Past

I will describe a simple macroscopic model describing the evolution of a cloud of particles confined in a magneto-optical trap. The behavior of the particles is mainly driven by self--consistent attractive forces. In contrast to the standard model of gravitational forces, the force field does not result from a potential; moreover, the nonlinear coupling is more singular than the coupling based on the Poisson equation.  In addition to existence of uniqueness results of the model PDE, I will discuss the convergence of the  particles description towards the solution of the PDE system in the mean field regime.

To fly is not to fall. How does an insect fly, why does it fly so well, and how can we infer its ‘thoughts’ from its flight dynamics?  We have been seeking  mechanistic explanations of the complex movement of insect flight. Starting from the Navier-Stokes equations governing the unsteady aerodynamics of flapping flight, a  theoretical framework for computing flight leads to new interpretations and predictions of the functions of an insect’s internal machinery that orchestrate its flight. The talk will discuss recent computational and experimental studies of the balancing act of dragonflies and fruit flies:  how a dragonfly recovers from falling upside-down and how a fly balances in air. In each case,  the physics of flight informs us about the neural feedback circuitries underlying their fast reflexes.

I will give a survey of known results about when two RAAGs are quasi-isometric, and will then describe a visual graph of groups decomposition of a RAAG (its JSJ tree of cylinders) that can often be used to determine whether or not two RAAGs are quasi-isometric.

Adaptive oblivious transfer (OT) is a protocol where a sender
initially commits to a database {M_i}_{i=1}^N . Then, a receiver can query the
sender up to k times with private indexes ρ_1, …, ρ_k so as to obtain
M_{ρ_1}, …, M_{ρ_k} and nothing else. Moreover, for each i ∈ [k], the receiver’s
choice ρ_i may depend on previously obtained messages {M_ρ_j}_{j<i} . Oblivious transfer
with access control (OT-AC) is a flavor of adaptive OT
where database records are protected by distinct access control policies
that specify which credentials a receiver should obtain in order to access
each M_i . So far, all known OT-AC protocols only support access policies
made of conjunctions or rely on ad hoc assumptions in pairing-friendly
groups (or both). In this paper, we provide an OT-AC protocol where access policies may consist of any branching program of polynomial length, which is sufficient to realize any access policy in NC^1. The security of
our protocol is proved under the Learning-with-Errors (LWE) and Short-
Integer-Solution (SIS) assumptions. As a result of independent interest,
we provide protocols for proving the correct evaluation of a committed
branching program on a committed input.

Joint work with Benoît Libert, San Ling, Khoa Nguyen and Huaxiong Wang.

By way of shameless advertising for a TCC course I hope to give next term on the theory of totally disconnected locally compact groups, I will present two interesting and illuminating examples of such groups: the full automorphism group of a regular tree, and Neretin's group of spheromorphisms
 

The talk is based on my joint work with Anand Pillay and Tomasz Rzepecki.

I will describe some connections between various objects from topological dynamics associated with a given first order theory and various Galois groups of this theory. One of the main corollaries is a natural presentation of the closure of the neutral element of the Lascar Galois group of any given theory $T$ (this closure is a group sometimes denoted by $Gal_0(T)$) as a quotient of a compact Hausdorff group by a dense subgroup.

As an application, I will present a very general theorem concerning the complexity of bounded, invariant equivalence relations (whose classes are sometimes called strong types) in countable theories, generalizing a theorem of Kaplan, Miller and Simon concerning Borel cardinalities of Lascar strong types and also later extensions of this result to certain bounded, $F_\sigma$ equivalence relations (which were obtained in a paper of Kaplan and Miller and, independently, in a paper of Rzepecki and myself). The main point of our general theorem says that in a countable theory, any bounded, invariant equivalence relation defined
on the set of realizations of a single complete type over $\emptyset$ is type-definable if and only if it is smooth (in the sense of descriptive set theory). If time permits, I will very briefly mention more recent developments in this direction (also based on the results from the first paragraph) which will appear in my future paper with Rzepecki.
 

