Past

Since their introduction by Gromov in the 80s, a wealth of tools have been developed to study hyperbolic groups. Thus, when studying a class of groups, a characterisation of those that are hyperbolic can be very useful. In this talk, we will turn to the class of one-relator groups. In previous work, we showed that a one-relator group not containing any Baumslag--Solitar subgroups is hyperbolic, provided it has a Magnus hierarchy in which no one-relator group with a so called `exceptional intersection' appears. I will define one-relator groups with exceptional intersection, discuss the aforementioned result and will then provide a characterisation of the hyperbolic one-relator groups with exceptional intersection. Finally, I will then discuss how this characterisation can be used to establish properties for all one-relator groups.

Rational functions are able to approximate functions containing branch point singularities with a root-exponential convergence rate. These appear for example in the solution of boundary value problems on domains containing corners or edges. Results from Newman in 1964 indicate that the poles of the optimal rational approximant are exponentially clustered near the branch point singularities. Trefethen and collaborators use this knowledge to linearize the approximation problem by fixing the poles in advance, giving rise to the Lightning approximation. The resulting approximation set is however highly ill-conditioned, which raises the question of stability. We show that augmenting the approximation set with polynomial terms greatly improves stability. This observation leads to a  decoupling of the approximation problem into two regimes, related to the singular and the smooth behaviour of the function. In addition, adding polynomial terms to the approximation set can result in a significant increase in convergence speed. The convergence rate is however very sensitive to the speed at which the clustered poles approach the singularity.

A Latin square of order n is an n by n grid filled with n different symbols so that every symbol occurs exactly once in each row and each column, while a transversal in a Latin square is a collection of cells which share no row, column or symbol. The Ryser-Brualdi-Stein conjecture states that every Latin square of order n should have a transversal with n-1 elements, and one with n elements if n is odd. In 2020, Keevash, Pokrovskiy, Sudakov and Yepremyan improved the long-standing best known bounds on this conjecture by showing that a transversal with n-O(log n/loglog n) elements exists in any Latin square of order n. In this talk, I will discuss how to show, for large n, that a transversal with n-1 elements always exists.

We outline Dirac cohomology for Lie algebras and Vogan’s conjecture. We then cover some basic material on Schur-Weyl duality and Arakawa-Suzuki functors. Finishing with current efforts and results on generalising Vogan’s conjecture to a Schur-Weyl duality setting. This would relate the centre of a Lie algebra with the centre of the relevant tantaliser algebra. We finish by considering a unitary module X and giving a bound on the action of the tantalizer algebra.

Most algorithms for computing a matrix function $f(A)$ are based on finding a rational (or polynomial) approximant $r(A)≈f(A)$ to the scalar function on the spectrum of $A$. These functions are often in a composite form, that is, $f(z)≈r(z)=r_k(⋯r_2(r_1(z)))$ (where $k$ is the number of compositions, which is often the iteration count, and proportional to the computational cost); this way $r$ is a rational function whose degree grows exponentially in $k$. I will review algorithms that fall into this category and highlight the remarkable power of composite (rational) functions.

Standard twistor theory involves a complex projective
3-space PT which naturally divides into two halves PT+
and PT–, joined by their common 5-real-dimensional
boundary PN. However, this splitting has two quite
different basic physical interpretations, namely
positive/negative helicity and positive/negative
frequency, which ought not to be confused in the
formalism, and the notion of “bi-twistors” is introduced
to resolve this issue. It is found that quantized bi-
twistors have a previously unnoticed G2* structure,
which enables the split-octonion algebra to be directly
formulated in terms of quantized bi-twistors, once the
appropriate complex structure is incorporated.

So-called Schauder estimates are a standard tool in the analysis of linear elliptic and parabolic PDE. They have been originally obtained by Hopf (1929, interior case), and by Schauder and Caccioppoli (1934, global estimates). The nonlinear case is a more recent achievement from the ’80s (Giaquinta & Giusti, Ivert, Lieberman, Manfredi). All these classical results hold in the uniformly elliptic framework. I will present the solution to the longstanding problem, open since the ‘70s, of proving estimates of such kind in the nonuniformly elliptic setting. I will also cover the case of nondifferentiable functionals and provide a complete regularity theory for a new double phase model. From joint work with Giuseppe Mingione (University of Parma).

