Past

Let X be a smooth projective curve over a finite field equipped with an action of a finite group G. I’ll first briefly introduce the corresponding Artin L-function and a certain equivariant Euler characteristic. The main result will be a precise relation between the epsilon constants appearing in the functional equations of Artin L-functions and that Euler characteristic if the projection X → X/G is at most weakly ramified. This generalises a theorem of Chinburg for the tamely ramified case. I’ll end with some speculations in the arbitrarily wildly ramified case. This is joint work with Helena Fischbacher-Weitz and with Adriano Marmora.

Polynomial preconditioning for linear solvers is well-known but not frequently used in current scientific applications.  Furthermore, polynomial preconditioning has long been touted as well-suited for parallel computing; does this claim still hold in our new world of GPU-dominated machines?  We give details of the GMRES polynomial preconditioner and discuss its simple implementation, its properties such as eigenvalue remapping, and choices such as the starting vector and added roots.  We compare polynomial preconditioned GMRES to related methods such as FGMRES and full GMRES without restarting. We conclude with initial evaluations of the polynomial preconditioner for parallel and GPU computing, further discussing how polynomial preconditioning can become useful to real-word applications.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome

This is a preparatory study for our ultimate goal of understanding the various instabilities associated with an electrodes-coated dielectric membrane that is subject to mechanical stretching and electric loading. Leaving out electric loading for the moment, we consider bifurcations from the homogeneous solution of a circular or square hyperelastic sheet that is subjected to equibiaxial stretching under either force- or displacement-controlled edge conditions. We derive the condition for axisymmetric necking and show, for the class of strain-energy functions considered, that the critical stretch for necking is greater than the critical stretch for the Treloar-Kearsley (TK) instability and less than the critical stretch for the limiting-point instability. Abaqus simulations are conducted to verify the bifurcation conditions and the expectation that the TK instability should occur first under force control, but when the edge displacement is controlled the TK instability is suppressed, and it is the necking instability that will be observed. It is also demonstrated that axisymmetric necking follows a growth/propagation process typical of all such localization problems.

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. 

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.

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.

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.

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.

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 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.

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.

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.  

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.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome

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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A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Connectivity and percolation are two well studied phenomena in random graphs. 

In this talk we will discuss higher-dimensional analogues of connectivity and percolation that occur in random simplicial complexes.

Simplicial complexes are a natural generalization of graphs that consist of vertices, edges, triangles, tetrahedra, and higher dimensional simplexes.

We will mainly focus on random geometric complexes. These complexes are generated by taking the vertices to be a random point process, and adding simplexes according to their geometric configuration.

Our generalized notions of connectivity and percolation use the language of homology - an algebraic-topological structure representing cycles of different dimensions.

In this talk we will discuss recent results analyzing phase transitions related to these topological phenomena. 

Previous joint work with Caroline Terry had identified model-theoretic stability as a sufficient condition for the existence of strong arithmetic regularity decompositions in finite abelian groups, pioneered by Ben Green around 2003. 
Higher-order arithmetic regularity decompositions, based on Tim Gowers’s groundbreaking work on Szemerédi’s theorem in the late 90s, are an essential part of today's arithmetic combinatorics toolkit.
In this talk, I will describe recent joint work with Caroline Terry in which we define a natural higher-order generalisation of stability and prove that it implies the existence of particularly efficient higher-order arithmetic regularity decompositions in the setting of finite elementary abelian groups. If time permits, I will briefly outline some analogous results we obtain in the context of hypergraph regularity decompositions.

We will record some surprising and lesser-known properties of free groups, and use these to give a model theoretic analysis of free group automorphisms and orbits under Aut(F). This will result in a neat geometric description of (a logic-flavoured analogue of) algebraic closures in a free group. An almost immediate corollary will be that elementary subgroups of a free group are free factors.

I will assume no familiarity with first-order logic and model theory - the beginning of the talk will be devoted to familiarize everyone with the few required notions.

In this talk, I will explain how to obtain the space of local operators of a 4d N=2 gauge theory using the category of line operators in the Kapustin twist (holomorphic topological twist). This category is given a precise definition by Cautis-Williams, as the category of equivariant coherent sheaves on the space of Braverman-Finkelberg-Nakajima. We compute the derived endomorphism of the monoidal unit in this category, and show that it coincides with the vacuum module of the Poisson vertex algebra of Oh-Yagi and Butson. The Euler character of this space reproduces the Schur index. I will also explain how to obtain the space of local operators at the junction of minimal Wilson-t’Hooft line operators. Its Euler character can be compared to the index formula of Cordova-Gaiotto-Shao. This is based on arXiv: 2112.12164.