We investigate Néron models of Jacobians of singular curves over strictly Henselian discretely valued fields, and their behaviour under tame base change. For a semiabelian variety, this behaviour is governed by a finite sequence of (a priori) real numbers between 0 and 1, called "jumps". The jumps are conjectured to be rational, which is known in some cases. The purpose of this paper is to prove this conjecture in the case where the semiabelian variety is the Jacobian of a geometrically integral curve with a push-out singularity. Along the way, we prove the conjecture for algebraic tori which are induced along finite separable extensions, and generalize Raynaud's description of the identity component of the Néron model of the Jacobian of a smooth curve (in terms of the Picard functor of a proper, flat, and regular model) to our situation. The main technical result of this paper is that the exact sequence which decomposes the Jacobian of one of our singular curves into its toric and Abelian parts extends to an exact sequence of Néron models. Previously, only split semiabelian varieties were known to have this property.

Classical algorithms for numerical integration (quadrature/cubature) proceed by approximating the integrand with a simple function (e.g. a polynomial), and integrate the approximant exactly. In high-dimensional integration, such methods quickly become infeasible due to the curse of dimensionality.

A common alternative is the Monte Carlo method (MC), which simply takes the average of random samples, improving the estimate as more and more samples are taken. The main issue with MC is its slow "sqrt(variance/#samples)" convergence, and various techniques have been proposed to reduce the variance.

In this work we reveal a numerical analyst's interpretation of MC: it approximates the integrand with a simple(st) function, and integrates that function exactly. This observation leads naturally to MC-like methods that combines MC with function approximation theory, including polynomial approximation and sparse grids. The resulting method can be regarded as another variance reduction technique for Monte Carlo.

Reed conjectured in 1998 that the chromatic number of a graph should be at most the average of the clique number (a trivial lower bound) and maximum degree plus one (a trivial upper bound); in support of this conjecture, Reed proved that the chromatic number is at most some nontrivial convex combination of these two quantities.  King and Reed later showed that a fraction of roughly 1/130000 away from the upper bound holds. Motivated by a paper by Bruhn and Joos, last year Bonamy, Perrett, and I proved that for large enough maximum degree, a fraction of 1/26 away from the upper bound holds. Then using new techniques, Delcourt and I showed that the list-coloring version holds; moreover, we improved the fraction for ordinary coloring to 1/13. Most recently, Kelly and I proved that a 'local' list version holds with a fraction of 1/52 wherein the degrees, list sizes, and clique sizes of vertices are allowed to vary.
 

We develop a general purpose solver for quadratic programs based on operator splitting. We introduce a novel splitting that requires the solution of a quasi-definite linear system with the same coefficient matrix in each iteration. The resulting algorithm is very robust, and once the initial factorization is carried out, division free; it also eliminates requirements on the problem data such as positive definiteness of the objective function or linear independence of the constraint functions. Moreover, it is able to detect primal or dual infeasible problems providing infeasibility certificates. The method supports caching the factorization of the quasi-definite system and warm starting, making it efficient for solving parametrized problems arising in finance, control, and machine learning. Our open-source C implementation OSQP has a small footprint and is library-free. Numerical benchmarks on problems arising from several application domains show that OSQP is typically 10x faster than interior-point methods, especially when factorization caching or warm start is used.

This is joint work with Goran Banjac, Paul Goulart, Alberto Bemporad and Stephen Boyd
 

Identifying clusters or "communities" of densely connected nodes in networks is an active area of research, with relevance to many applications. Recent advances in the field have focused especially on temporal, multiplex, and other kinds of multilayer networks.

One method for detecting communities in multilayer networks is to maximise a generalised version of an objective function known as modularity. Writing down multilayer modularity requires the specification of two types of resolution parameters, and choosing appropriate values is crucial for uncovering meaningful community structure. In the simplest case, there are just two parameters, one controlling the sizes of detected communities, and the other influencing how much communities change from layer to layer. By establishing an equivalence between modularity optimisation and a multilayer maximum-likelihood approach to community detection, we are able to determine statistically optimal values for these two parameters. 

When applied to existing multilayer benchmarks, our optimized approach performs significantly better than using parameter choices guided by heuristics. We also apply the method to supermarket data, revealing changes in consumer behaviour over time.

