Past

Using results by Eisenbud, Schoutens and Zilber I will propose a model theoretic structure that aims to capture the algebra (or geometry) of a non reduced scheme over an algebraically closed field. 

In this seminar, we present a fast fully homomorphic encryption (FHE) based on GSW and its ring variants. The cryptosystem relies on the hardness of lattice problems in the unique domain (e.g. the LWE family). After a brief presentation of these lattice problems, with a few notes on their asymptotic and practical average case hardness, we will present our homomorphic cryptosystem TFHE, based on a ring variant of GSW. TFHE can operate in two modes: The first one is a leveled homomorphic mode, which has the ability to evaluate deterministic automata (or branching programs) at a rate of 1 transition every 50microseconds. For the second mode, we also show that this scheme can evaluate its own decryption in only 20milliseconds, improving on the the construction by Ducas-Micciancio, and of Brakerski-Perlman. This makes the scheme fully homomorphic by Gentry's bootstrapping principle, and for instance, suitable for representing fully dynamic encrypted databases in the cloud.

An expander is a family of finite graphs of uniformly bounded degree, increasing number of vertices and Cheeger constant bounded away from zero. They occur throughout mathematics and computer science; the most famous constructions of expanders rely on powerful results in geometric group theory and number theory, while expanders are used in everything from error-correcting codes, through disproving the strongest version of the Baum-Connes conjecture, to affine sieve theory and the twin prime, Mersenne prime and Hardy-Littlewood conjectures.

However, very little was known about how different the geometry of two expanders could be. This question was raised by Ostrovskii in 2013, and a year later Mendel and Naor gave the first example of two 'distinct' expanders.

In this talk I will construct a continuum of expanders which are, in a certain sense, geometrically incomparable. Once the existence of a single expander is accepted, the remainder of the proof is a heady mix of counting, addition, multiplication, and just for the experts, a little bit of division. Two very different - and very interesting - continuums of 'distinct' expanders have since been constructed by Khukhro-Valette and Das.

We consider two types of actions on moduli spaces of quiver representations over a field k and we decompose their fixed loci using group cohomology. First, for a perfect field k, we study the action of the absolute Galois group of k on the points of this quiver moduli space valued in an algebraic closure of k; the fixed locus is the set of k-rational points and we obtain a decomposition of this fixed locus indexed by the Brauer group of k. Second, we study algebraic actions of finite groups of quiver automorphisms on these moduli spaces; the fixed locus is decomposed using group cohomology and each component has a modular interpretation. If time permits, we will describe the symplectic and holomorphic geometry of these fixed loci in hyperkaehler quiver varieties. This is joint work with Florent Schaffhauser.

We prove tight upper and lower bounds on an observable of the antiferromagnetic Potts model. From this we deduce the case d=3 of a conjecture of Galvin and Tetali on maximising the number of proper colourings in d-regular graphs.

I will discuss recent results in finite simple groups. These include growth, generation (with a number theoretic flavour), and conjectures of Gowers and Viola on mixing and complexity whose proof requires representation theory as a main tool.
 

Joint work with Wolfgang Hackbusch and Daniel Kressner

I will talk about connections between the compressible and incompressible Navier-Stokes systems. In case of the compressible model, as the bulk (volume) viscosity is very high, the divergence of the velocity becomes small, in the limit it is zero and we arrive at the case of incompressible system. An important role here is played by the inhomogeneous version of the classical Navier-Stokes equations. I plan to discuss analytical obstacle appearing within the analysis. The considerations are done in the framework of regular solutions in Besov and Sobolev spaces. The results which will be discussed are joint with Raphael Danchin from Paris.

