Past

Movie Me would like offer dynamical pricing for movie tickets, considering consumer’s demand for the movie, showtime and lead time before the show begins, such that the overall quantity of tickets sold is maximized. We encourage all interested party to join us and especially those interested in data science, optimization and mathematical finance.

If $Q(x_0,x_1,x_2)$ is a quadratic form, how many solutions, of size at most $B$, does $Q=0$ have? How does this depend on $Q$? We apply the answers to the surface $y_0 Q_0 +y_1 Q_1 = 0$ in $P^1 x P^2$. (Joint work with Dan Loughran.)
 

This talk presents a new framework for Merton’s optimal investment problem which uses the theory of Meyer $\sigma$-fields to allow for signals that possibly warn the investor about impending jumps. With strategies no longer predictable, some care has to be taken to properly define wealth dynamics through stochastic integration. By means of dynamic programming, we solve the problem explicitly for power utilities. In a case study with Gaussian jumps, we find, for instance, that an investor may prefer to disinvest even after a mildly positive signal. Our setting also allows us to investigate whether, given the chance, it is better to improve signal quality or quantity and how much extra value can be generated from either choice.
This talk is based on joint work with Peter Bank.

Physics-informed neural networks (PINNs) [2] are a solution method for solving boundary value problems based on differential equations (PDEs). The key idea of PINNs is to incorporate the residual of the PDE as well as boundary conditions into the loss function of the neural network. This provides a simple and mesh-free approach for solving problems relating to PDEs. However, a key limitation of PINNs is their lack of accuracy and efficiency when solving problems with larger domains and more complex, multi-scale solutions. 

In a more recent approach, Finite Basis Physics-Informed Neural Networks (FBPINNs) [1], the authors use ideas from domain decomposition to accelerate the learning process of PINNs and improve their accuracy in this setting. In this talk, we show how Schwarz-like additive, multiplicative, and hybrid iteration methods for training FBPINNs can be developed. Furthermore, we will present numerical experiments on the influence on convergence and accuracy of these different variants. 

This is joint work with Alexander Heinlein (Delft) and Benjamin Moseley (Oxford).

References 
1.    [1]  B. Moseley, A. Markham, and T. Nissen-Meyer. Finite basis physics- informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations. arXiv:2107.07871, 2021. 
2.    [2]  M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686–707, 2019.

We will be joined by Charlotte Turner-Smith to discuss issues surrounding harassment and mental health, and how the department is helping to tackle these.

Many microorganisms must navigate strange biological environments whose physics are unique and counter-intuitive, with wide-ranging consequences for evolutionary biology and human health. Mucus, for instance, behaves like both a fluid and an elastic solid. This can affect locomotion dramatically, which can be highly beneficial (e.g. for mammalian spermatozoa swimming through cervical fluid) or extremely problematic (e.g. the Lyme disease spirochete B. burgdorferi swimming through the extracellular matrix of human skin). Mathematical modeling and numerical simulations continue to provide new fundamental insights about the biological world in and around us and point toward new possibilities in biomedical engineering. These complex fluid phenomena can either enhance or retard a microorganism's swimming speed, and can even change the direction of swimming, depending on the body geometry and the properties of the fluid. We will discuss analytical and numerical insights into swimming through model viscoelastic (Oldroyd-B) and liquid-crystalline (Ericksen-Leslie) fluids, with a special focus on the important and in some cases dominant roles played by the presence of nearby boundaries.

We review a few instances in which the first $\ell^2$ Betti number of a group is a profinite invariant and we discuss some applications and open problems.

From each point of a Poisson point process start growing a balloon at rate 1. When two balloons touch, they pop and disappear. Will balloons reach the origin infinitely often or not? We answer this question for various underlying spaces. En route we find a new(ish) 0-1 law, and generalize bounds on independent sets that are factors of IID on trees. Joint work with Omer Angel and Gourab Ray.