I will start with a quick reminder of what we have learned so far about
transplanckian-energy collisions of particles, strings and branes.
I will then address the (so-far unsolved) problem of gravitational
bremsstrahlung from massless particle collisions at leading order in the
gravitational deflection angle.
Two completely different calculations, one classical and one quantum, lead
to the same final, though somewhat puzzling, result.

For liquid films with a thickness in the order of 10¹−10³ molecule layers, classical models of continuum mechanics do not always give a precise description of thin-film evolution: While morphologies of film dewetting are captured by thin-film models, discrepancies arise with respect to time-scales of dewetting.

In this talk, we study stochastic thin-film equations. By multiplicative noise inside an additional convective term, these stochastic partial differential equations differ from their deterministic counterparts, which are fourth-order degenerate parabolic. First, we present some numerical simulations which indicate that the aforementioned discrepancies may be overcome under the influence of noise.

In the main part of the talk, we prove existence of almost surely nonnegative martingale solutions. Combining spatial semi-discretization with appropriate stopping time arguments, arbitrary moments of coupled energy/entropy functionals can be controlled.

Having established Hölder regularity of approximate solutions, the convergence proof is then based on compactness arguments - in particular on Jakubowski’s generalization of Skorokhod’s theorem - weak convergence methods, and recent tools for martingale convergence.

The results have been obtained in collaboration with K. Mecke and M. Rauscher and with J. Fischer, respectively

This talk concerns a twenty-thousand-year old mistake: The natural numbers record only the result of counting and not the process of counting. As algebra is rooted in the natural numbers, the higher algebra of Joyal and Lurie is rooted in a more basic notion of number which also records the process of counting. Long advocated by Waldhausen, the arithmetic of these more basic numbers should eliminate denominators. Notable manifestations of this vision include the Bökstedt-Hsiang-Madsen topological cyclic homology, which receives a denominator-free Chern character, and the related Bhatt-Morrow-Scholze integral p-adic Hodge theory, which makes it possible to exploit torsion cohomology classes in arithmetic geometry. Moreover, for schemes smooth and proper over a finite field, the analogue of de Rham cohomology in this setting naturally gives rise to a cohomological interpretation of the Hasse-Weil zeta function by regularized determinants as envisioned by Deninger.

We consider L^2-approximations of white noise within the framework of regularity structures. Possible applications include support theorems for SPDEs driven by degenerate noises and numerics. Joint work with Ilya Chevyrev, Peter Friz and Tom Klose. 

We give new obstructions to the existence of planar open books on contact structures, in terms of the homology of their fillings. I will talk about applications to links of surface singularities, Seifert fibred spaces, and integer homology spheres. No prior knowledge of contact or symplectic topology will be assumed. This is joint work with Paolo Ghiggini and Olga Plamenevskaya.
 

This talk is based on a joint work with Dmitry Beliaev.

We study the volume distribution of nodal domains of families of naturally arising Gaussian random field on generic manifolds, namely random band-limited functions. It is found that in the high energy limit a typical instance obeys a deterministic universal law, independent of the manifold. Some of the basic qualitative properties of this law, such as its support, monotonicity and continuity of the cumulative probability function, are established.

We discuss mathematical studies on the Vafa-Witten theory, one of topological twists of N=4 super Yang-Mills theory in four dimensions, from the viewpoints of both differential and algebraic geometry. After mentioning backgrounds and motivation, we describe some issues to construct mathematical theory of this Vafa-Witten one, and explain possible ways to sort them out by analytic and algebro-geometric methods, the latter is joint work with Richard Thomas.

In this year’s Simonyi Lecture, Geoffrey West discusses the universal laws that govern everything from the growth of plants and animals to cities and corporations. These laws help us to answer big, urgent questions about global sustainability, population explosion, urbanization, ageing, cancer, human lifespans and the increasing pace of life.

Why can we live for 120 years but not for a thousand? Why do mice live for just two or three years and elephants for up to 75? Why do companies behave like mice, and are they all destined to die? Do cities, companies and human beings have natural, pre-determined lifespans?

Geoffrey West is a theoretical physicist whose primary interests have been in fundamental questions in physics and biology. West is a Senior Fellow at Los Alamos National Laboratory and a distinguished professor at the Sante Fe Institute, where he served as the president from 2005-2009. In 2006 he was named to Time’s list of The 100 Most Influential People in the World.