Motivated by a problem in quasiconformal mapping, we introduce a new type of problem in complex analysis, with its roots in the mathematical physics of the Bose-Einstein condensates in superconductivity.The problem will be referred to as \emph{geometric zero packing}, and is somewhat analogous to studying Fekete point configurations.The associated quantity is a density, denoted  $\rho_\C$ in the planar case, and $\rho_{\mathbb{H}}$ in the case of the hyperbolic plane.We refer to these densities as \emph{discrepancy densities for planar and hyperbolic zero packing}, respectively, as they measure the impossibility of atomizing the uniform planar and hyperbolic area measures.The universal asymptoticvariance $\Sigma^2$ associated with the boundary behavior of conformal mappings with quasiconformal extensions of small dilatation is related to one of these discrepancy densities: $\Sigma^2= 1-\rho_{\mathbb{H}}$.We obtain the estimates$2.3\times 10^{-8}<\rho_{\mathbb{H}}\le0.12087$, where the upper estimate is derived from the estimate from below on $\Sigma^2$ obtained by Astala, Ivrii, Per\"al\"a,  and Prause, and the estimate from below is much more delicate.In particular, it follows that $\Sigma^2<1$, which in combination with the work of Ivrii shows that the maximal fractal dimension of quasicircles conjectured by Astala cannot be reached.Moreover, along the way, since the universal quasiconformal integral means spectrum has the asymptotics$\mathrm{B}(k,t)\sim\frac14\Sigma^2 k^2|t|^2$ for small $t$ and $k$, the conjectured formula $\mathrm{B}(k,t)=\frac14k^2|t|^2$ is not true.As for the actual numerical values of the discrepancy density $\rho_\C$, we obtain the estimate from above $\rho_\C\le0.061203\ldots$ by using the equilateral triangular planar zero packing, where the assertion that equality should hold can be attributed to Abrikosov. The values of $\rho_{\mathbb{H}}$ is expected to be somewhat close to the value of $\rho_\C$.

Elliptic cohomology is a family of generalised cohomology theories
$Ell_E^*$ parametrised by an elliptic curve $E$ (over some ring $R$).
Just like many other cohomology theories, elliptic cohomology admits
equivariant versions. In this talk, I will recall an old conjectural
description of elliptic cohomology, due to G. Segal, S. Stolz and P.
Teichner. I will explain how that conjectural description led me to
suspect that there should exist a generalisation of equivariant
elliptic cohomology, where the group of equivariance gets replaced by
a fusion category. Finally, I will construct $C$-equivariant elliptic
cohomology when $C$ is a fusion category, and $R$ is a ring of
characteristc zero.

One of the challenges of 21st-century science is to model the evolution of complex systems.  One example of practical importance is urban structure, for which the dynamics may be described by a series of non-linear first-order ordinary differential equations.  Whilst this approach provides a reasonable model of urban retail structure, it is somewhat restrictive owing to uncertainties arising in the modelling process.

We address these shortcomings by developing a statistical model of urban retail structure, based on a system of stochastic differential equations.   Our model is ergodic and the invariant distribution encodes our prior knowledge of spatio-temporal interactions.  We proceed by performing inference and prediction in a Bayesian setting, and explore the resulting probability distributions with a position-specific metrolpolis-adjusted Langevin algorithm.

Twistor spaces were originally devised as a way to use techniques of complex geometry to study 4-dimensional Riemannian manifolds. In this talk I will show that they also make it possible to apply techniques from symplectic geometry.  In the first part of the talk I will explain that when the 4-manifold satisfies a certain curvature inequality, its twistor space carries a natural symplectic structure. In the second part of the talk I will discuss some results in Riemannian geometry which can be proved via the symplectic geometry of the twistor space. Finally, if there is time, I will end with some speculation
about potential future applications, involving Poincaré—Einstein 4-manifolds, minimal surfaces and distinguished closed curves in their conformal infinities

We study D3 brane theories that are described as deformations of N=2 SCFTs. They arise at the self-intersection of a 7-brane in F-Theory. As we shall explain, the associated string junctions and their monodromies can be studied via torus knots or links. The monodromy reduces (potentially different) flavor algebras of dual deformations of N=2 theories and projects out charged states, leading to N=1 SCFTs. We propose an explanation for these effects in terms of an electron-monopole-dyon condensate.