The classification by K-theory and traces of the category of simple, separable, nuclear, Z-stable C*-algebras satisfying the UCT is an extraordinary feat of mathematics. What's more, it provides powerful machinery for the analysis of the internal structure of these regular C*-algebras. In this talk, I will explain one such application of classification: In the subclass of classifiable C*-algebras consisting of those for which the simplex of tracial states is nonempty, with extremal boundary that is compact and has the structure of a connected topological manifold, automorphisms can be shown to be generically tracially chaotic. Using similar ideas, I will also show how certain stably projectionless C*-algebras can be described as crossed products.

In 1983 Thomassen conjectured that every graph of sufficiently large average degree has a subgraph of large girth and large average degree. While this conjecture remains open, recent evidence suggests that something stronger might be true; perhaps the subgraph can be made induced when a clique and biclique are forbidden. We overview our proof for removing 4-cycles from $K_{t,t}$-free bipartite graphs. Moreover, we discuss consequences to tau-boundedness, which is an analog of chi-boundedness.

The Calogero-Painlevé systems were introduced in 2001 by K. Takasaki as a natural generalization of the classical Painlevé equations to the case of the several Painlevé “particles” coupled via the Calogero type interactions. In 2014, I. Rumanov discovered a remarkable fact that a particular case of the Calogero– Painlevé II equation describes the Tracy-Widom distribution function for the general $\beta$-ensembles with the even values of parameter $\beta$. in 2017 work of M. Bertola, M. Cafasso , and V. Rubtsov, it was proven that all Calogero-Painlevé systems are Lax integrable, and hence their solutions admit a Riemann-Hilbert representation. This important observation has opened the door to rigorous asymptotic analysis of the Calogero-Painlevé equations which in turn yields the possibility of rigorous evaluation of the asymptotic behavior of the Tracy-Widom distributions for the values of $\beta$ beyond the classical $\beta =1, 2, 4$. In the talk these recent developments will be outlined with a special focus on the Calogero-Painlevé system corresponding to $\beta = 6$. This is a joint work with Andrei Prokhorov.

I will discuss some new invariants of 2-complexes. They are inspired by recent developments in the theory of one-relator groups, but also have the potential to unify the theories of many well-studied families including small-cancellation presentation complexes, CAT(0) 2-complexes and 3-manifold spines, in addition to the motivating examples of one-relator presentation complexes. The fundamental result is that these invariants are the extrema of explicit linear-programming problems, and in particular are rational, computable and realised. The definitions suggest a conjectural “map” of 2-complexes, which I will attempt to describe.
 

Spectral methods offer a tractable, global framework for clustering in graphs via eigenvector computations on graph matrices. Hypergraph data, in which entities interact on edges of arbitrary size, poses challenges for matrix representations and therefore for spectral clustering. We study spectral clustering for arbitrary hypergraphs based on the hypergraph nonbacktracking operator. After reviewing the definition of this operator and its basic properties, we prove a theorem of Ihara-Bass type which allows eigenpair computations to take place on a smaller matrix, often enabling faster computation. We then propose an alternating algorithm for inference in a hypergraph stochastic blockmodel via linearized belief-propagation which involves a spectral clustering step, again using nonbacktracking operators. We provide proofs related to this algorithm that both formalize and extend several previous results. We pose several conjectures about the limits of spectral methods and detectability in hypergraph stochastic blockmodels in general, supporting these with in-expectation analysis of the eigeinpairs of our studied operators. We perform experiments with real and synthetic data that demonstrate the benefits of hypergraph methods over graph-based ones when interactions of different sizes carry different information about cluster structure.

Joint work with Nicole Eikmeier (Grinnell) and Jamie Haddock (Harvey Mudd).

I will discuss algebraic conditions that for a given group guarantee or characterize the vanishing of cohomology in a given degree with coefficients in any unitary representation. These conditions will be expressed in terms positivity of certain elements over group algebras, where positivity is meant as being a sum of hermitian squares. I will explain how conditions like this can be used to give computer-assisted proofs of vanishing of cohomology. 

The Chabauty-Kim method is an effective algorithm for finding the $S$-integral points of hyperbolic curves by directly using the hyperbolicity in group-cohomological arguments. Central objects in the theory are affine spaces known as a Selmer schemes. We'll introduce the CK method and Selmer schemes, and demonstrate some additional structures possessed by Selmer schemes which can aid in implementing the CK method.
 