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.

In the late 1980s, Berger and Coburn showed that the Hankel operator $H_f$ on the Segal-Bargmann space of Gaussian square-integrable entire functions is compact if and only if $H_{\bar f}$ is compact using C*-algebra and Hilbert space techniques. I will briefly discuss this and three other proofs, and then consider the question of whether an analogous phenomenon holds for Schatten class Hankel operators. 

I will present a collection of moment computations over the unitary, symplectic and special orthogonal matrix ensembles that I've done throughout my thesis. I will focus on the methods used, the motivation from number theory, the relationship to Painlev\'e equations, and directions for future work.

Measures of discrete curvature are a recent addition to the toolkit of network analysts and data scientists. At the basis lies the idea that networks and other discrete objects exhibit distinct geometric properties that resemble those of smooth objects like surfaces and manifolds, and that we can thus find inspiration in the tools of differential geometry to study these discrete objects. In this talk, I will introduce how this has lead to the development of notions of discrete curvature, and what they are good for. Furthermore, I will discuss our latest results on a new notion of curvature on graphs, based on the effective resistance. These new "resistance curvatures" are related to other well-known notions of discrete curvature (Ollivier, Forman, combinatorial curvature), we find evidence for convergence to continuous curvature in the case of Euclidean random graphs and there is a naturally associated discrete Ricci flow.

A preprint on this work is available on arXiv: https://arxiv.org/abs/2201.06385

The placenta is the essential organ of maternal-fetal interactions, where nutrient, oxygen, and waste exchange occur. In recent studies, differences in the morphology of the placental chorionic surface vascular network (PCSVN) have been associated with developmental disorders such as autism. This suggests that the PCSVN could potentially serve as a biomarker for the early diagnosis and treatment of autism. Studying PCSVN features in large cohorts requires a reliable and automated mechanism to extract the vascular networks. In this talk, we present a method for PCSVN extraction. Our algorithm builds upon a directional multiscale mathematical framework based on a combination of shearlets and Laplacian eigenmaps and can isolate vessels with high success in high-contrast images such as those produced in CT scans. 

I’ll review a new, simpler explanation for the large-scale properties of the
cosmos, presented with L. Boyle in our recent preprint arXiv:2201.07279. The
basic ingredients are elementary and well-known, namely Einstein’s theory of
gravity and Hawking’s method of computing gravitational entropy. The new
twist is provided by the boundary conditions we proposed for big bang-type
singularities, allowing conformal zeros but imposing CPT symmetry and

analyticity at the bang. These boundary conditions, which have significant
overlap with Penrose’s Weyl curvature hypothesis, allow gravitational
instantons for universes with Lambda, massless radiation and space
curvature, of either sign, from which we are able to infer a gravitational
entropy. We find the gravitational entropy can exceed the de Sitter entropy
and that, to the extent that it does, the most probable large-scale geometry
for the universe is flat, homogeneous and isotropic. I will briefly
summarise our earlier work showing how the gauge-fermion Lagrangian of the
standard model may be reconciled with Weyl symmetry and a small cosmological
constant, at leading order, provided there are precisely three generations
of fermions. The same mechanism generates scale-invariant primordial
perturbations. The cosmic dark matter consists of a right-handed neutrino.
In summary, we have taken significant steps towards a new, highly principled
and testable theory of cosmology.

The soap bubble theorem says that a closed, embedded surface of the Euclidean space with constant mean curvature must be a round sphere. Especially in real-life problems it is of importance whether and to what extent this phenomenon is stable, i.e. when a surface with almost constant mean curvature is close to a sphere. This problem has been receiving lots of attention until today, with satisfactory recent solutions due to Magnanini/Poggesi and Ciraolo/Vezzoni.
The purpose of this talk is to discuss further problems of this type and to provide two approaches to their solutions. The first one is a new general approach based on stability of the so-called "Nabelpunktsatz". The second one is of variational nature and employs the theory of curvature flows. 