Data science: The secret to unlocking operational performance within the UK’s largest retail supply chain

Chris Reddick was instrumental in setting up the InFoMM CDT. After helping secure the EPSRC funding he chaired the Industrial Engagement Committee and supported the CDT in all its Industrial relations. The success of the CDT, as evidenced by the current size of the industrial partnership and the vibrant programme we have developed, is in no small part due to Chris' charm, vision, and tenacity.

This session will look at the range of courses available to early career researchers and graduate students from MPLS. It will also discuss the benefits of training and development for researchers and how it can help you in enhancing your career inside and outside academia.
 

In this talk, I wish to address the problem of evaluating an integral on an n-dimensional complex vector space whose n-form of integration has poles along a union of (affine) hyperplanes, following the work of Heckman and Opdam. Such situation arise often in the harmonic analysis of a reductive group and when that is the case, the singular hyperplane arrangement in question is dictated by the root system of the group. I will then try to explain how we can relate the intersection lattice of the hyperplane arrangement with nilpotent orbits of a complex Lie algebra related to the root system in question.

We discuss vector bundles with numerically stable reduction on smooth complete varieties over a p-adic number field and sketch the construction of associated p-adic representations of the geometric fundamental group. On projective varieties, such bundles are semistable with respect to every polarization and have vanishing Chern classes. One of the main problems in the construction consisted in getting rid of infinitely many obstruction classes. This is achieved by adapting a theory of Bhatt based on de Jongs's alteration method. One also needs control over numerically flat bundles on arbitrary singular varieties over finite fields. The singular Riemann Roch Theorem of Baum Fulton Macpherson is a key ingredient for this step. This is joint work with Annette Werner.
 

In recent years, ideas from the world of partial differential equations (PDEs) have found their way into the arena of graph and network problems. In this talk I will discuss how techniques based on nonlinear PDE models, such as the Allen-Cahn equation and the Merriman-Bence-Osher threshold dynamics scheme can be used to (approximately) detect particular structures in graphs, such as densely connected subgraphs (clustering and classification, minimum cuts) and bipartite subgraphs (maximum cuts). Such techniques not only often lead to fast algorithms that can be applied to large networks, but also pose interesting theoretical questions about the relationships between the graph models and their continuum counterparts, and about connections between the different graph models.

The share of market making conducted by high-frequency trading (HFT) firms has been rising steadily. A distinguishing feature of HFTs is that they trade intraday, ending the day flat. To shed light on the economics of HFTs, and in a departure from existing market making theories, we model an HFT that has access to unlimited leverage intraday but must fund any end-of-day inventory at an exogenously determined cost. Even though the inventory costs only occur at the end of the day, they impact intraday price and liquidity dynamics. This gives rise to an intraday endogenous price impact mechanism. As time approaches the end of the trading day, the sensitivity of prices to inventory levels intensifies, making price impact stronger and widening bid-ask spreads. Moreover, imbalances of buy and sell orders may catalyze hikes and drops of prices, even under fixed supply and demand functions. Empirically, we show that these predictions are borne out in the U.S. Treasury market, where bid-ask spreads and price impact tend to rise towards the end of the day. Furthermore, price movements are negatively correlated with changes in inventory levels as measured by the cumulative net trading volume.
 

(based on joint work with Tobias Adrian, Erik Vogt, and Hongzhong Zhang)

We will present a very general framework for unconstrained stochastic optimization which is based on standard trust region framework using  random models. In particular this framework retains the desirable features such step acceptance criterion, trust region adjustment and ability to utilize of second order models. We make assumptions on the stochasticity that are different from the typical assumptions of stochastic and simulation-based optimization. In particular we assume that our models and function values satisfy some good quality conditions with some probability fixed, but can be arbitrarily bad otherwise. We will analyze the convergence and convergence rates of this general framework and discuss the requirement on the models and function values. We will will contrast our results with existing results from stochastic approximation literature. We will finish with examples of applications arising the area of machine learning. 
 

The non-local Fisher KPP equation is used to model non-local interaction and competition in a population. I will discuss recent work on solutions of this equation with a compactly supported initial condition, which strengthens results on the spreading speed obtained by Hamel and Ryzhik in 2013. The proofs are probabilistic, using a Feynman-Kac formula and some ideas from Bramson's 1983 work on the (local) Fisher KPP equation.