Ideal triangulations were introduced by Thurston as a tool for studying hyperbolic three-manifolds.  Taut ideal triangulations were introduced by Lackenby as a tool for studying "optimal" representatives of second homology classes.

After these applications in geometry and topology, it is time for dynamics. Veering triangulations (taut ideal triangulations with certain decorations) were introduced by Agol to study the mapping tori of pseudo-Anosov homeomorphisms.  Gueritaud gave an alternative construction, and then Agol and Gueritaud generalised it to find veering triangulations of three-manifolds admitting pseudo-Anosov flows (without perfect fits).

We prove the converse of their result: that is, from any veering triangulation we produce a canonical dynamic pair of branched surfaces (in the sense of Mosher).  These give flows on appropriate Dehn fillings of the original manifold.  Furthermore, our construction and that of Agol--Gueritaud are inverses.  This then gives a "perfect" combinatorialisation of pseudo-Anosov flow (without perfect fits).

This is joint work with Henry Segerman.

In this talk, I will discuss the (non-)construction of the focusing Gibbs measures and the associated dynamical problems. This study was initiated by Lebowitz, Rose, and Speer (1988) and continued by Bourgain (1994), Brydges-Slade (1996), and Carlen-Fröhlich-Lebowitz (2016). In the one-dimensional setting, we consider the mass-critical case, where a critical mass threshold is given by the mass of the ground state on the real line. In this case, I will show that the Gibbs measure is indeed normalizable at the optimal mass threshold, thus answering an open question posed by Lebowitz, Rose, and Speer (1988).

In the three dimensional-setting, I will first discuss the construction of the $\Phi^3_3$-measure with a cubic interaction potential. This problem turns out to be critical, exhibiting a phase transition:normalizability in the weakly nonlinear regime and non-normalizability in the strongly nonlinear regime. Then, I will discuss the dynamical problem for the canonical stochastic quantization of the $\Phi^3_3$-measure, namely, the three-dimensional stochastic damped nonlinear wave equation with a quadratic nonlinearity forced by an additive space-time white noise (= the hyperbolic $\Phi^3_3$-model). As for the local theory, I will describe the paracontrolled approach to study stochastic nonlinear wave equations, introduced in my work with Gubinelli and Koch (2018). In the globalization part, I introduce a new, conceptually simple and straightforward approach, where we directly work with the (truncated) Gibbs measure, using the variational formula and ideas from theory of optimal transport.

The first part of the talk is based on a joint work with Philippe Sosoe (Cornell) and Leonardo Tolomeo (Bonn/Edinburgh), while the second part is based on a joint work with Mamoru Okamoto (Osaka) and Leonardo Tolomeo (Bonn/Edinburgh).

For a family of finite sets whose cardinalities are naturally called class numbers, the Gauss problem asks to determine the subfamily in which every member has class number one. We study the Siegel moduli space of abelian varieties in characteristic $p$ and solve the Gauss problem for the family of central leaves, which are the loci consisting of points whose associated $p$-divisible groups are isomorphic. Our solution involves mass formulae, computations of automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus $4$. This geometric Gauss problem is closely related to an arithmetic Gauss problem for genera of positive-definite quaternion Hermitian lattices, which we also solve.

I will talk about homological enumerative invariants for vector bundles on algebraic curves. These invariants were defined by Joyce, and encode rich information about the moduli space of semistable vector bundles, such as its volume and intersection numbers, which were computed by Witten, Jeffrey and Kirwan in previous work. I will define a notion of regularization of divergent infinite sums, and I will express the invariants explicitly as such a divergent sum in a vertex algebra.

While there have been substantial successes of actor-critic methods in continuous control, simpler critic-only methods such as Q-learning often remain intractable in the associated high-dimensional action spaces. However, most actor-critic methods come at the cost of added complexity: heuristics for stabilisation, compute requirements as well as wider hyperparameter search spaces. To address this limitation, we demonstrate in two stages how a simple variant of Deep Q Learning matches state-of-the-art continuous actor-critic methods when learning from simpler features or even directly from raw pixels. First, we take inspiration from control theory and shift from continuous control with policy distributions whose support covers the entire action space to pure bang-bang control via Bernoulli distributions. And second, we combine this approach with naive value decomposition, framing single-agent control as cooperative multi-agent reinforcement learning (MARL). We finally add illustrative examples from control theory as well as classical bandit examples from cooperative MARL to provide intuition for 1) when action extrema are sufficient and 2) how decoupled value functions leverage state information to coordinate joint optimization.

