Past

Deficiency is a measure of how complicated the presentations of a particular group need to be; it is defined as the maximum of the number of generators minus the number of relators (over all finite presentations of the group). This talk will introduce the basics of deficiency, give a deft example of Swan which illustrates why our understanding of deficiency is deficient, and conclude with some new examples that defy this defeatism: finite $p$-groups can have any deficiency you could (reasonably) wish for.

The $r$-neighbour bootstrap process on a graph $G$ starts with an initial set of "infected" vertices and, at each step of the process, a healthy vertex becomes infected if it has at least $r$ infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of $G$ becomes infected during the process, then we say that the initial set percolates.

In this talk I will discuss the proof of a conjecture of Balogh and Bollobás: for fixed $r$ and $d\to\infty$, the minimum cardinality of a percolating set in the $d$-dimensional hypercube is $\frac{1+o(1)}{r}\binom{d}{r-1}$. One of the key ideas behind the proof exploits a connection between bootstrap percolation and weak saturation. This is joint work with Jonathan Noel.

Let G be a split reductive group over a finite extension k of Q_p. Reeder and Yu have given a new construction of supercuspidal representations of G(k) using geometric invariant theory. Their construction is uniform for all p but requires as input stable vectors in certain representations coming from Moy-Prasad filtrations. In joint work, Jessica Fintzen and I have classified the representations of this kind which contain stable vectors; as a corollary, the construction of Reeder-Yu gives new representations when p is small. In my talk, I will give an overview of this work, as well as explicit examples for the case when G = G_2. For these examples, I will explicitly describe the locus of all stable vectors, as well as the Langlands parameters which correspond under the local Langlands correspondence to the representations of G(k). 

The past few years have seen a surge of interest in nonconvex reformulations of convex optimizations using nonlinear reparameterizations of the optimization variables. Compared with the convex formulations, the nonconvex ones typically involve many fewer variables, allowing them to scale to scenarios with millions of variables. However, one pays the price of solving nonconvex optimizations to global optimality, which is generally believed to be impossible. In this talk, I will characterize the nonconvex geometries of several low-rank matrix optimizations. In particular, I will argue that under reasonable assumptions, each critical point of the nonconvex problems either corresponds to the global optimum of the original convex optimizations, or is a strict saddle point where the Hessian matrix has a negative eigenvalue. Such a geometric structure ensures that many local search algorithms can converge to the global optimum with random initializations. Our analysis is based on studying how the convex geometries are transformed under nonlinear parameterizations.

I will give an overview of the complexes used in algebraic topology using the language of abstract complexes.

This is a lunch seminar, so feel free to bring your lunch along!

In this presentation, we give an overview of the numerical methods used in commercial oil and gas reservoir simulation. The models are described by flow through porous media and are solved using a series of nested numerical methods. Most of the computational effort resides in solving large linear systems resulting from Newton iterations. Therefore, we will go in greater detail about the iterative linear solvers and preconditioning techniques.

Note: This talk will cover similar topics to the InFoMM group meeting talks on Friday 28th April, but I will discuss more mathematical details for this JAMS talk.

In 1997, Maxim Kontsevich gave a universal formula for the
quantization of Poisson brackets.  It can be viewed as a perturbative
expansion in a certain two-dimensional topological field theory.  While the
formula is explicit, it is currently impossible to compute in all but the
simplest cases, not least because the values of the relevant Feynman
integrals are unknown.  In forthcoming joint work with Peter Banks and Erik
Panzer, we use Francis Brown's approach to the periods of the moduli space
of genus zero curves to give an algorithm for the computation of these
integrals in terms of multiple zeta values.  It allows us to calculate the
terms in the expansion on a computer for the first time, giving tantalizing
evidence for several open conjectures concerning the convergence and sum of
the series, and the action of the Grothendieck-Teichmuller group by gauge
transformations.

Computing the solutions $u$ of an equation $f(u, \lambda) = 0$ as the parameter $\lambda$ is varied is a central task in applied mathematics and engineering. In this talk I will present a new algorithm, deflated continuation, for this task.

Deflated continuation has three main advantages. First, it is capable of computing disconnected bifurcation diagrams; previous algorithms only aimed to compute that part of the bifurcation diagram continuously connected to the initial data. Second, its implementation is extremely simple: it only requires a minor modification to any existing Newton-based solver. Third, it can scale to very large discretisations if a good preconditioner is available.