Given a Shimura variety, I will show how to define a corresponding two-sorted structure. Based on work of Chris Daw and Adam Harris, we will study what is needed for the class of this structures to be categorical. Of course, an introduction to Shimura varieties will be given.
 

Given a Shimura variety, I will show how to define a corresponding two-sorted structure. Based on work of Chris Daw and Adam Harris, we will study what is needed for the class of this structures to be categorical. Of course, an introduction to Shimura varieties will be given.

 Anyone who has worked in $\beta $N will not be surprised to learn that some of the algebraically defined subsets of $\beta N$ are not topologically simple, even though their algebraic definition may be very simple.  I shall show that the following subsets of $\beta N$ are not Borel: $N^*+N^*$; the smallest ideal of $\beta N$; the set of idempotents in $\beta N$; any semiprincipal right ideal in $\beta N$; the set of idempotents in any left ideal in $\beta N$.

Asymptotic dimension is a large-scale analogue of Lebesgue covering dimension. I will give a gentle introduction to asymptotic dimension, prove some basic propeties and give some applications to group theory. I will then define coarse homology and explain how when defined, virtual cohomological dimension gives a lower bound on asymptotic dimension.

Abstract:  Anyone who has worked in $\beta $N will not be surprised to learn that some of the algebraically defined subsets of $\beta N$ are not topologically simple, even though their algebraic definition may be very simple.  I shall show that the following subsets of $\beta N$ are not Borel: $N^*+N^*$; the smallest ideal of $\beta N$; the set of idempotents in $\beta N$; any semiprincipal right ideal in $\beta N$; the set of idempotents in any left ideal in $\beta N$.

We present “Ouroboros,” the first blockchain protocol based on proof of stake with rigorous security guarantees. We establish security properties for the protocol comparable to those achieved by the bitcoin blockchain protocol. As the protocol provides a “proof of stake” blockchain discipline, it offers qualitative efficiency advantages over blockchains based on proof of physical resources (e.g., proof of work). We showcase the practicality of our protocol in real world settings by providing experimental results on transaction processing time obtained with a prototype implementation in the Amazon cloud. We also present a novel reward mechanism for incentivizing the protocol and we prove that given this mechanism, honest behavior is an approximate Nash equilibrium, thus neutralizing attacks such as selfish mining. 

Joint work with  Alexander Russell and Bernardo David and Roman Oliynykov

I'll be talking about my masters' research in Quantum Gravity in a way that is accessible to mathematicians.

Ambitious mathematical models of highly complex natural phenomena are challenging to analyse, and more and more computationally expensive to evaluate. This is a particularly acute problem for many tasks of interest and numerical methods will tend to be slow, due to the complexity of the models, and potentially lead to sub-optimal solutions with high levels of uncertainty which needs to be accounted for and subsequently propagated in the statistical reasoning process. This talk will introduce our contributions to an emerging area of research defining a nexus of applied mathematics, statistical science and computer science, called "probabilistic numerics". The aim is to consider numerical problems from a statistical viewpoint, and as such provide numerical methods for which numerical error can be quantified and controlled in a probabilistic manner. This philosophy will be illustrated on problems ranging from predictive policing via crime modelling to computer vision, where probabilistic numerical methods provide a rich and essential quantification of the uncertainty associated with such models and their computation. 

The cobordism hypothesis gives a correspondence between the
framed local topological field theories with values in C and a fully
dualizable objects in C.  Changing framing gives an O(n) action on the
space of local TFTs, and hence by the cobordism hypothesis it gives a
(homotopy coherent) action of O(n) on the space of fully dualizable
objects in C.  One example of this phenomenon is that O(3) acts on the
space of fusion categories.  In fact, O(3) acts on the larger space of
finite tensor categories.  I'll describe this action explicitly and
discuss its relationship to the double dual, Radford's theorem,
pivotal structures, and spherical structures.  This is part of work in
progress joint with Chris Douglas and Chris Schommer-Pries.
 