This lecture will take place at the Oxford Playhouse, Beaumont Street. Book here

I will describe our research on numerical methods for atmospheric dynamical cores based on compatible finite element methods. These methods extend the properties of the Arakawa C-grid to finite element methods by using compatible finite element spaces that respect the elementary identities of vector-calculus. These identities are crucial in demonstrating basic stability properties that are necessary to prevent the spurious numerical degradation of geophysical balances that would otherwise make numerical discretisations unusable for weather and climate prediction without the introduction of undesirable numerical dissipation. The extension to finite element methods allow these properties to be enjoyed on non-orthogonal grids, unstructured multiresolution grids, and with higher-order discretisations. In addition to these linear properties, for the shallow water equations, the compatible finite element structure can also be used to build numerical discretisations that respect conservation of energy, potential vorticity and enstrophy; I will survey these properties. We are currently developing a discretisation of the 3D compressible Euler equations based on this framework in the UK Dynamical Core project (nicknamed "Gung Ho"). The challenge is to design discretisation of the nonlinear operators that remain stable and accurate within the compatible finite element framework. I will survey our progress on this work to date and present some numerical results.

Feedback control is found extensively in many natural and technological systems. Indeed, many biological processes use feedback
to regulate key processes – examples include bacterial chemotaxis and negative autoregulation in genetic circuits. Despite the prevalence of
feedback in natural systems, its design and implementation in a Synthetic Biological context is much harder.  In this talk I will give
examples of how we implemented feedback systems in three different biological systems. The first one concerns the design of a synthetic
recombinase-based feedback loop, which results into robust expression. The second describes the use of small RNAs to post-transcriptionally
regulate gene expression through interaction with messenger RNA (mRNA). The third involves the introduction of negative feedback in a
two-component signalling system through a controllable phosphatase.  Closing, I will outline the challenges posed by the design of such
systems, both theoretical and on their implementation.

Rita Maria del Rio Chanona:

Global financial contagion on a Multiplex Network

We explore the global financial system, in particular the risk of global financial contagion through network theory. Although there is extensive literature on contagion in networks, we argue that it is important to consider different channels of contagion. Therefore we deem into the multilayer framework, where nodes are countries and each layer represents a different type of financial obligation. The multiplex network is built using data provided by collaborators in the IMF. We study contagion with a percolation model and conclude that financial shocks can be amplified considerably when the multilayer structure is taken into account.


Johannes Wiesel:

Robust Superhedging vs Robust Statistics

In this talk I try to reconcile the different understanding of robustness in mathematical finance and statistics. Motivated by recent advances in the estimation of risk measures, I present estimators for the superhedging price of a claim given a history of observed prices. I discuss weak efficiency and convergence speed of these estimators. Besides I explain how to apply classical notions of sensitivity for the estimation procedure. This talk is based on ongoing work with Jan Obloj.

 

Railway traffic management is the combination of monitoring the progress of trains, forecasting of the likely future progression of trains, and evaluating the impact of intervention options in near real time in order to make traffic adjustments that minimise the combined delay of trains when measured against the planned timetable.

In a time of increasing demand for rail travel, the desire to maximise the usage of the available infrastructure capacity competes with the need for contingency space to allow traffic management when disruption occurs. Optimisation algorithms and decision support tools therefore need to be increasingly sophisticated and traffic management has become a crucial function in meeting the growing expectations of rail travellers for punctuality and quality of service.

Resonate is a technology company specialising in rail and connected transport solutions. We have embarked on a drive to maximise capacity and performance through the use of mathematical, statistical, data-driven and machine learning based methods driving decision support and automated traffic management solutions.

Manifolds, the main objects of study in Differential Geometry, do not have nice categorical properties. For example, the category of manifolds with smooth maps does not contain all fibre products.
The algebraic counterparts to this (varieties and schemes) do have nice categorical properties. 

A method to ‘fix’ these categorical issues is to consider C^infinity schemes, which generalise the category of manifolds using algebraic geometry techniques. I will explain these concepts, and how to translate to manifolds with corners, which is joint work with my supervisor Professor Dominic Joyce.