Two CW-complexes are simple homotopy equivalent if they are related by a sequence of collapses and expansions of cells. It implies homotopy equivalent as is implied by homeomorphic. This notion proved extremely useful in manifold topology and is central to the classification of non-simply connected manifolds up to homeomorphism. I will present the first examples of two 4-manifolds which are homotopy equivalent but not simple homotopy equivalent, as well as in all higher even dimensions. The examples are constructed using surgery theory and the s-cobordism theorem, and are distinguished using methods from algebraic number theory and algebraic K-theory. I will also discuss a number of new directions including progress on classifying the possible fundamental groups for which examples exist. This is joint work with Csaba Nagy and Mark Powell.

In this talk, we present recent results on an anisotropic variant of the Kardar-Parisi-Zhang equation, the Anisotropic KPZ equation (AKPZ), in the critical spatial dimension d=2. This is a singular SPDE which is conjectured to capture the behaviour of the fluctuations of a large family of random surface growth phenomena but whose analysis falls outside of the scope not only of classical stochastic calculus but also of the theory of Regularity Structures and paracontrolled calculus. We first consider a regularised version of the AKPZ equation which preserves the invariant measure and prove the conjecture made in [Cannizzaro, Erhard, Toninelli, "The AKPZ equation at stationarity: logarithmic superdiffusivity"], i.e. we show that, at large scales, the correlation length grows like t1/2 (log t)1/4 up to lower order correction. Second, we prove that in the so-called weak coupling regime, i.e. the equation regularised at scale N and the coefficient of the nonlinearity tuned down by a factor (log N)-1/2, the AKPZ equation converges to a linear stochastic heat equation with renormalised coefficients. Time allowing, we will comment on how some of the techniques introduced can be applied to other SPDEs and physical systems at and above criticality. 

Labourie proved that every Hitchin representation into PSL(n,R) gives rise to an equivariant minimal surface in the corresponding symmetric space. He conjectured that uniqueness holds as well (this was known for n=2,3), and explained that if true, then the Hitchin component admits a mapping class group equivariant parametrization as a holomorphic vector bundle over Teichmüller space.

In this talk, we will define Hitchin representations, Higgs bundles, and minimal surfaces, and give the background for the Labourie conjecture. We will then explain that the conjecture fails for n at least 4, and point to some future questions and conjectures.

To be a mathematicus in 15th- and 16th-century Europe often meant practising as an astrologer. Far from being an unwelcome obligation, or simply a means of paying the rent, astrology frequently represented a genuine form of mathematical engagement. This is most clearly seen by examining changing definitions of one of the key elements of horoscope construction: the astrological houses. These twelve houses are divisions of the zodiac circle and their character fundamentally affects the significance of the planets which occupy them at any particular moment in time. While there were a number of competing systems for defining the houses, one system was standard throughout medieval Europe. However, the 16th-century witnessed what John North referred to as a “minor revolution”, as a different technique first developed in the Islamic world but adopted and promoted by Johannes Regiomontanus became increasingly prevalent. My paper reviews this shift in astrological practice and investigates the mathematical values it represents – from aesthetics and geometrical representation to efficiency and computational convenience.

Quantum field theories (QFTs) in d dimensions that posses a (d-1)-form symmetry are conjectured to decompose into disjoint “universes”, each of which is itself a (local and unitary) QFT. I will give an overview of our current understanding of decomposition, and then discuss how this phenomenon occurs in the fusion of condensation defects of certain 3d QFTs. This gives a “microscopic” explanation of why in these instances, the fusion coefficient can be taken as an integer rather than a general TQFT.

Academic research can be challenging and can bring with it difficulties in maintaining good mental health. This session will be led by Rebecca Reed, Mental Health First Aid (MHFA) Instructor, Meditation & Yoga Teacher and Personal Development Coach and owner of wellbeing company Siendo. Rebecca will talk about how we can maintain good mental fitness, recognizing good practices to ensure we avoid mental-health difficulties before they begin. We have deliberately set this session to be at the beginning of the academic year in this spirit. We will also talk about maintaining good mental health specifically in the academic community.   