Identifying correlations within multiple streams of high-volume time series is a general but challenging problem.  A simple exact solution has cost that is linear in the dimensionality of the data, and quadratic in the number of streams.  In this work, we use dimensionality reduction techniques (sketches), along with ideas derived from coding theory and fast matrix multiplication to allow fast (subquadratic) recovery of those pairs that display high correlation.

Joint work with Jacques Dark

Symplectic duality is an equivalence of mathematical structures associated to pairs of hyper-Kahler cones. All known examples arise as the `Higgs branch’ and `Coulomb branch' of a 3d superconformal quantum field theory. In particular, there is a rich class of examples where the Higgs branch is a Nakajima quiver variety and the Coulomb branch is a moduli spaceof singular magnetic monopoles. In this case, I will show that the equivariant cohomology of the moduli space of based quasi-maps to the Higgs branch transforms as a Verma module for the deformation quantisation of the Coulomb branch

Mirror symmetry predicts surprising geometric correspondences between distinct families of algebraic varieties. In some cases, these correspondences have arithmetic consequences. Among the arithmetic correspondences predicted by mirror symmetry are correspondences between point counts over finite fields, and more generally between factors of their Zeta functions. In particular, we will discuss our results on a common factor for Zeta functions alternate families of invertible polynomials. We will also explore closed formulas for the point counts for our alternate mirror families of K3 surfaces and their relation to their Picard–Fuchs equations. Finally, we will discuss how all of this relates to hypergeometric motives. This is joint work with: Charles Doran (University of Alberta, Canada), Tyler Kelly (University of Cambridge, UK), Steven Sperber (University of Minnesota, USA), John Voight (Dartmouth College, USA), and Ursula Whitcher (American Mathematical Society, USA).

Self-organization is observed in systems driven by the “social engagement” of agents with their local neighbors. Prototypical models are found in opinion dynamics, flocking, self-organization of biological organisms, and rendezvous in mobile networks.

We discuss the emergent behavior of such systems. Two natural questions arise in this context. The underlying issue of the first question is how different rules of engagement influence the formation of clusters, and in particular, the emergence of 'consensus'. Different paradigms of emergence yield different patterns, depending on the propagation of connectivity of the underlying graphs of communication.  The second question involves different descriptions of self-organized dynamics when the number of agents tends to infinity. It lends itself to “social hydrodynamics”, driven by the corresponding tendency to move towards the local means. 

We discuss the global regularity of social hydrodynamics for sub-critical initial configurations.

The Skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of Brownian motion stopped at a chosen stopping time. In this talk, we consider an extension of the weak formulation of the optimal Skorokhod embedding problem. Using the classical convex duality approach together with the optimal stopping theory, we establish some duality. Moreover, based on the duality, we provide an alternative proof of the monotonicity principle proved by Beiglbock, Cox and Huesmann.

In 1923, Dulac published a proof of the claim that every real analytic vector field on the plane has only finitely many limit cycles (now known as Dulac's Problem). In the mid-1990s, Ilyashenko completed Dulac's proof; his completion rests on the construction of a quasianalytic class of functions. Unfortunately, this class has very few known closure properties. For various reasons I will explain, we are interested in constructing a larger quasianalytic class that is also a Hardy field. This can be achieved using Ilyashenko's idea of superexact asymptotic expansion.  (Joint work with Tobias Kaiser)

Hyperbolic groups were introduced by Gromov and generalize the fundamental groups of closed hyperbolic manifolds. Since a closed hyperbolic manifold is aspherical, it is a classifying space for its fundamental group, and a hyperbolic group will also admit a compact classifying space in the torsion-free case. After an introduction to this and other topological finiteness properties of hyperbolic groups and their subgroups, we will meet a construction of R. Kropholler, building on work of Brady and Lodha. The construction gives an infinite family of hyperbolic groups with finitely-presented subgroups which are non-hyperbolic by virtue of their finiteness properties. We conclude with progress towards determining minimal examples of the "sizeable" graphs which are needed as input to the construction.