Past

The number of solutions of a given algebro-geometric configuration, when it is finite, does not change upon a small perturbation of the parameters; this persists 
even upon specializations that change the topology.    The precise formulation of this principle of Poncelet and Schubert   required, i.a., the notions of   algebraically closed fields, flatness, completenesss, multiplicity.     I will explain a model-theoretic version, presented in   quite different terms.  It applies notably to difference equations involving the Galois-Frobenius automorphism $x^p$, uniformly in a prime $p$.   In fixed positive characteristic, interesting technical problems arise that I will discuss if time permits.  

Let $\mathbb K$ be a field, and $\mathcal M$ be the “projective linear" moduli stack of objects in a suitable $\mathbb K$-linear abelian category  $\mathcal A$ (such as the coherent sheaves coh($X$) on a smooth projective $\mathbb K$-scheme $X$) or triangulated category $\mathcal T$ (such as the derived category $D^b$coh($X$)). I will explain how to define a Lie bracket [ , ] on the homology $H_*({\mathcal M})$ (with a nonstandard grading), making $H_*({\mathcal M})$ into a graded Lie algebra. This is a new variation on the idea of Ringel-Hall algebra.
 There is also a differential-geometric version of this: if $X$ is a compact manifold with a geometric structure giving instanton-type equations (e.g. oriented Riemannian 4-manifold, $G_2$-manifold, Spin(7)-manifold) then we can define Lie brackets both on the homology of the moduli spaces of all $U(n)$ or $SU(n)$ connections on $X$ for all $n$, and on the homology of the moduli spaces of instanton $U(n)$ or $SU(n)$ connections on $X$ for all $n$.
 All this is (at least conjecturally) related to enumerative invariants, virtual cycles, and wall-crossing formulae under change of stability condition.
 Several important classes of invariants in algebraic and differential geometry — (higher rank) Donaldson invariants of 4-manifolds (in particular with $b^2_+=1$), Mochizuki invariants counting semistable coherent sheaves on surfaces, Donaldson-Thomas type invariants for Fano 3-folds and CY 4-folds — are defined by forming virtual classes for moduli spaces of “semistable” objects, and integrating some cohomology classes over them. The virtual classes live in the homology of the “projective linear" moduli stack. Yuuji Tanaka and I are working on a way to define virtual classes counting strictly semistables, as well as just stables / stable pairs. 
 I conjecture that in all these theories, the virtual classes transform under change of stability condition by a universal wall-crossing formula (from my previous work on motivic invariants) in the Lie algebra $(H_*({\mathcal M}), [ , ])$. 

The Gyárfás-Sumner conjecture says that every graph with huge (enough) chromatic number and bounded clique number contains any given forest as an induced subgraph. The Erdős-Hajnal conjecture says that for every graph H, all graphs not containing H as an induced subgraph have a clique or stable set of polynomial size. This talk is about a third problem related to both of these, the following. Say an n-vertex graph is "c-coherent" if every vertex has degree <cn, and every two disjoint vertex subsets of size at least cn have an edge between them. To prove a given graph H satisfies the Erdős-Hajnal conjecture, it is enough to prove H satisfies the conjecture in all c-coherent graphs and their complements, where c>0 is fixed and as small as we like. But for some graphs H, all c-coherent graphs contain H if c is small enough, so half of the task is done for free. Which graphs H have this property? Paths do (a theorem of Bousquet, Lagoutte, and Thomassé), and non-forests don't. Maybe all forests do? In other words, do all c-coherent graphs with c small enough contain any given forest as an induced subgraph? That question is the topic of the talk. It looks much like the Gyárfás-Sumner conjecture, but it seems easier, and there are already several pretty results. For instance the conjecture is true for all subdivided caterpillars (which is more than we know for Gyárfás-Sumner), and all trees of radius two. Joint work with Maria Chudnovsky, Jacob Fox, Anita Liebenau, Marcin Pilipczuk, Alex Scott and Sophie Spirkl.

We explore the use of applying multiple preconditioners for solving linear systems arising in simulations of incompressible two-phase flow. In particular, we use a selective MPGMRES algorithm, for which the search space grows linearly throughout the iterative solver, and block preconditioners based on Schur complement approximations

We introduce a discontinuous Galerkin finite element method (DGFEM) for Hamilton–Jacobi–Bellman equations on piecewise curved domains, and prove that the method is consistent, stable, and produces optimal convergence rates. Upon utilising a long standing result due to N. Krylov, we may characterise the Monge–Ampère equation as a HJB equation; in two dimensions, this HJB equation can be characterised further as uniformly elliptic HJB equation, allowing for the application of the DGFEM

We are all familiar with the need for continuum mechanics-based models in physical applications. In this case, we are interested in large-scale water-wave problems, such as coastal flows and dam breaks.
When modelling these problems, we inevitably wish to solve them on a finite domain, and require boundary conditions to do so. Ideally, we would recreate the semi-infinite nature of a coastline by allowing any generated waves to flow out of the domain, as opposed to them reflecting off the far-field boundary and disrupting the remainder of our simulation. However, applying an appropriate boundary condition is not as straightforward as we might think.
In this talk, we aim to evaluate alternatives to so-called 'active boundary condition' absorption. We will derive a toy model of a shallow-water wavetank, and consider the implementation and efficacy of two 'passive' absorption techniques.
 