Argyres—Douglas (AD) theories are 4d N=2 SCFTs which have some unusual features, and until recently, explicit holographic duals of these theories were unknown. We will consider a concrete class of these theories obtained by wrapping the 6d N=(2,0) ADE theories on a (twice) punctured sphere: one irregular and one regular puncture, and construct their holographic duals. The novel aspects of these solutions require a relaxation of the regularity conditions of the usual Gaiotto—Maldacena framework and to allow for brane singularities. We show how to construct the dictionary between the AdS(5) solutions and the field theory and match observables between the two. If time allows, I will comment on some on-going work about further compactifying the AD theories on spindles, or the 6d theories on four-dimensional orbifolds. 

What should we be thinking about when we're making a diagram for a paper? How do we help it to express the right things? Or make it engaging? What kind of colour palette is appropriate? What software should we use? And how do we make this process as painless as possible? Join Joshua Bull and Christoph Dorn for a lively Fridays@4 session on illustrating mathematics, as they share tips, tricks, and their own personal experiences in bringing mathematics to life via illustrations.

Models of animal brains are increasingly common and mapped in increasing detail. To simplify analysis of their function, we consider subregions and show that they perform well as classifiers of overall activity, with only a fraction of the neurons. The uniqueness of such ''reliable'' regions seems to be related to the types of connections that pairs of neurons form in them. By focusing on topologically significant structures and reciprocally connected neurons we find even stronger classification results. This is ongoing work across several institutions, including EPFL, the Blue Brain Project, and the University of Aberdeen.

Isostasy is one of the earliest quantitative geophysical theories still in current use. It explains why observed gravity anomalies are generally much weaker than what is inferred from visible topography, and why planetary crusts can support large mountains without breaking up. At large scale, most topography (including bathymetry) is in isostatic equilibrium, meaning that surface loads are buoyantly supported by crustal thickness variations or density variations within the crust and lithosphere, in such a way that deeper layers are hydrostatic. On Earth, examples of isostasy are the average depth of the oceans, the elevation of the Himalayas, and the subsidence of ocean floor away from mid-ocean ridges, which are respectively attributed to the crust-ocean thickness difference, to crustal thickening under mountain belts, and to the density increase due to plate cooling. Outside Earth, isostasy is useful to constrain the crustal thickness of terrestrial planets and the shell thickness of icy moons with subsurface oceans.

Given the apparent simplicity of the isostatic concept – buoyant support of mountains by iceberg-like roots – it is surprising that a debate has been going on for over a century about its various implementations. Classical isostasy is indeed not self-consistent, neglects internal stresses and geoid contributions to topographical support, and yields ambiguous predictions of geoid anomalies at the planetary scale. In the last few years, these problems have attracted renewed attention when applying isostasy to planetary bodies with an unbroken crustal shell. In this talk I will discuss isostatic models based on the minimization of stress, on time-dependent viscous evolution, and on stationary viscous flow. I will show that these new isostatic approaches are mostly equivalent and discuss their implications for the structure of icy moons.

As one of the largest retailers in the world, Tesco relies on automated forecasting to help with decision making. A common issue with forecasts is that of the cold start problem; that we must make forecasts for new products that have no history to learn from. Lack of historical data becomes a real problem as it prevents us from knowing how products react to events, and if their sales react to the time of year. We might consider using similar products as a way to produce a starting forecast, but how should we define what ‘similar’ means, and how should we evolve this model as we start getting real live data? We’ll present some examples to hopefully start a fruitful discussion.