Minimal Lagrangian maps play an important role in Teichmüller theory, with important existence and uniqueness results for hyperbolic surfaces obtained by Labourie, Schoen, Bonsante-Schlenker, Toulisse and others. In positive curvature, it is thus natural to ask whether one can find minimal Lagrangian diffeomorphisms between two spherical surfaces with cone points. In this talk we will show that the answer is negative, unless the two surfaces are isometric. As an application, we obtain a generalization of Liebmann’s theorem for branched immersions of constant curvature in Euclidean space. This is joint work with Christian El Emam.

Let $\Gamma$ be a finite subgroup of $SL(2, \mathbb{C})$. We can attach several different moduli spaces to the action of $\Gamma$ on $\mathbb{C}^2$, and we show how Nakajima's quiver varieties provide constructions of them. The definition of such a quiver variety depends on a stability parameter, and we are especially interested in what happens when this parameter moves into a specific ray in its associated wall-and-chamber structure. Some of the resulting quiver varieties can be understood as moduli spaces of certain framed sheaves on an appropriate stacky compactification of the Kleinian singularity $\mathbb{C}^2/\Gamma$. As a special case, this includes the punctual Hilbert schemes of $\mathbb{C}^2/\Gamma$.

Much of this is joint work with A. Craw, Á. Gyenge, and B. Szendrői.

Blind deconvolution problems are ubiquitous in many areas of imaging and technology and have been the object of study for several decades. Recently, motivated by the theory of compressed sensing, a new viewpoint has been introduced, motivated by applications in wireless application, where a signal is transmitted through an unknown channel. Namely, the idea is to randomly embed the signal into a higher dimensional space before transmission. Due to the resulting redundancy, one can hope to recover both the signal and the channel parameters. In this talk we analyze convex approaches based on lifting as they have first been studied by Ahmed et al. (2014). We show that one encounters a fundamentally different geometric behavior as compared to generic bilinear measurements. Namely, for very small levels of deterministic noise, the error bounds based on common paradigms no longer scale linearly in the noise level, but one encounters dimensional constants or a sublinear scaling. For larger - arguably more realistic - noise levels, in contrast, the scaling is again near-linear.

This is joint work with Yulia Kostina (TUM) and Dominik Stöger (KU Eichstätt-Ingolstadt).

Great progress has been made recently in exploiting categorical/topological/higher symmetries in quantum field theory. I will explain how the same structure is realised directly in the lattice models of statistical mechanics, generalizing Kramers-Wannier duality to a wide class of models. In particular, I will give an overview of my work with Aasen and Mong on using fusion categories to find and analyse lattice topological defects in two and 1+1 dimensions.  These defects possess a variety of remarkable properties. Not only is the partition function is independent of deformations of their path, but they can branch and fuse in a topologically invariant fashion.  The universal behaviour under Dehn twists gives exact results for scaling dimensions, while gluing a topological defect to a boundary allows universal ratios of the boundary g-factor to be computed exactly on the lattice.  I also will describe how terminating defect lines allows the construction of fractional-spin conserved currents, giving a linear method for Baxterization, I.e. constructing integrable models from a braided tensor category.

We will consider line defects in d-dimensional CFTs. The ambient CFT places nontrivial constraints on renormalization group flows on such line defects. We will see that the flow on line defects is consequently irreversible and furthermore a canonical decreasing entropy function exists. This construction generalizes the g theorem to line defects in arbitrary dimensions. We will demonstrate this generalization in some concrete examples, including a flow between Wilson loops in 4 dimensions, and an O(3) bosonic theory coupled to an impurity in the large spin representation of the bulk global symmetry.

Topological data analysis is starting to establish itself as a powerful and effective framework in machine learning , supporting the analysis of neural networks, but also driving the development of novel algorithms that incorporate topological characteristics. As a problem class, graph representation learning is of particular interest here, since graphs are inherently amenable to a topological description in terms of their connected components and cycles. This talk will provide
an overview of how to address graph learning tasks using machine learning techniques, with a specific focus on how to make such techniques 'topology-aware.' We will discuss how to learn filtrations for graphs and how to incorporate topological information into modern graph neural networks, resulting in provably more expressive algorithms. This talk aims to be accessible to an audience of TDA enthusiasts; prior knowledge of machine learning is helpful but not required.

The Bruhat-Tits building is a mysterious combinatorial gadget that encodes key information about the structure and representation theory of a p-adic group. In this talk we will talk about apartments, buildings, and all the furnishings therein to hopefully demystify this beautiful subject.