 I will start this talk with a brief introduction and summary of the outcome of a joint work with James Gabe. An important special case of the main result is that for any countable discrete amenable group G, any two outer G-actions on stable Kirchberg algebras are cocycle conjugate precisely when they are equivariantly KK-equivalent. In the main body of the talk, I will outline the key arguments toward a special case of the 'uniqueness theorem', which is one of the fundamental ingredients in our theory: Suppose we have two G-actions on A and B such that B is a stable Kirchberg algebra and the action on B is outer and equivariantly O_2-absorbing. Then any two cocycle embeddings from A to B are approximately unitarily equivalent. If time permits, I will provide a (very rough) sketch of how this leads to the dynamical O_2-embedding theorem, which implies that such cocycle embeddings always exist in the first place.

A common problem in data science is "use this function defined over this small set to generate predictions over that larger set." Extrapolation, interpolation, statistical inference and forecasting all reduce to this problem. The Kan extension is a powerful tool in category theory that generalizes this notion. In this work we explore several applications of Kan extensions to data science. We begin by deriving simple classification and clustering algorithms as Kan extensions and experimenting with these algorithms on real data. Next, we build more complex and resilient algorithms from these simple parts.

We show that the category of vector bundles over the vertices of a locally finite $\tilde{A}_{n-1}$ building $\Delta$, $Vec(\Delta)$, has the structure of a $Tilt(SL_{2k+1})$ module category. This module category is the $q$-analogue of the $Tilt(SL_{2k+1})$ action on vector bundles over the $sl_n$ weight lattice.  Our construction of the $Tilt(SL_{2k+1})$ action on $Vec(\Delta)$ extends to $Vec(\Delta)^{G}$, its equivariantization, which gives us a class of non-standard $Tilt(SL_{2k+1})$ module categories. When $G$ acts simply transitively, this recovers the fiber functors of Jones.

Rivers choose their size and shape, and spontaneously organize into ramified networks. Yet, they are essentially a channelized flow of water that carries sediment. Based on laboratory experiments, field measurements and simple theory, we will investigate the basic mechanisms by which rivers form themselves, and carve the landscapes that surround us.

The overall goal of the Sherwood lab is to advance genomic and precision medicine applications through high-throughput, multi-disciplinary science. In this talk, I will review a suite of high-throughput genomic and cellular perturbation platforms using CRISPR-based genome editing that the lab has developed to improve our understanding of genetic disease, gene regulation, and genome editing outcomes.

This talk will focus on recent efforts using combined analysis of rare coding variants from the UK Biobank and genome-scale CRISPR-Cas9 knockout and activation screening to improve the identification of genes, coding variants, and non-coding variants whose alteration impacts serum LDL cholesterol (LDL-C) levels. Through these efforts, we show that dysfunction of the RAB10 vesicle transport pathway leads to hypercholesterolemia in humans and mice by impairing surface LDL receptor levels. Further, we demonstrate that loss of function of OTX2 leads to robust reduction in serum LDL-C levels in mice and humans by increasing cellular LDL-C uptake. Finally, we unveil an activity-normalized base editing screening framework to better understand the impacts of coding and non-coding variation on serum LDL-C levels, altogether providing a roadmap for further efforts to dissect complex human disease genetics.

Classical ramification theory deals with complete discrete valuation fields k((X)) with perfect residue fields k. Invariants such as the Swan conductor capture important information about extensions of these fields. Many fascinating complications arise when we allow non-discrete valuations and imperfect residue fields k. Particularly in positive residue characteristic, we encounter the mysterious phenomenon of the defect (or ramification deficiency). The occurrence of a non-trivial defect is one of the main obstacles to long-standing problems, such as obtaining resolution of singularities in positive characteristic.

Degree p extensions of valuation fields are building blocks of the general case. In this talk, we will present a generalization of ramification invariants for such extensions and discuss how this leads to a better understanding of the defect. If time permits, we will briefly discuss their connection with some recent work (joint with K. Kato) on upper ramification groups.

