Past

I will present Pila's Modular Zilber-Pink with Derivatives (MZPD) conjecture, which is a Zilber-Pink type statement for the j-function and its derivatives, and discuss some weak and functional/differential analogues. In particular, I will define special varieties in each setting and explain the relationship between them. I will then show how one can prove the aforementioned weak/functional/differential MZPD statements using the Ax-Schanuel theorem for the j-function and its derivatives and some basic complex analytic geometry. Note that I gave a similar talk in Oxford last year (where I discussed a differential MZPD conjecture and proved it assuming an Existential Closedness conjecture for j), but this talk is going to be significantly different from that one (the approach presented in this talk will be mostly complex analytic rather than differential algebraic, and the results will be unconditional).

A graph of groups decomposition is a way of splitting a group into smaller and hopefully simpler groups. A natural thing to try and do is to keep splitting until you can't split anymore, and then argue that this decomposition is unique. This is the idea behind JSJ decompositions, although, as we shall see, the strength of the uniqueness statement for such a decomposition varies depending on the class of groups that we restrict our edge groups to

We consider Schroedinger operators on the whole Euclidean space or on the half-space, subject to real Robin boundary conditions. I will present the construction of a non-real potential that decays at infinity so that the corresponding Schroedinger operator has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum. This proves that the Lieb-Thirring inequalities, crucial in quantum mechanics for the proof of stability of matter, do no longer hold in the non-selfadjoint case.

I will discuss some recent results around Hilbert schemes of points on singular surfaces, obtained in joint work with Craw, Gammelgaard and Gyenge, and their connection to combinatorics (of coloured partitions) and representation theory (of affine Lie algebras and related algebras such as their W-algebra). 

The study of 'moments' of random matrices (expectations of traces of powers of the matrix) is a rich and interesting subject, with fascinating connections to enumerative geometry, as discovered by Harer and Zagier in the 1980’s. I will give some background on this and then describe some recent work which offers some new perspectives (and new results). This talk is based on joint work with Fabio Deelan Cunden, Francesco Mezzadri and Nick Simm.

Ocean biogeochemical models used in climate change predictions are very computationally expensive and heavily parameterised. With derivatives too costly to compute, we optimise the parameters within one such model using derivative-free algorithms with the aim of finding a good optimum in the fewest possible function evaluations. We compare the performance of the evolutionary algorithm CMA-ES which is a stochastic global optimization method requiring more function evaluations, to the Py-BOBYQA and DFO-LS algorithms which are local derivative-free solvers requiring fewer evaluations. We also use initial Latin Hypercube sampling to then provide DFO-LS with a good starting point, in an attempt to find the global optimum with a local solver. This is joint work with Coralia Cartis and Samar Khatiwala.
 

A mathematical structure is `axiomatizable' if it is completely determined by some family of sentences in a suitable first-order language. This idea has been explored for various kinds of structure, but I will concentrate on groups. There are some general results (not many) about which groups are or are not axiomatizable; recently there has been some interest in the sharper concept of 'finitely axiomatizable' or FA - that is, when only a finite set of sentences (equivalently, a single sentence) is allowed.

While an infinite group cannot be FA, every finite group is so, obviously. A profinite group is kind of in between: it is infinite (indeed, uncountable), but compact as a topological group; and these groups share many properties of finite groups, though sometimes for rather subtle reasons. I will discuss some recent work with Andre Nies and Katrin Tent where we prove that certain kinds of profinite group are FA among profinite groups. The methods involve a little model theory, and quite a lot of group theory.

The variational data assimilation (VarDA) problem is usually solved using a method equivalent to Gauss-Newton (GN) to obtain the initial conditions for a numerical weather forecast. However, GN is not globally convergent and if poorly initialised, may diverge such as when a long time window is used in VarDA; a desirable feature that allows the use of more satellite data. To overcome this, we apply two globally convergent GN variants (line search & regularisation) to the long window VarDA problem and show when they locate a more accurate solution versus GN within the time and cost available.
Joint work with Coralia Cartis, Amos S. Lawless, Nancy K. Nichols.

We consider the problem of global minimization with bound constraints. The problem is known to be intractable for large dimensions due to the exponential increase in the computational time for a linear increase in the dimension (also known as the “curse of dimensionality”). In this talk, we demonstrate that such challenges can be overcome for functions with low effective dimensionality — functions which are constant along certain linear subspaces. Such functions can often be found in applications, for example, in hyper-parameter optimization for neural networks, heuristic algorithms for combinatorial optimization problems and complex engineering simulations.