Among other problems, we will apply this to a famous singularly perturbed ODE, Carrier's problem. The computations reveal a striking and beautiful bifurcation diagram, with an infinite sequence of alternating pitchfork and fold bifurcations as the singular perturbation parameter tends to zero. The analysis yields a novel and complete taxonomy of the solutions to the problem, and demonstrates that a claim of Bender & Orszag (1999) is incorrect. We will also use the algorithm to calculate distinct local minimisers of a topology optimisation problem via the combination of deflated continuation and a semismooth Newton method.

Character varieties have been studied largely by means of their correspondence to the moduli space of Higgs bundles. In this talk we will report on a method to study their Hodge structure, in particular to compute their E- polynomials. Moreover, we will explain some applications of the given method such as, the study of the topology of the moduli space of doubly periodic instantons. This is joint work with A. González, V.Muñoz and P. Newstead.

Permutations of finitely many elements are often drawn as permutation diagrams. We take this point of view as a motivation to construct and describe more complicated algebras arising for instance from differential operators, from operators acting on (co)homologies, from invariant theory, or from Hecke algebras. The surprising fact is that these diagrams are elementary and simple to describe, but at the same time describe relations between cobordisms as well as categories of represenetations of p-adic groups. The goal of the talk is to give some glimpses of these phenomena and indicate which role categorification plays here.
 

The management of natural resources, from fisheries and climate change to gut bacteria colonies, all require the development of ecological models that represent the full spectrum of population interactions, from competition, through mixotrophy and mutualism, to predation.

Mixotrophic plankton, that both photosynthesise and eat other plankton, underpin all marine food webs and help regulate climate by facilitating gas exchange between the ocean and atmosphere. We show the recent discovery that their feeding preferences change with increasing temperature implies climate change could dramatically alter the structure of marine food webs.

We describe a theoretical framework that reveals the key role of mixotrophy in facilitating transitions between trophic interactions. Mixotrophy smoothly and stably links competition to predation, and extends this linkage to include mutualism in both facultative and obligate forms. Such smooth stable transitions further allow the development of eco-evolutionary theory at the population level through quantitative trait modelling.

Transseries arise naturally when solving differential equations around essential singularities. Just like most Taylor series are not convergent, most transseries do not converge to real functions, even when using advanced summation techniques.

On the other hand, we can show that all classical transseries induce analytic functions on the surreal line. In fact, this holds for an even larger (proper) class of series which we call "omega-series".

Omega-series can be composed and differentiated, like LE-series, and they form a differential subfield of surreal numbers equipped with the simplest derivation. This raises once again the question whether all surreal numbers can be also interpreted as functions. Unfortunately, it turns out that the simplest derivation is in fact incompatible with this goal.

This is joint work with A. Berarducci.

In this talk, I will give a brief overview of the Langlands program and Langlands functoriality with reference to the examples of Galois representations attached to cusp forms and the Jacquet-Langlands correspondence for $\mathrm{GL}_2$. I will then explain how one can generalise this idea, sketching a proof of a Jacquet-Langlands type correspondence from $\mathrm{U}_n(B)$, where $B$ is a quaternion algebra, to $\mathrm{Sp}_{2n}$ and showing that one can attach Galois representations to regular algebraic cuspidal automorphic representations of $\mathrm{Sp}_{2n}$.
 

In this lecture I intend to review a few selected recent results on numerical approximations for high-dimensional nonlinear parabolic partial differential equations (PDEs), nonlinear stochastic ordinary differential equations (SDEs), and high-dimensional nonlinear forward-backward stochastic ordinary differential equations (FBSDEs). Such equations are key ingredients in a number of pricing models that are day after day used in the financial engineering industry to estimate prices of financial derivatives. The lecture includes content on lower and upper error bounds, on strong and weak convergence rates, on Cox-Ingersoll-Ross (CIR) processes, on the Heston model, as well as on nonlinear pricing models for financial derivatives. We illustrate our results by several numerical simulations and we also calibrate some of the considered derivative pricing models to real exchange market prices of financial derivatives on the stocks in the American Standard & Poor's 500 (S&P 500) stock market index.

Many systems in nature consist of stochastically interacting agents or particles. Stochastic processes have been widely used to model such systems, yet they are notoriously difficult to analyse. In this talk I will show how ideas from statistics can be used to tackle some challenging problems in the field of stochastic processes.

In the first part, I will consider the problem of inference from experimental data for stochastic reaction-diffusion processes. I will show that multi-time distributions of such processes can be approximated by spatio-temporal Cox processes, a well-studied class of models from computational statistics. The resulting approximation allows us to naturally define an approximate likelihood, which can be efficiently optimised with respect to the kinetic parameters of the model. 