Phytoplankton moving in the ocean, spermatozoa making their way  through the female reproductive tract and harmful bacteria that form biofilms on implanted medical devices interact with a surrounding fluid. Their length scales are small enough so that viscous effects dominate inertial effects allowing the resulting fluid dynamics to be described by the linear Stokes equations. However,  nonlinear behavior can occur because these structures are flexible and their form evolves with the flow. In addition, the fluid environment may also  be complex because of embedded microstructures that further complicate the dynamics.  We will discuss recent successes and challenges in describing these elastohydrodynamic systems.

This talk will report on the definition of some motivic cohomology classes and the proof that they satisfy the norm relations expected of Euler systems, emphasizing a connection with the local Gan-Gross-Prasad conjecture.

 In the first part of the talk, we present some recent and new developments in the theory of control and optimal stopping with nonlinear expectations. We first introduce an optimal stopping game with nonlinear expectations (Generalized Dynkin Game) in a non-Markovian framework and study its links with nonlinear doubly reflected BSDEs. We then present some new results (which are part of an ongoing work) on mixed stochastic stochastic control/optimal stopping problems (as well as stochastic control/optimal stopping game problems) in a non-Markovian framework and their relation with constrained reflected BSDEs with lower obstacle (resp. upper obstacle). These results are obtained using some technical tools of stochastic analysis. In the second part of the talk, we discuss applications to the $\cal{E}^g$ pricing of American options and Game options in complete and incomplete markets (based on joint works with M.C.Quenez and Agnès Sulem).
 

Because of atmospheric turbulence, images of objects in outer space acquired via ground-based telescopes are usually blurry.  One way to estimate the blurring kernel or point spread function (PSF) is to make use of the aberration of wavefront received at the telescope, i.e., the phase. However only the low-resolution wavefront gradients can be collected by wavefront sensors. In this talk, I will discuss how to use regularization methods to reconstruct high-resolution phase gradients and then use them to recover the phase and the PSF in high accuracy. I will end by relating the problem to high-resolution image reconstruction and methods for solving it.
Joint work with Rui Zhao and research supported by HKRGC.

In this talk I will discuss acoustic and electromagnetic transmission problems; i.e. problems where the wave speed jumps at an interface. I will focus on what is known mathematically about resonances and trapped waves (e.g. When do these occur? When can they be ruled out? What do we know in each case?). This is joint work with Andrea Moiola (Pavia).

Can mathematics really help us in our fight against infectious disease? Join Julia Gog as we explore some exciting current research areas where mathematics is being used to study pandemics, viruses and everything in between, with a particular focus on influenza.

Julia Gog is Professor of Mathematical Biology, University of Cambridge and David N Moore Fellow at Queens’ College, Cambridge.

Please email: @email to regsiter

Classifying line arrangements on the plane is a problem that has been around for a long time. There has been a lot of work from the perspective of incidence geometry, but after a paper of Hirzebruch in in 80's, it has also attracted the attention of algebraic geometers for the applications that it has on classifying complex algebraic surfaces of general type. In this talk I will present various results around this problem, I will show some specific questions that are still open, and I will explain how it relates to complex surfaces of general type. 
 

It is well known that the theory of differentially closed fields of characteristic 0 has prime models and that they are unique up to isomorphism. One can ask the same question for the theory ACFA of existentially closed difference fields (recall that a difference field is a field with an automorphism).

In this talk, I will first give the trivial reasons of why this question cannot have a positive answer. It could however be the case that over certain difference fields prime models (of the theory ACFA) exist and are unique. Such a prime model would be called a difference closure of the difference field K. I will show by an example that the obvious conditions on K do not suffice.

I will then consider the class of aleph-epsilon saturated models of ACFA, or of kappa-saturated models of ACFA. There are natural notions of aleph-epsilon prime model and kappa-prime model. It turns out that for these stronger notions, if K is an algebraically closed difference field of characteristic 0, with fixed subfield F aleph-epsilon saturated, then there is an aleph-epsilon prime model over K, and it is unique up to K-isomorphism. A similar result holds for kappa-prime when kappa is a regular cardinal.

None of this extends to positive characteristic.
 