We study the winding mode sector of recently discovered string theories, which were, until now, believed to describe only conventional field theories in target space. We discover that upon compactification winding modes allows the string to acquire an oscillator spectrum giving rise to an infinite tower of massive higher-spin modes. We study the spectra, S-matrices, T-duality and high-energy behaviour of the bosonic and supersymmetric models. In the tensionless limit, we obtain formulae for amplitudes based on the scattering equations. The windings decouple from the scattering equations but remain in the integrands. The existence of this winding sector shows that these new theories do have stringy aspects and describe non-conventional field theories.  This talk is based on https://arxiv.org/abs/1710.01241.

Systemic risk arises as a multi-layer network phenomenon. Layers represent direct financial exposures of various types, including interbank liabilities, derivative or foreign exchange exposures. Another network layer of systemic risk emerges through common asset holdings of financial institutions. Strongly overlapping portfolios lead to similar exposures that are caused by price movements of the underlying financial assets. Based on the knowledge of portfolio holdings of financial agents we quantify systemic risk of overlapping portfolios. We present an optimization procedure, where we minimize the systemic risk in a given financial market by optimally rearranging overlapping portfolio networks, under the constraints that the expected returns and risks of the individual portfolios are unchanged. We explicitly demonstrate the power of the method on the overlapping portfolio network of sovereign exposure between major European banks by using data from the European Banking Authority stress test of 2016. We show that systemic-risk-efficient allocations are accessible by the optimization. In the case of sovereign exposure, systemic risk can be reduced by more than a factor of two, without any detrimental effects for the individual banks. These results are confirmed by a simple simulation of fire sales in the government bond market. In particular we show that the contagion probability is reduced dramatically in the optimized network.
 

Existence results in large for fully non-linear compressible multi-fluid models are in the mathematical literature in a short supply (if not non-existing). In this talk, we shall recall the main ideas of Lions' proof of the existence of weak solutions to the compressible (mono-fluid) Navier-Stokes equations in the barotropic regime. We shall then eplain how this approach can be adapted to the construction of weak solutions to some simple multi-fluid models. The main tools in the proofs are renormalization techniques for the continuity and transport equations. They will be discussed in more detail.

Developments in geometry and low dimensional topology have given renewed vigour to the following classical question: to what extent do the finite images of a finitely presented group determine the group? I'll survey what we know about this question in the context of 3-manifolds, and I shall present recent joint work with McReynolds, Reid and Spitler showing that the fundamental groups of certain hyperbolic orbifolds are distingusihed from all other finitely generated groups by their finite quotients.

Smooth Gaussian functions appear naturally in many areas of mathematics. Most of the talk will be about two special cases: the random plane model and the Bargmann-Fock ensemble. Random plane wave are conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator in a generic domain. The Bargmann-Fock ensemble appears in quantum mechanics and is the scaling limit of the Kostlan ensemble, which is a good model for a `typical' projective variety. It is believed that these models, despite very different origins have something in common: they have scaling limits that are described be the critical percolation model. This ties together ideas and methods from many different areas of mathematics: probability, analysis on manifolds, partial differential equation, projective geometry, number theory and mathematical physics. In the talk I will introduce all these models, explain the conjectures relating them, and will talk about recent progress in understanding these conjectures.

TBC

 The basic algebra-geometry dictionary for finitely generated k-algebras is one of the triumphs of 19th and early 20th century mathematics.  However, classes of related rings, such as their k-subalgebras, lack clean general properties or organizing principles, even when they arise naturally in problems of smooth projective geometry.  “Stabilization” in smooth topology and symplectic geometry, achieved by products with Euclidean space, substantially simplifies many
problems.  We discuss an analog in the more rigid setting of algebraic and arithmetic geometry, which, among other things (e.g., applications to counting rational points), gives some structure to the study of k-subalgebras.  We focus on the case of the moduli space of stable rational n-pointed curves to illustrate.

Wondering about how to organise your DPhil? How to make the most of your supervision meetings?

In this session we will explore these and other questions related to what makes a successful DPhil with help from faculty members, postdocs and DPhil students.

