Past

It is sometimes possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope $P$ is defined to be the minimum number of facets in a (possibly higher-dimensional) polytope from which $P$ can be obtained as a (linear) projection. In this talk, we discuss some results on the extension complexity of random $d$-dimensional polytopes (obtained as convex hulls of random points on either on the unit sphere or in the unit ball), and on the extension complexity of polygons with all vertices on a common circle. Joint work with Matthew Kwan and Yufei Zhao

How to evaluate  meromorphic germs at their poles while preserving a
locality principle reminiscent of locality in QFT is a    question that
lies at the heart of  pQFT. It further  arises in other disguises in
number theory, the combinatorics on cones and toric geometry. We
introduce an abstract notion of locality and a related notion of
mutually independent meromorphic germs in several variables. Much in the
spirit of Speer's generalised evaluators in the framework of analytic
renormalisation, the question then amounts to extending the ordinary
evaluation at a point  to  certain algebras of meromorphic germs, in
such a way that the extension  factorises  on mutually independent
germs. In the talk, we shall describe a family of such extended
evaluators  and show that modulo a Galois type  transformation, they
amount to a minimal subtraction scheme in several variables.
This talk is based on ongoing joint work with Li Guo and Bin Zhang.
 

The CLE_4 Conformal Loop Ensemble in a 2D simply connected domain is a random countable collection of fractal Jordan curves that satisfies a statistical conformal invariance and appears, or is conjectured to appear, as a scaling limit of interfaces in various statistical physics models in 2D, for instance in the double dimer model. The CLE_4   is also related to the 2D Gaussian free field. Given a simply connected domain D and a point z in D, we consider the CLE_4 loop that surrounds z and study the extremal distance between the loop and the boundary of the domain, and the conformal radius of the interior surrounded by the loop seen from z. Because of the confomal invariance, the joint law of this two quantities does not depend (up to a scale factor) on the choice of the domain D and the point z in D. The law of the conformal radius alone has been known since the works of Schramm, Sheffield and Wilson. We complement their result by deriving the joint law of (extremal distance, conformal radius). Both quantities can be read on the same 1D Brownian path, by tacking a last passage time and a first hitting time. This joint law, together with some distortion bounds, provides some exponents related to the CLE_4. This is a joint work with Juhan Aru and Avelio Sepulveda.

In this talk, I will talk about two seemingly disjoint topics - Vinogradov’s mean value theorem, a classically important topic of study in additive number theory concerning solutions to a specific system of diophantine equations, and Incidence geometry, a collection of combinatorial results which focus on estimating the number of incidences between an arbitrary set of points and curves. I will give a brief overview of these two topics along with some basic proofs and applications, and then point out how these subjects connect together.

Hierarchically Hyperbolic Groups (HHGs) were introduced by Behrstock—Hagen—Sisto to provide a common framework to study several groups of interest in geometric group theory, and have been an object of great interest in the area ever since. The goal of the talk is to provide an introduction to the theory of HHGs and discuss the advantages of the unified approach that they provide. If time permits, we will conclude with applications to growth and asymptotic cones of groups.

In the talk I shall discuss an approach to the localisation technique, for spaces satisfying the curvature-dimension condition, by means of L1-optimal transport. Moreover, I shall present recent work on a generalisation of the technique to multiple constraints setting. Applications of the theory lie in functional and geometric inequalities, e.g. in the Lévy-Gromov isoperimetric inequality.

NOTE: unusual time! 

We discuss a class of 5-dimensional supersymmetric non-Lorentzian Lagrangians with an SU(1,3) conformal symmetry. These theories arise from reduction of 6-dimensional CFT's on a comformally compactified spacetime. We use the SU(1,3) Ward identities to find the form of the correlation functions which have a rich structure. Furthermore we show how these can be used to reconstruct  6-dimensional  CFT correlators. 
 