We consider a generalization of degeneracy loci of morphisms between vector bundles based on orbit closures of algebraic groups in their linear representations. Using a certain crepancy condition on the orbit closure we gain some control over the canonical sheaf in a preferred class of examples. This is notably the case for Richardson nilpotent orbits and partially decomposable skew-symmetric three-forms in six variables. We show how these techniques can be applied to construct Calabi-Yau manifolds and Fano varieties of dimension three and four.

This is a joint work with Vladimiro Benedetti, Laurent Manivel and Fabio Tanturri.

We provide the rate of convergence of general monotone numerical schemes for parabolic Hamilton-Jacobi-Bellman equations in bounded domains with Dirichlet boundary conditions. The so-called "shaking coefficients" technique introduced by Krylov is used. This technique is based on a perturbation of the dynamics followed by a regularization step by convolution. When restricting the equation to a domain, the perturbed problem may not satisfy such a restriction, so that a special treatment near the boundary is necessary. 

In my talk I will explain how to relate 1-dimensional representations of finite W-algebras with multiplicity free primitive ideals of universal enveloping algebras and representations of minimal dimension of the corresponding reduced enveloping algebras (Humphreys' conjecture). I will also mention some open problems in the field.

The phenomenon of poor algorithmic scalability is a critical problem in large-scale machine learning and data science. This has led to a resurgence in the use of first-order (Hessian-free) algorithms from classical optimisation. One major drawback is that first-order methods tend to converge extremely slowly. However, there exist techniques for efficiently accelerating them.
    
The topic of this talk is the Dual Regularisation Nonlinear Acceleration algorithm (DRNA) (Geleta, 2017) for nonconvex optimisation. Numerical studies using the CUTEst optimisation problem set show the method to accelerate several nonconvex optimisation algorithms, including quasi-Newton BFGS and steepest descent methods. DRNA compares favourably with a number of existing accelerators in these studies.
    
DRNA extends to the nonconvex setting a recent acceleration algorithm due to Scieur et al. (Advances in Neural Information Processing Systems 29, 2016). We have proven theorems relating DRNA to the Kylov subspace method GMRES, as well as to Anderson's acceleration method and family of multi-secant quasi-Newton methods.
 

Superradiance in black hole spacetimes is a phenomenon by which a field of spin 0 or 1 can extract energy from the background. Typically, one can imagine sending a wave packet with a given energy towards a black hole and receiving in return a superposition of wave packets carrying a total amount of energy that is larger than the energy sent in. It can be caused by rotation or by interaction between the charges of the black hole and the field. In the first case, the region where superradiance takes place (the ergoregion) has a clear geometrical localization depending only on the physical parameters of the black hole. For charge induced superradiance, this is not the case and we have a generalized ergoregion depending also on the physical properties of the field (mass, charge, angular momentum). In the most severe cases, the generalized ergoregion may cover the whole exterior of the black hole. We focus on charge-induced superradiance for spin 0 fields in spherically symmetric situations. Alain Bachelot wrote a thorough theoretical study of this question in 2004, which, to my knowledge, is the only work of its kind. When I was in Bordeaux, he and I discussed the possibility of investigating superradiance numerically. Over the years it became an actual research project, involving Laurent Di Menza and more recently Mathieu Pellen, of which this talk is an account. The idea was to observe numerically some superradiant behaviours and gain a more precise understanding of the phenomenon. We shall show an exact analogue of the Penrose process with the superradiance of wave packets and a slightly different behaviour for fields "emerging" inside the ergoregion. We shall also explore the related question of black hole bombs and present some recent observations. 