In the second part, we consider more general path properties of a certain class of stochastic processes. Specifically, we consider the problem of computing first-passage times for Markov jump processes, which are used to describe systems where the spatial locations of particles can be ignored.  I will show that this important class of generally intractable problems can be exactly recast in terms of a Bayesian inference problem by introducing auxiliary observations. This leads us to derive an efficient approximation scheme to compute first-passage time distributions by solving a small, closed set of ordinary differential equations.

Almost all real-world applications involve a degree of uncertainty. This may be the result of noisy measurements, restrictions on observability, or simply unforeseen events. Since many models in both engineering and the natural sciences make use of partial differential equations (PDEs), it is natural to consider PDEs with random inputs. In this context, passing from modelling and simulation to optimization or control results in stochastic PDE-constrained optimization problems. This leads to a number of theoretical, algorithmic, and numerical challenges.

 From a mathematical standpoint, the solution of the underlying PDE is a random field, which in turn makes the quantity of interest or the objective function an implicitly defined random variable. In order to minimize this distributed objective, one can use, e.g., stochastic order constraints, a distributionally robust approach, or risk measures. In this talk, we will make use of risk measures.

After motivating the approach via a model for the mitigation of an airborne pollutant, we build up an analytical framework and introduce some useful risk measures. This allows us to prove the existence of solutions and derive optimality conditions. We then present several approximation schemes for handling non-smooth risk measures in order to leverage existing numerical methods from PDE-constrained optimization. Finally, we discuss solutions techniques and illustrate our results with numerical examples.

We consider the Euler–Voigt equations in a bounded domain as an approximation for the 3D Euler equations. We adopt suitable physical conditions and show that the solutions of the Voigt equations are global, do not smooth out the solutions and converge to the solutions of the Euler equations, hence they represent a good model.

 This talk will discuss the so-called ``generic cohomology’’ of function fields over algebraically closed fields, from the point of view of motives and/or Zariski geometry. In particular, I will describe some interesting connections between cup products, algebraic dependence, and (geometric) valuation theory. As an application, I will mention a new result which reconstructs higher-dimensional function fields from their generic cohomology, endowed with some additional motivic data. 

   Everyone welcome!
 

We give a construction of a boundary (the Morse boundary) which can be assigned to any proper geodesic metric space and which is rigid, in the sense that a quasi-isometry of spaces induces a homeomorphism of boundaries. To obtain a more workable invariant than the homeomorphism type, I will introduce the metric Morse boundary and discuss notions of capacity and conformal dimensions of the metric Morse boundary. I will then demonstrate that these dimensions give useful invariants of relatively hyperbolic and mapping class groups. This is joint work with Matthew Cordes (Technion).

The chromatic number of a graph is trivially bounded from above by the maximum degree plus one, and from below by the size of a largest clique. Reed proved in 1998 that compared to the trivial upper bound, we can always save a number of colors proportional to the gap between the maximum degree and the size of a largest clique. A key step in the proof deals with how to spare colors in a graph whose every vertex "sees few edges" in its neighborhood. We improve the existing approach, and discuss its applications to Reed's theorem and strong edge coloring.  This is joint work with Thomas Perrett (Technical University of Denmark) and Luke Postle (University of Waterloo).

We will discuss the Chapter "Cohomology" from the book "Elementary applied topology" by Robert Ghrist (available at https://www.math.upenn.edu/~ghrist/notes.html).

This is a lunch seminar, so feel free to bring your lunch along!

This talk is about a 4d "(ambi-)twistor string" formula for sEYM tree-level scattering amplitudes and is work in progress. The formula sheds some new light on the translation between CHY formulae and (ambi-)twistor formulae in general, potentially even at loop level.

In the mathematical theory of liquid crystals, a hedgehog is a universal equilibrium solution for Frank's elastic free-energy functional. It is characterized by a radial defect for the nematic director, reminiscent of the way spines are arranged in the spiny mammal. For certain choices of Frank's elastic constants, the free energy stored in a ball subject to radial boundary conditions for the director is minimized by a hedgehog with its defect in the centre of the ball. For other choices of Frank's constants, it is known that a radial hedgehog cannot be a minimizer for this variational problem. We shall gather evidence supporting the conjecture that a "twisted" hedgehog takes the place of a radial hedgehog as an energy minimizer (and we shall not fail to say in which sense it is "twisted"). We shall also show that a twisted hedgehog often accompanies, unseen, a radial hedgehog, as its virtual double, ready to beat its energy as a certain elastic anisotropy is reached.

I will described how ideas from constructive quantum field theory can be adapted to produce a systematic approach for analytic renormalization in the theory of regularity structures.