We consider the Plebanski-Demianski family of solutions of minimal gauged supergravity in D=4, which describes an accelerating, rotating and charged black-hole in AdS4. The 4d metric has conical singularities, but we show that it can uplifted to a completely regular solution of D=11 supergravity. We focus on the supersymmetric and extremal case, where the near-horizon geometry is AdS2x\Sigma, where \Sigma is a spindle, or weighted projective space. We argue that this is dual to a d=1, N=(2,0) SCFT which is the IR limit of a 3d SCFT compactified on a spindle. This, in turn, should be realized holographically by wrapping a stack of M2-branes on a spindle. Such construction displays two interesting features: 1) supersymmetry is realized in a novel way, which is not the topological twist, and 2) the R-symmetry of the d=1 SCFT mixes with the U(1) isometry of the spindle, even in the absence of rotation. A similar idea also applies to a class of AdS3x\Sigma solutions of minimal gauged supergravity in D=5.

For those who do not have login access to the Mathematical Institute website, please email @email to receive the link to this session.

The pandemic has forced all of us to assess our mental well being and the way in which we care for ourselves. We have learnt that good mental health is not a state but a constant evolution, and that it is natural that changes will take place on a daily and weekly timescale.
In this very timely session, Dr Tim Knowlson, Counselling Psychologist and University of Oxford Peer Support Programme Manager will discuss how we can care for our mental health and how we can develop resilience using current evidence-based research for tackling change and uncertainty that will serve us not only in the current pandemic but also provide us with tips that will serve us long into the future.

Ollivier Ricci curvature is a notion originated from Riemannian Geometry and suitable for applying on different settings from smooth manifolds to discrete structures such as (directed) hypergraphs. In the past few years, alongside Forman Ricci curvature, this curvature as an edge based measure, has become a popular and powerful tool for network analysis. This notion is defined based on optimal transport problem (Wasserstein distance) between sets of probability measures supported on data points and can nicely detect some important features such as clustering and sparsity in their structures. After introducing this notion for (directed) hypergraphs and mentioning some of its properties, as one of the main recent applications, I will present the result of implementation of this tool for the analysis of chemical reaction networks. 

There is a connection between certain smooth representations of a reductive p-adic group and the representations of the Iwahori-Hecke algebra of this p-adic group. This Iwahori-Hecke algebra is a specialisation of a more general affine Hecke algebra. In this talk, we will discuss affine Hecke algebras and graded Hecke algebras. We will state a result from Lusztig (1989) that relates the representation theory of an affine Hecke algebra and a particular graded Hecke algebra and we will present a simple example of this relation.

Deep-marine volcanism drives Earth's most energetic transfers of heat and mass between the crust and the oceans. Yet little is known of the primary source and intensity of the energy release that occurs during seafloor volcanic events owing to the lack of direct observations. Seafloor magmatic activity has nonetheless been correlated in time with the appearance of massive plumes of hydrothermal fluid known as megaplumes. However, the mechanism by which megaplumes form remains a mystery. By utilising observations of pyroclastic deposits on the seafloor, we show that their dispersal required an energy discharge that is sufficiently powerful (1-2 TW) to form a hydrothermal discharge with characteristics that align precisely with those of megaplumes observed to date. The result produces a conclusive link between tephra production, magma extrusion, tephra dispersal and megaplume production. However, the energy flux is too high to be explained by a purely volcanic source (lava heating), and we use our constraints to suggest other more plausible mechanisms for megaplume creation. The talk will cover a combination of new fluid mechanical fundamentals in volcanic transport processes, inversion methods and their implications for volcanism in the deep oceans.

Bacteria use intercellular signalling, or quorum sensing (QS), to share information and respond collectively to aspects of their surroundings. The autoinducers that carry this information are exposed to the external environment. Consequently, they are affected by factors such as removal through fluid flow, a ubiquitous feature of bacterial habitats ranging from the gut and lungs to lakes and oceans.

We develop and apply a general theory that identifies and quantifies the conditions required for QS activation in fluid flow by systematically linking cell- and population-level genetic and physical processes. We predict that cell-level positive feedback promotes a robust collective response, and can act as a low-pass filter at the population level in oscillatory flow, responding only to changes over slow enough timescales. Moreover, we use our model to hypothesize how bacterial populations can discern between increases in cell density and decreases in flow rate.