Zarankiewicz's problem for hypergraphs asks for upper bounds on the number of edges of a hypergraph that has no complete sub-hypergraphs of a given size. Let M be an o-minimal structure. Basit-Chernikov-Starchenko-Tao-Tran (2021) proved that the following are equivalent:

(1) "linear Zarankiewicz's bounds" hold for hypergraphs whose edge relation is induced by a fixed relation definable in M

(2) M does not define an infinite field.

We prove that the following are equivalent:

(1') linear Zarankiewicz bounds hold for sufficiently "distant" hypergraphs whose edge relation is induced by a fixed relation definable in M

(2') M does not define a full field (that is, one whose domain is the whole universe of M).

This is joint work (in progress) with Aris Papadopoulos.

After a brief introduction, I elucidate a technique, dubbed "bootstrap'', which generates an infinite family of D_p(G) theories, where for a given arbitrary group G and a parameter b, each theory in the same family has the same number of mass parameters, same number of marginal deformations, same 1-form symmetry, and same 2-group structure. This technique is utilized to establish the presence or absence of the 2-group symmetries in several classes of D_p(G) theories. I, then, argue that we found the presence of 2-group symmetries in a class of Argyres-Douglas theories, called D_p(USp(2N)), which can be realized by Z_2-twisted compactification of the 6d N=(2,0) of the D-type on a sphere with an irregular twisted puncture and a regular twisted full puncture. I will also discuss the 3d mirror theories of general D_p(USp(2N)) theories that serve as an important tool to study their flavor symmetry and Higgs branch.

For the first time, a method has become available for fast computation of near-best rational approximations on arbitrary sets in the real line or complex plane: the AAA algorithm (Nakatsukasa-Sète-T. 2018).  After a brief presentation of the algorithm this talk will focus on twenty demonstrations of the kinds of things we can do, all across applied mathematics, with a black-box rational approximation tool.
 

The formulation of Mean Field Games (MFG) via partial differential equations typically requires continuous differentiability of the Hamiltonian in order to determine the advective term in the Kolmogorov--Fokker--Planck equation for the density of players. However, in many cases of practical interest, the underlying optimal control problem may exhibit bang-bang controls, which typically lead to nondifferentiable Hamiltonians. In this talk we will present results on the analysis and numerical approximation of stationary second-order MFG systems for the general case of convex, Lipschitz, but possibly nondifferentiable Hamiltonians. In particular, we will propose a generalization of the MFG system as a Partial Differential Inclusion (PDI) based on interpreting the derivative of the Hamiltonian in terms of subdifferentials of convex functions. We present results that guarantee the existence of unique weak solutions to the stationary MFG PDI under a monotonicity condition similar to one that has been considered previously by Lasry and Lions. Moreover, we will propose a monotone finite element discretization of the weak formulation of the MFG PDI, and present results that confirm the strong H^1-norm convergence of the approximations to the value function and strong L^q-norm convergence of the approximations to the density function. The performance of the numerical method will be illustrated in experiments featuring nonsmooth solutions. This talk is based on joint work with my supervisor Iain Smears.

In this talk I will discuss our recent work on two problems. The first problem concerns with capillary rise between rough structures, a fundamental wetting phenomenon that is functionalised in biological organisms and prevalent in geological or man-made materials. Predicting the liquid rise height is more complex than currently considered in the literature because it is necessary to couple two wetting phenomena: capillary rise and hemiwicking. Experiments, simulations and analytic theory demonstrate how this coupling challenges our conventional understanding and intuitions of wetting and roughness. For example, the critical contact angle for hemiwicking becomes separation-dependent so that hemiwicking can vanish for even highly wetting liquids. The rise heights for perfectly wetting liquids can also be different in smooth and rough systems. The second problem concerns with droplets (or condensates) formed via a liquid-liquid phase separation process in biological cells. Despite the widespread importance of surface tension for the interactions between these droplets and other cellular components, there is currently no reliable technique for their measurement in live cells. To address this, we develop a high-throughput flicker spectroscopy technique. Applying it to a class of cellular droplets known as stress granules, we find their interface fluctuations cannot be described by surface tension alone. It is necessary to consider elastic bending deformation and a non-spherical base shape, suggesting that stress granules are viscoelastic droplets with a structured interface, rather than simple Newtonian liquids. Moreover, given the broad distributions of surface tension and bending rigidity observed, different types of stress granules can only be differentiated via large-scale surveys, which was not possible previously and our technique now enables.