Extending the idea of random subspace embeddings in Wang et al. (2013), we introduce a new framework (called REGO) compatible with any global min- imization algorithm. Within REGO, a new low-dimensional problem is for- mulated with bound constraints in the reduced space. We provide probabilistic bounds for the success of REGO; these results indicate that the success is depen- dent upon the dimension of the embedded subspace and the intrinsic dimension of the function, but independent of the ambient dimension. Numerical results show that high success rates can be achieved with only one embedding and that rates are independent of the ambient dimension of the problem.

 I will review some of the major research themes in Quantum Chaos over the past 50 years, and some of the questions currently attracting attention in the mathematics and physics literatures.

Cities and their inter-connected transport networks form part of the fundamental infrastructure developed by human societies. Their organisation reflects a complex interplay between many natural and social factors, including inter alia natural resources, landscape, and climate on the one hand, combined with business, commerce, politics, diplomacy and culture on the other. Nevertheless, despite this complexity, there has been some success in capturing key aspects of city growth and network formation in relatively simple models that include non-linear positive feedback loops. However, these models are typically embedded in an idealised, homogeneous space, leading to regularly-spaced, lattice-like distributions arising from Turing-type pattern formation. Here we argue that the geographical landscape plays a much more dominant, but neglected role in pattern formation. To examine this hypothesis, we evaluate the weighted distance between locations based on a least cost path across the natural terrain, determined from high-resolution digital topographic databases for Italy. These weights are included in a co-evolving, dynamical model of both population aggregation in cities, and movement via an evolving transport network. We compare the results from the stationary state of the system with current population distributions from census data, and show a reasonable fit, both qualitatively and quantitatively, compared with models in homogeneous space. Thus we infer that that addition of weighted topography from the natural landscape to these models is both necessary and almost sufficient to reproduce the majority of the real-world spatial pattern of city sizes and locations in this example.

Arakelov geometry studies schemes X over ℤ, together with the Hermitian complex geometry of X(ℂ).
Most notably, it has been used to give a proof of Mordell's conjecture (Faltings's Theorem) by Paul Vojta; curves of genus greater than 1 have at most finitely many rational points.
In this talk, we'll introduce some of the ideas behind Arakelov theory, and show how many results in Arakelov theory are analogous—with additional structure—to classic results such as intersection theory and Riemann Roch.

In the seminar I will present a recent work joint with  S. Suhr (Bochum) giving an optimal transport formulation of the full Einstein equations of general relativity, linking the (Ricci) curvature of a space-time with the cosmological constant and the energy-momentum tensor. Such an optimal transport formulation is in terms of convexity/concavity properties of the Shannon-Bolzmann entropy along curves of probability measures extremizing suitable optimal transport costs. The result gives a new connection between general relativity and  optimal transport; moreover it gives a mathematical reinforcement of the strong link between general relativity and thermodynamics/information theory that emerged in the physics literature of the last years.

I will report on recent joint work with Karen Vogtmann on the Euler characteristic of $Out(F_n)$ and the moduli space of graphs. A similar study has been performed in the seminal 1986 work of Harer and Zagier on the Euler characteristic of the mapping class group and the moduli space of curves. I will review a topological field theory proof, due to Kontsevich, of Harer and Zagier´s result and illustrate how an analogous `renormalized` topological field theory argument can be applied to $Out(F_n)$.

Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed one such family of models, which includes versions of the physical processes described above. An intriguing property of their model is a conjectured phase transition between models that converge to growing disks, and 'turbulent' non-disk like models. In this talk I will describe a natural generalisation of the Hastings-Levitov family in which the location of each successive particle is distributed according to the density of harmonic measure on the cluster boundary, raised to some power. In recent joint work with Norris and Silvestri, we show that when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability. This phase transition in fluctuations can be interpreted as the beginnings of a macroscopic phase transition from disks to non-disks analogous to that present in the Hastings-Levitov family.
 