Knots and links naturally appear in the neighbourhood of the singularity of a complex curve; this creates a bridge between algebraic geometry and differential topology. I will discuss a topological approach to the study of 1-parameter families of singular curves, using correction terms in Heegaard Floer homology. This is joint work with József Bodnár and Daniele Celoria.

We will give an overview of the Soliton Resolution Conjecture, focusing mainly on the Wave Maps Equation. This is a program about understanding the formation of singularities for a variety of critical hyperbolic/dispersive equations, and stands as a remarkable topic of research in modern PDE theory and Mathematical Physics. We will be presenting our contributions to this field, elaborating on the required background, as well as discussing some of the latest results by various authors.

The Hastings-Levitov models describe the growth of random sets (or clusters) in the complex plane as the result of iterated composition of random conformal maps. The correlations between these maps are determined by the harmonic measure density profile on the boundary of the clusters. In this talk I will focus on the simplest case, that of i.i.d. conformal maps, and obtain a description of the local fluctuations of the harmonic measure density around its deterministic limit, showing that these are Gaussian. This is joint work with James Norris.

Calabi-Yau 3-folds play a large role in string theory.  Cohomology of sheaves on such varieties has many uses in string theory, including counting the number of particles or fields in a theory, as well as to help identify terms in the superpotential that determines the equations of motion of the corresponding string theory, and many other uses as well.  As a computational algebraic geometer, string theory provides a rich source of new computational problems to solve.

In this talk, we focus on the search for rigid divisors on these Calabi-Yau hypersurfaces of toric varieties.  We have had methods to compute sheaf cohomology on these varieties for many years now (Eisenbud-Mustata-Stillman, around 2000), but these methods fail for many of the examples of interest, in that they take a very long time, or the software (wisely) refuses to try!

We provide techniques and formulas for the sheaf cohomology of certain divisors of interest in string theory, that other current methods cannot handle.  Along the way, we describe a Macaulay2 package for computing with these objects, and show its use on examples.

This is joint work with Andreas Braun, Cody Long, Liam McAllister, and Benjamin Sung.

Research takes a long time while the attention span of the world is apparently decreasing, so today's researchers need to be able to get their message across quickly and succinctly. In this session we'll share some tips on how to communicate the key messages of your work in just a few minutes, and give you a chance to have a go yourself.  This will be helpful for job and funding applications and interviews, and also for public engagement. In September there will be an opportunity to do it for real, for our alumni, when we'll showcase Oxford Mathematics at the Alumni Weekend.

We study vortex sheets for the relativistic Euler equations in three-dimensional Minkowski spacetime. The problem is a nonlinear hyperbolic problem with a characteristic free boundary. The so-called Lopatinskii condition holds only in a weak sense, which yields losses of derivatives. A necessary condition for the weak stability is obtained by analyzing roots of the Lopatinskii determinant associated to the linearized problem. Under such stability condition,  we prove short-time existence and nonlinear stability of relativistic vortex sheets by the Nash-Moser iterative scheme.

We consider the free boundary problem for 2D current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. 
In this talk we present our results about the well-posedness of the problem in the sense of Hadamard, under a suitable stability condition, that is the 
local-in-time existence in Sobolev spaces and uniqueness of smooth solutions to the Cauchy problem, and the strong continuous dependence on the data in the same topology.
Joint works with: Alessandro Morando and Paola Trebeschi.
 

 I will start with a motivation of what algebraic and model-theoretic properties an algebraically closed field of characteristic 1 is expected to have. Then I will explain how these properties forces one to follow the route of Hrushovski's construction leading to a a 'pseudo-analytic' structure which we identify as an algebraically closed field of characteristic 1 . Then I am able to formulate very precise axioms that such a field must satisfy.  The main theorem then states that under the axioms the structure has the desired algebraic and analytic properties. The axioms have a form of statements about existence of solutions to systems of equations in terms of a 'multi-dimensional' valuation theory and the validity of these statements is an open problem to be discussed. 
This is a joint work with Alex Cruz Morales.
 

1.Kremnitzer. I will explain an approach to constructing geometries relative to a symmetric monoidal 
category. I will then introduce the category of normed sets as a possible analytic geometry over 
the field with one element. I will show that the Fargues-Fontaine curve from p-adic Hodge theory and 
the Connes-Bost system are naturally interpreted in this geometry. This is joint work with Federico Bambozzi and 
Oren Ben-Bassat.
 

In this talk, I discuss the positive geometry of the Wilson Loop Diagrams appearing in SYM N-4 theory. In particular, I define an algorithm for associating Wilson Loop diagrams to convex cells of the positive Grassmannians. Using combinatorics of these cells, I then consider the geometry of N^2MHV diagrams on 6 points.