• In the first half of the session Dan Ciubotaru and Philip Maini will give short talks on their experiences as PhD students and supervisors.
• The second part of the session will be a panel discussion with final-year Dphil students and early postdocs.

The panel will consist of Thomas Wasserman, Renee Hoekzema, Jaroslav Fowkes and Carolina Matte Gregory. Senior faculty members will be kindly asked to leave the lecture theatre to ensure that students feel comfortable discussing their experiences with other students and postdocs without any senior faculty present.

In his landmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. He proved a great deal about these congruences, and also posed some questions: how many cusp forms of a given level are congruent to the Eisenstein series? How big is the extension generated by their coefficients? In joint work with Preston Wake, we give an answer to these questions in terms of cup products (and Massey products) in Galois cohomology. Time permitting, we may be able to indicate some partial generalisations of Mazur's results to square-free level.

We study a dynamic multi-asset economy with private information, a stock and a derivative. There are informed and uninformed investors as well as bounded rational investors trading on noise. The noisy rational expectations equilibrium is obtained in closed form. The equilibrium stock price follows a non-Markovian process, is positive and has stochastic volatility. The derivative cannot be replicated, except at rare endogenous times. At any point in time, the derivative price adds information relative to the stock price, but the pair of prices is less informative than volatility, the residual demand or the history of prices. The rank of the asset span drops at endogenous times causing turbulent trading activity. The effects of financial innovation are discussed. The equilibrium is fully revealing if the derivative is not traded: financial innovation destroys information.

Network models may be applied to describe many complex systems, and in the era of online social networks the study of dynamics on networks is an important branch of computational social science.  Cascade dynamics can occur when the state of a node is affected by the states of its neighbours in the network, for example when a Twitter user is inspired to retweet a message that she received from a user she follows, with one event (the retweet) potentially causing further events (retweets by followers of followers) in a chain reaction. In this talk I will review some simple models that can help us understand how social contagion (the spread of cultural fads and the viral diffusion of information) depends upon the structure of the social network and on the dynamics of human behaviour. Although the models are simple enough to allow for mathematical analysis, I will show examples where they can also provide good matches to empirical observations of cascades on social networks.

Solving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the non standard interface conditions are naturally defined at the boundary between elements. In this work we introduce the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. We present the detailed analysis of the hybrid discontinuous Galerkin method for the Stokes problem with non standard boundary conditions. This analysis is supported by numerical evidence. In addition, the advantage of the new preconditioners over more classical choices is also supported by numerical experiments.

This work was done in collaboration with G. Barrenechea, M. Bosy (Univ. Strathclyde) and F. Nataf, P-H Tournier (Univ of Paris VI)

We will discuss nonlinear cross-diffusion models describing cell motility of two distinct populations. The continuum PDE model is derived systematically from a stochastic discrete model consisting of impenetrable diffusing spheres. In this talk, I will outline the derivation of the cross-diffusion model, discuss some of its features such as the gradient-flow structure, and show numerical results comparing the discrete stochastic system to the derived model.

Stable commutator length (scl) is a well established invariant of group elements g  (write scl(g)) and  has both geometric and algebraic meaning.

It is a phenomenon that many classes of non-positively curved groups have a gap in stable commutator length: For every non-trivial element g, scl(g) > C for some C>0. Such gaps may be found in hyperbolic groups, Baumslag-solitair groups, free products, Mapping class groups, etc. 
However, the exact size of this gap usually unknown, which is due to a lack of a good source of “quasimorphisms”.

In this talk I will construct a new source of quasimorphisms which yield optimal gaps and show that for Right-Angled Artin Groups and their subgroups the gap of stable commutator length is exactly 1/2. I will also show this gap for certain amalgamated free products.

With the evolution towards the IoT and cyber-physical systems, the role that the underlying hardware plays in securing an application is becoming more prominent. Hardware can be used constructively, e.g. for accelerating computationally- intensive cryptographic algorithms. Hardware can also be used destructively, e.g., for physical attacks or transistor-level Trojans which are virtually impossible to detect. In this talk, we will present case studies for high-speed cryptography used in car2x communication and recent research on low-level hardware Trojans. 

A famous theorem of Mantel from 1907 states that every n-vertex graph with more than n^2/4 edges contains at least one triangle. In the 50s, Erdős asked for a quantitative version of this statement: for every n and e, how many triangles must an n-vertex e-edge graph contain?

This question has received a great deal of attention, and a long series of partial results culminated in an asymptotic solution by Razborov, extended to larger cliques by Nikiforov and Reiher. Currently, an exact solution is only known for a small range of edge densities, due to Lovász and Simonovits. In this talk, I will discuss the history of the problem and recent work which gives an exact solution for almost the entire range of edge densities. This is joint work with Hong Liu and Oleg Pikhurko.