I will introduce the concept of forward rank-dependent performance processes, extending the original notion to forward criteria that incorporate probability distortions and, at the same time, accommodate “real-time” incoming market information. A fundamental challenge is how to reconcile the time-consistent nature of forward performance criteria with the time-inconsistency stemming from probability distortions. For this, I will first propose two distinct definitions, one based on the preservation of performance value and the other on the time-consistency of policies and, in turn, establish their equivalence. I will then fully characterize the viable class of probability distortion processes, providing a bifurcation-type result. This will also characterize the candidate optimal wealth process, whose structure motivates the introduction of a new, distorted measure and a related dynamic market. I will, then, build a striking correspondence between the forward rank-dependent criteria in the original market and forward criteria without probability distortions in the auxiliary market. This connection provides a direct construction method for forward rank-dependent criteria with dynamic incoming information. Furthermore, a direct by-product of our work are new results on the so-called dynamic utilities and time-inconsistent problems in the classical (backward) setting. Indeed, it turns out that open questions in the latter setting can be directly addressed by framing the classical problem as a forward one under suitable information rescaling.

In the talk I will discuss the  following result and related analytic and geometric questions:   On the boundary of any convex body in the Euclidean space there exists at least one infinite geodesic.

We show the existence of an infinite family of complex saddle-points at large N, for the matrix model of the superconformal index of SU(N) N = 4 super Yang-Mills theory on S3 × S1 with one chemical potential τ. The saddle-point configurations are labelled by points (m,n) on the lattice Λτ = Z τ + Z with gcd(m, n) = 1. The eigenvalues at a given saddle are uniformly distributed along a string winding (m, n) times along the (A, B) cycles of the torus C/Λτ . The action of the matrix model extended to the torus is closely related to the Bloch-Wigner elliptic dilogarithm, and its values at (m,n) saddles are determined by Fourier averages of the latter along directions of the torus. The actions of (0,1) and (1,0) agree with that of pure AdS5 and the Gutowski-Reall AdS5 black hole, respectively. The actions of the other saddles take a surprisingly simple form. Generically, they carry non vanishing entropy. The Gutowski-Reall black hole saddle dominates the canonical ensemble when τ is close to the origin, and other saddles dominate when τ approaches rational points. 

In this talk, linear algebra for persistence modules will be introduced, together with a generalization of persistent homology. This theory permits us to handle the Mayer-Vietoris spectral sequence for persistence modules, and solve any extension problems that might arise. The result of this approach is a distributive algorithm for computing persistent homology. That is, one can break down the underlying data into different covering subsets, compute the persistent homology for each cover, and join everything together. This approach has the added advantage that one can recover extra geometrical information related to the barcodes. This addresses the common complaint that persistent homology barcodes are 'too blind' to the geometry of the data.

Despite the stereotype of the lone genius working by themselves, most professional mathematicians collaborate with others.  But when you're learning maths as a student, is it OK to work with other people, or is that cheating?  And if you're not used to collaborating with others, then you might feel shy about discussing your ideas when you're not confident about them.  In this session, we'll explore ways in which you can get the most out of collaborations with your fellow students, whilst avoiding inadvertently passing off other people's work as your own.  This session will be suitable for undergraduate and MSc students at any stage of their degree who would like to increase their confidence in collaboration.  Please bring a pen or pencil!

The space of complete collineations is an important and beautiful chapter of algebraic geometry, which has its origins in the classical works of Chasles, Schubert and many others, dating back to the 19th century. It provides a 'wonderful compactification' (i.e. smooth with normal crossings boundary) of the space of full-rank maps between two (fixed) vector spaces. More recently, the space of complete collineations has been studied intensively and has been used to derive groundbreaking results in diverse areas of mathematics. One such striking example is L. Lafforgue's compactification of the stack of Drinfeld's shtukas, which he subsequently used to prove the Langlands correspondence for the general linear group. 

In joint work with M. Kapranov, we look at these classical spaces from a modern perspective: a complete collineation is simply a spectral sequence of two-term complexes of vector spaces. We develop a theory involving more full-fledged (simply graded) spectral sequences with arbitrarily many terms. We prove that the set of such spectral sequences has the structure of a smooth projective variety, the 'variety of complete complexes', which provides a desingularization, with normal crossings boundary, of the 'Buchsbaum-Eisenbud variety of complexes', i.e. a 'wonderful compactification' of the union of its maximal strata.
 

In this talk we consider a mean field optimal control problem with an aggregation-diffusion constraint, where agents interact through a potential, in the presence of a Gaussian noise term. Our analysis focuses on a PDE system coupling a Hamilton-Jacobi and a Fokker-Planck equation, describing the optimal control aspect of the problem and the evolution of the population of agents, respectively. We will discuss the existence and regularity of solutions for the aforementioned system. We notice this model is in close connection with the theory of mean-field games systems. However, a distinctive feature concerns the nonlocal character of the interaction; it affects the drift term in the Fokker-Planck equation as well as the Hamiltonian of the system, leading to new difficulties to be addressed.