The "Weyl fermion" was discovered in a topological semimetal in
2015. Its mathematical characterisation turns out to involve deep and subtle
results in differential topology. I will outline this theory, and explain
some connections to Euler structures, torsion of manifolds,
and Seiberg-Witten invariants. I also propose interesting generalisations
with torsion topological charges arising from Kervaire semicharacteristics
and ``Quaternionic'' characteristic classes.

The technique of scanning, or the parameterised Pontrjagin--Thom construction, has been extraordinarily successful in calculating the cohomology of configuration spaces (McDuff), moduli spaces of Riemann surfaces (Madsen, Tillmann, Weiss), moduli spaces of graphs (Galatius), and moduli spaces of manifolds of higher dimension (Galatius, R-W, Botvinnik, Perlmutter), with constant coefficients. In each case the method also works to study the cohomology of moduli spaces of objects equipped with a "tangential structure". I will explain how choosing an auxiliary highly-symmetric tangential structure often lets one calculate the cohomology of these moduli spaces with large families of twisted coefficients, by exploiting the symmetries of the tangential structure and using a little representation theory.

 

 Markus Szymik and I computed the homology of the Higman-Thompson groups by first showing that they stabilize (with slope 0), and then computing the stable homology. I will in this talk give a new point of view on the computation of the stable homology using Thumann's "operad groups". I will also give an idea of how scanning methods can enter the picture. (This is partially joint work with Søren Galatius.) 

It is hard to detect the exotic nature of an exotic n-sphere M 
in homotopical features of the diffeomorphism group Diff(M). The well 
known reason is that Diff(M) contains a big topological subgroup H which 
is identified with the group of diffeomorphisms rel boundary of the 
n-disk, with a small coset space Diff(M)/H which is invariably homotopy 
equivalent to O(n+1). Therefore it seems that our only chance to detect 
the exotic nature of M in homotopical features of Diff(M) is to see 
something in this extension.  (To make sense of "homotopical features of 
Diff(M)" one should think of Diff(M) as a space with a multiplication 
acting on an n-sphere.) I am planning to report on PhD work of O Sommer 
and calculations due to myself and Sommer which, if all goes well, would 
show that Diff(M) has some exotic homotopical properties in the case 
where M is the 7-dimensional exotic sphere of Kervaire-Milnor fame which 
bounds a compact smooth framed 8-manifold of signature 8. The 
theoretical work is based on classical smoothing theory and the 
calculations would be based on ever-ongoing (>30 years) joint work 
Weiss-Williams, and might give me and Williams another valuable 
incentive to finish it.

We construct the diffeomorphism-equivariant “scanning map” associated to the configuration spaces of manifolds with twisted summable labels. The scanning map is also functorial with respect to embeddings of manifolds. To adapt P. Salvatore's idea of non-commutative summation into twisted setting, we define a bundle of Fulton-MacPherson operads over a manifold M whose fibres are built within tangent spaces of M.

In 1956 Milnor published a paper proving that there are manifolds homeomorphic to the 7-sphere but not diffeomorphic to it. Seeking to generalise this example, he was led in around 1960 to introduce a construction for  killing homotopy groups of manifolds. When this was generalised to killing relative homotopy groups it became a general and powerful method of construction. An obstruction arises to killing the last group, and the analysis of this obstruction in general leads to a new theory.

Categorification of knot polynomials -- Daniele Celoria

Classically, the most powerful and versatile knot invariants take the form of polynomials. These can usually be defined by simple recursive equations, known as skein relations; after giving the main examples of polynomial knot invariants (Alexander and Jones polynomials), we are going to informally introduce categorifications. Finally we are going to present the Knot Floer and the Khovanov homologies, and show that they provide a categorification of the aforementioned polynomial knot invariants.

Network science for online social media: an x-ray or a stethoscope for society -- Mariano Beguerisse

The abundance of data from social media outlets such as Twitter provides the opportunity to perform research at a societal level at a scale unforeseen. This has spurred the development of mathematical and computational methods such as network science, which uses the formalism and language of graph theory to study large systems of interacting agents. In this talk, I will provide a sketch of network science and its application to study online social media. A number of different networks can be constructed from Twitter data, which can be used to ask questions about users, ranging from the structural (an 'x-ray' to see how societies are connected online) to the topical ('stethoscope' to feel how users interact in the context of specific event). I will provide concrete examples from the UK riots of 2011, applications to medical anthropology, and political referenda, and will also highlight distinct challenges such as the directionality of connections, the size of the network, the use of temporal information and text, all of which are active areas of research.