Automated Market Makers (AMMs) are a new prototype of 
trading venues which are revolutionising the way market participants 
interact. At present, the majority of AMMs are Constant Function 
Market Makers (CFMMs) where a deterministic trading function 
determines how markets are cleared. A distinctive characteristic of 
CFMMs is that execution costs for liquidity takers, and revenue for 
liquidity providers, are given by closed-form functions of price, 
liquidity, and transaction size. This gives rise to a new class of 
trading problems. We focus on Constant Product Market Makers with 
Concentrated Liquidity and show how to optimally take and make 
liquidity. We use Uniswap v3 data to study price and liquidity 
dynamics and to motivate the models.

For liquidity taking, we describe how to optimally trade a large 
position in an asset and how to execute statistical arbitrages based 
on market signals. For liquidity provision, we show how the wealth 
decomposes into a fee and an asset component. Finally, we perform 
consecutive runs of in-sample estimation of model parameters and 
out-of-sample trading to showcase the performance of the strategies.

European option payoffs can be generated by combinations of hockeystick payoffs or of monomials. Interestingly, path dependent options can be generated by combinations of signatures, which are the building blocks of path dependence. We focus on the case of 1 asset together with time, typically the evolution of the price x as a function of the time t. The signature of a path for a given word with letters in the alphabet {t,x} (sometimes called augmented signature of dimension 1) is an iterated Stratonovich integral with respect to the letters of the word and it plays the role of a monomial in a Taylor expansion. For a given time horizon T the signature elements associated to short words are contained in the linear space generated by the signature elements associated to longer words and we construct an incremental basis of signature elements. It allows writing a smooth path dependent payoff as a converging series of signature elements, a result stronger than the density property of signature elements from the Stone-Weierstrass theorem. We recall the main concepts of the Functional Itô Calculus, a natural framework to model path dependence and draw links between two approximation results, the Taylor expansion and the Wiener chaos decomposition. The Taylor expansion is obtained by iterating the functional Stratonovich formula whilst the Wiener chaos decomposition is obtained by iterating the functional Itô formula applied to a conditional expectation. We also establish the pathwise Intrinsic Expansion and link it to the Functional Taylor Expansion.

It is known that the Brauer group of a smooth, projective surface
defined by an equality of two homogeneous polynomials in characteristic 0, is
finite up to constants. I will report on new methods to determine these Brauer
groups, or at least their algebraic parts, as long as the coefficients are in a
certain sense generic. This generalises previous results obtained over the
years by Colliot-Thélène--Kanevsky--Sansuc, Bright, Uematsu and Santens.
(Joint work with A. N. Skorobogatov.)

I will explain how the model-theoretic algebraic closure in Zilber’s pseudo-exponential field can be described in terms of the self-sufficient closure. I will sketch a proof and show how the Mordell-Lang conjecture for algebraic tori comes into play. If time permits, I’ll also talk about the characterisation of strongly minimal sets and their geometries. This is joint work (still in progress) with Jonathan Kirby.

Solving sparse linear systems is omnipresent in scientific computing. Direct approaches based on matrix factorization are very robust, and since they can be used as a black-box, it is easy for other software to use them. However, the memory requirement of direct approaches scales poorly with the problem size, and the algorithms underpinning sparse direct solvers software are poorly suited to parallel computation. Multilevel Domain decomposition (MDD) methods are among the most efficient iterative methods for solving sparse linear systems. One of the main technical difficulties in using efficient MDD methods (and most other efficient preconditioners) is that they require information from the underlying problem which prohibits them from being used as a black-box. This was the motivation to develop the widely used algebraic multigrid for example. I will present a series of recently developed robust and fully algebraic MDD methods, i.e., that can be constructed given only the coefficient matrix and guarantee a priori prescribed convergence rate. The series consists of preconditioners for sparse least-squares problems, sparse SPD matrices, general sparse matrices, and saddle-point systems. Numerical experiments illustrate the effectiveness, wide applicability, scalability of the proposed preconditioners. A comparison of each one against state-of-the-art preconditioners is also presented.

The classification of anti de Sitter black holes is an open problem of central importance in holography. In this talk, I will present new advances in classification of supersymmetric solutions to five-dimensional minimal gauged supergravity. In particular, we prove a black hole uniqueness theorem within a ‘Calabi-type’ subclass of solutions with biaxial symmetry. This subclass includes all currently known black hole solutions within this theory.