It is well-known that only a limited number of the fluid flow problems can be solved (or approximated) by the solutions in the explicit form. To derive such solutions, we usually need to start with (over)simplified mathematical models and consider ideal geometries on the flow domains with no distortions introduced. However, in practice, the boundary of the fluid domain can contain various small irregularities (rugosities, dents, etc.) being far from the ideal one. Such problems are challenging from the mathematical point of view and, in most cases, can be treated only numerically. The analytical treatments are rare because introducing the small parameter as the perturbation quantity in the domain boundary forces us to perform tedious change of variables. Having this in mind, our goal is to present recent analytical results on the effects of a slightly perturbed boundary on the fluid flow through a channel filled with a porous medium. We start from a rectangular domain and then perturb the upper part of its boundary by the product of the small parameter $\varepsilon$ and arbitrary smooth function. The porous medium flow is described by the Darcy-Brinkman model which can handle the presence of a boundary on which the no-slip condition for the velocity is imposed. Using asymptotic analysis with respect to $\varepsilon$, we formally derive the effective model in the form of the explicit formulae for the velocity and pressure. The obtained asymptotic approximation clearly shows the nonlocal effects of the small boundary perturbation. The error analysis is also conducted providing the order of accuracy of the asymptotic solution. We will also address the problem of the solute transport through a semi-infinite channel filled with a fluid saturated sparsely packed porous medium. A small perturbation of magnitude $\varepsilon$ is applied on the channel's walls on which the solute particles undergo a first-order chemical reaction. The effective model for solute concentration in the small-Péclet-number-regime is derived using asymptotic analysis with respect to $\varepsilon$. The obtained mathematical model clearly indicates the influence of the porous medium, chemical reaction and boundary distortion on the effective flow.

This is a joint work with Eduard Marušić-Paloka (University of Zagreb).

Fix a loop group LG, a level k∈ℕ, and let Repᵏ(LG) be corresponding category of positive energy representations.
For any pair of pants Σ (with complex structure in the interior and parametrized boundary), there is an associated functor Repᵏ(LG) × Repᵏ(LG) → Repᵏ(LG): (H,K) ↦ H⊠K, called the fusion product.

It had been widely expected (but never proven) that this operation should be unitary. Namely, that the choice of LG-invariant inner products on H and on K should induce an LG-invariant inner product on H⊠K. We show that this is not the case: there is an anomaly.
More precisely, there is an ℝ₊-torsor canonically associated to Σ. It is only after trivialising of this ℝ₊-torsor that the fusion product acquires an LG-invariant inner product. (The same statement applies when Σ is an arbitrary Riemann surface with boundary.)
Joint work with James Tener.

Having made sense of differential equations driven by rough paths, we now have a new set of models available but when it comes to calibrating them to data, the tools are still underdeveloped. I will present some results and discuss some challenges related to building these tools.

Aharoni and Berger conjectured that in any bipartite multigraph that is properly edge-coloured by n colours with at least n+1 edges of each colour there must be a matching that uses each colour exactly once (such a matching is called rainbow). This conjecture recently have been proved asymptotically by Pokrovskiy. In this talk, I will consider the same question without the bipartiteness assumption. It turns out that in any multigraph with bounded edge multiplicities, that is properly edge-coloured by n colours with at least n+o(n) edges of each colour, there must be a matching of size n-O(1) that uses each colour at most once. This is joint work with Peter Keevash.

Consider the Loewner equation associated to the upper-half plane. This is an equation originated from an extremal problem in complex analysis. Nowadays, it attracts a lot of attention due to its connection to probability. Normally this equation is driven by a real-valued function. In this talk, we will show that the equation still makes sense when being driven by a complex-valued function. We will relate this situation to the classical situation and also to complex dynamics. 

We show that a closed almost Kähler 4-manifold of globally constant holomorphic sectional curvature k<=0 with respect to the canonical Hermitian connection is automatically Kähler. The same result holds for k < 0 if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern–Weil theory implies useful integral formulas, which are then combined with results from Seiberg–Witten theory.

I will talk about a generalisation of the Seiberg-Witten equations introduced by Taubes and Pidstrygach, in dimension 3 and 4 respectively, where the spinor representation is replaced by a hyperKahler manifold admitting certain symmetries. I will discuss the 4-dimensional equations and their relation with the almost-Kahler geometry of the underlying 4-manifold. In particular, I will show that the equations can be interpreted in terms of a PDE for an almost-complex structure on 4-manifold. This generalises a result of Donaldson. 

In recognition of a lifetime's contribution across the mathematical sciences, we are initiating a series of annual Public Lectures in honour of Roger Penrose. The first lecture will be given by his long-time collaborator and friend Stephen Hawking.

Unfortunately the lecture is now sold out and we have a full waiting list. However, we will be podcasting the lecture live (and also via the University of Oxford Facebook page).