We survey the theory of $\ell^2$-invariants, their applications in group theory and topology, and introduce a positive characteristic version of $\ell^2$-theory. We also discuss the Atiyah and Lück approximation conjectures, two of the central problems in this area.

Equivariant Jiang-Su stability is an important regularity property for group actions on C*-algebras.  In this talk, I will explain this property and how it arises naturally in the context of the classification of C*-algebras and their actions. Depending on the time, I will then explain a bit more about the nature of equivariant Jiang- Su stability and the kind of techniques that are used to study it, including a recent result of Gábor Szabó and myself establishing an equivalence with equivariant property Gamma under certain conditions.
 

Neural networks are the most practically successful class of models in modern machine learning, but there are considerable gaps in the current theoretical understanding of their properties and success. Several authors have applied models and tools from random matrix theory to shed light on a variety of aspects of neural network theory, however the genuine applicability and relevance of these results is in question. Most works rely on modelling assumptions to reduce large, complex matrices (such as the Hessians of neural networks) to something close to a well-understood canonical RMT ensemble to which all the sophisticated machinery of RMT can be applied to yield insights and results. There is experimental work, however, that appears to contradict these assumptions. In this talk, we will explore what can be derived about neural networks starting from RMT assumptions that are much more general than considered by prior work. Our main results start from justifiable assumptions on the local statistics of neural network Hessians and make predictions about their spectra than we can test experimentally on real-world neural networks. Overall, we will argue that familiar ideas from RMT universality are at work in the background, producing practical consequences for modern deep neural networks.

Consider a random group $\Gamma$ with $k$ generators and $r$ random relators of large length $N$. We study the geometry of the character variety of $\Gamma$ with values in $\SL(2,\C)$ or more generally any semisimple Lie group $G$. This is the moduli space of group homomorphisms from $\Gamma$ to $G$ up to conjugation. We are in particular able to determine its dimension, number of components and Galois group, with an excellent control on the probability of exceptions. The proofs use effective Chebotarev type theorems as well as new spectral gap bounds  for Cayley graphs of finite simple groups. They are also conditional on GRH. Joint work with Peter Varju and Oren Becker.

A 1-independent bond percolation model on a graph $G$ is a probability distribution on the spanning subgraphs of $G$ in which, for all vertex-disjoint sets of edges $S_1$ and $S_2$, the states (i.e. present or not present) of the edges in $S_1$ are independent of the states of the edges in $S_2$. Such models typically arise in renormalisation arguments applied to independent percolation models, or percolation models with finite range dependencies. A 1-independent model is said to percolate if the random subgraph has an infinite component with positive probability. In 2012 Balister and Bollobás defined $p_{\textrm{max}}(G)$ to be the supremum of those $p$ for which there exists a 1-independent bond percolation model on $G$ in which each edge is present in the random subgraph with probability at least $p$ but which does not percolate. A fundamental and challenging problem in this area is to determine, or give good bounds on, the value of $p_{\textrm{max}}(G)$ when $G$ is the lattice graph $\mathbb{Z}^2$. Since $p_{\textrm{max}}(\mathbb{Z}^n)\leq p_{\textrm{max}}(\mathbb{Z}^{n-1})$, it is also of interest to establish the value of $\lim_{n\to\infty}p_{\textrm{max}}(\mathbb{Z}^n)$.

In this talk we will present a significantly improved upper bound for this limit as well as improved upper and lower bounds for $p_{\textrm{max}}(\mathbb{Z}^2)$. We will also show that with high confidence we have $p_{\textrm{max}}(\mathbb{Z}^n)<p_{\textrm{max}}(\mathbb{Z}^2)$ for large $n$ and discuss some open problems concerning 1-independent models on other graphs.

