bound obtained by arithmetic considerations. We show how this technique can be applied to the following problem of Lang: given an irreducible polynomial f(x,y) in C[x,y], if for infinitely many pairs of roots of unity (a,b) we have f(a,b)=0, then f(x,y) is either of the form x^my^n-c or x^m-cy^n for c a root of unity.

Let $A$ be an $n$ by $n$ matrix or a bounded linear operator on a complex Hilbert space $(H, \langle \cdot , \cdot \rangle , \| \cdot \|)$. A closed set $\Omega \subset \mathbb{C}$ is a $K$-spectral set for $A$ if the spectrum of $A$ is contained in $\Omega$ and if, for all rational functions $f$ bounded in $\Omega$, the following inequality holds:
\[\| f(A) \| \leq K \| f \|_{\Omega} ,\]
where $\| \cdot \|$ on the left denotes the norm in $H$ and $\| \cdot \|_{\Omega}$ on the right denotes the $\infty$-norm on $\Omega$. A simple way to obtain a $K$ value for a given set $\Omega$ is to use the Cauchy integral formula and replace the norm of the integral by the integral of the resolvent norm:
\[f(A) = \frac{1}{2 \pi i} \int_{\partial \Omega} ( \zeta I - A )^{-1}
f( \zeta )\,d \zeta \Rightarrow
\| f(A) \| \leq \frac{1}{2 \pi} \left( \int_{\partial \Omega}
\| ( \zeta I - A )^{-1} \|~| d \zeta | \right) \| f \|_{\Omega} .\]
Thus one can always take
\[K = \frac{1}{2 \pi} \int_{\partial \Omega} \| ( \zeta I - A )^{-1} \| | d \zeta | .\]
In M. Crouzeix and A. Greenbaum, Spectral sets: numerical range and beyond, SIAM J. Matrix Anal. Appl., 40 (2019), pp. 1087-1101, different bounds on $K$ were derived.  I will show how these compare to that from the Cauchy integral formula for a variety of applications.  In case $A$ is a matrix and $\Omega$ is simply connected, we can numerically compute what we believe to be the optimal value for $K$ (and, at least, is a lower bound on $K$).  I will show how these values compare with the proven bounds as well.

(joint with  Michel Crouzeix and Natalie Wellen)
 

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