This is joint work with Tom Johnston and Alex Scott.

Harish-Chandra's Lefschetz principle suggests that representations of real and p-adic split reductive groups are closely related, even though the methods used to study these groups are quite different. The local Langlands correspondence (as formulated by Vogan) indicates that these representation theoretic relationships stem from geometric relationships between real and p-adic Langlands parameters. In this talk we will discuss how the geometric structure of real and p-adic Langlands parameters lead to functorial relationships between representations of real and p-adic groups. I will describe work in progress which applies this functoriality to the study of unitary representations and signatures of invariant hermitian forms for GL(n). The main result expresses signatures of invariant hermitian forms on graded affine Hecke algebra modules in terms of signature characters of Harish-Chandra modules, which are computable via the unitary algorithm for real reductive groups by Adams-van Leeuwen-Trapa-Vogan.

Molecules can be broken apart with a high-powered laser or an electron beam. The position of charged fragments can then be detected on a screen. From the mass to charge ratio, the identity of the fragments can be determined. The covariance of two fragments then gives us the projection of a distribution related to the initial scattering distribution. We formulate the mathematical transformation from the scattering distribution to the covariance distribution obtained from experiments. We expand the scattering distribution in terms of basis functions to obtain a linear system for the coefficients, which we use to solve the inverse problem. Finally, we show the result of our method on three examples of test data, and also with experimental data.

The synthetic theory of Ricci curvature lower bounds introduced more than 15 years ago by Lott-Sturm-Villani has been largely succesful in describing the geometry of metric measure spaces. However, this theory fails to include sub-Riemannian manifolds (an important class of metric spaces, the simplest example being the so-called Heisenberg group). Motivated by Villani's ``great unification'' program, in this talk we propose an extension of Lott-Sturm-Villani's theory, which includes sub-Riemannian geometry. This is a joint work with Barilari (Padua) and Mondino (Oxford). The talk is intended for a general audience, no previous knowledge of optimal transport or sub-Riemannian geometry is required.

The Balog-Szemerédi-Gowers theorem is a powerful tool in additive combinatorics, that allows one to roughly convert any “large energy” estimate into a “small sumset” estimate. This has found applications in a lot of results in additive combinatorics and other areas. In this talk, we will provide a friendly introduction and overview of this result, and then discuss some proof ideas. No hardcore additive combinatorics pre-requisites will be assumed.

I will present classification results for 4-manifolds with boundary and infinite cyclic fundamental group, obtained in joint work with Anthony Conway and with Conway and Lisa Piccirillo.  Time permitting, I will describe applications to knotted surfaces in simply connected 4-manifolds, and to investigating the difference between the relations of homotopy equivalence and stable homeomorphism. These will also draw on work with Patrick Orson and with Conway,  Diarmuid Crowley, and Joerg Sixt.

In this talk, we consider differential equations perturbed by multiplicative fractional Brownian noise. Depending on the value of the Hurst parameter $H$, the resulting equation is pathwise viewed as an ordinary ($H>1$), Young  ($H \in (1/2, 1)$) or rough  ($H \in (1/3, 1/2)$) differential equation. In all three regimes we show regularisation by noise phenomena by proving the strongest kind of well-posedness  for equations with irregular drifts: strong existence and path-by-path uniqueness. In the Young and smooth regime $H>1/2$ the condition on the drift coefficient is optimal in the sense that it agrees with the one known for the additive case.

In the rough regime $H\in(1/3,1/2)$ we assume positive but arbitrarily small drift regularity for strong 
well-posedness, while for distributional drift we obtain weak existence. 

This is a joint work with Máté Gerencsér.

We study the problem of determining when a compact group can be realized as the group of isometries of a norm on a finite dimensional real vector space.  This problem turns out to be difficult to solve in full generality, but we manage to understand the connected groups that arise as connected components of isometry groups. The classification we obtain is related to transitive actions on spheres (Borel, Montgomery-Samelson) on the one hand and to prehomogeneous spaces (Vinberg, Sato-Kimura) on the other. (joint work with Martin Liebeck, Assaf Naor and Aluna Rizzoli)