Abstract: We reconsider the idea of trend-based predictability using methods that flexibly learn price patterns that are most predictive of future returns, rather than testing hypothesized or pre-specified patterns (e.g., momentum and reversal). Our raw predictor data are images—stock-level price charts—from which we elicit the price patterns that best predict returns using machine learning image analysis methods. The predictive patterns we identify are largely distinct from trend signals commonly analyzed in the literature, give more accurate return predictions, translate into more profitable investment strategies, and are robust to a battery of specification variations. They also appear context-independent: Predictive patterns estimated at short time scales (e.g., daily data) give similarly strong predictions when applied at longer time scales (e.g., monthly), and patterns learned from US stocks predict equally well in international markets.

This is based on joint work with Jingwen Jiang and Bryan T. Kelly.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Suspensions are composed of mixtures of particles and fluid and are
ubiquitous in industrial processes (e.g. waste disposal, concrete,
drilling muds, metalworking chip transport, and food processing) and in
natural phenomena (e.g. flows of slurries, debris, and lava). The
present talk focusses on the rheology of concentrated suspensions of
non-colloidal particles. It addresses the classical shear viscosity of
suspensions but also non-Newtonian behaviour such as normal-stress
differences and shear-induced migration. The rheology of dense
suspensions can be tackled via a diversity of approaches that are
introduced. In particular, the rheometry of suspensions can be
undertaken at an imposed volume fraction but also at imposed values of
particle normal stress, which is particularly well suited to yield
examination of the rheology close to the jamming transition. The
influences of particle roughness and shape are discussed.

The cohomology of a manifold classifies geometric structures over it. One instance of this principle is the classification of line bundles via Chern classes. The classifying space BG associated to a (Lie) group G is a simplicial manifold which encodes the group structure. Its cohomology hence classifies geometric objects over G which play well with its multiplication. These are known as characteristic classes, and yield invariants of G-principal bundles.
I will introduce multiplicative gerbes and show how they realise classes in H^4(BG) when G is compact. Along the way, we will meet different versions of Lie group cohomology, smooth 2-groups and a few spectral sequences.

Link: https://teams.microsoft.com/l/meetup-join/19%3ameeting_ZGRiMTM1ZjQtZWNi…

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.  

A major tool used to understand manifolds is understanding how different measures of complexity relate to one another. One particularly combinatorial measure of the complexity of a 3-manifold M is the minimal number of tetrahedra in a simplicial complex homeomorphic to M, called the triangulation complexity of M. A natural question is whether we can relate this with more geometric measures of the complexity of a manifold, especially understanding these relationships as combinatorial complexity grows.

In the case when the manifold fibres over the circle, a recent theorem of Marc Lackenby and Jessica Purcell gives both an upper and lower bound on the triangulation complexity in terms of a geometric invariant of the gluing map (its translation length in the triangulation graph). We will discuss this result as well as a new result concerning what happens when we alter the gluing map by a Dehn twist.

Symbol alphabets of n-particle amplitudes in N=4 super-Yang-Mills theory are known to contain certain cluster variables of Gr(4,n) as well as certain algebraic functions of cluster variables. In this talk we suggest an algorithm for computing these symbol alphabets from plabic graphs by solving matrix equations of the form C.Z = 0 to associate functions on Gr(m,n) to parameterizations of certain cells of Gr_+ (k,n) indexed by plabic graphs. For m=4 and n=8 we show that this association precisely reproduces the 18 algebraic symbol letters of the two-loop NMHV eight-point amplitude from four plabic graphs. We further show that it is possible to obtain all rational symbol letters (in fact all cluster variables) by solving C.Z = 0 if one allows C to be an arbitrary cluster parameterization of the top cell of Gr_+ (n-4,n).

Let $X$ and $Y$ be two $n$-vertex graphs. Identify the vertices of $Y$ with $n$ people, any two of whom are either friends or strangers (according to the edges and non-edges in $Y$), and imagine that these people are standing one at each vertex of $X$. At each point in time, two friends standing at adjacent vertices of $X$ may swap places, but two strangers may not. The friends-and-strangers graph $FS(X,Y)$ has as its vertex set the collection of all configurations of people standing on the vertices of $X$, where two configurations are adjacent when they are related via a single friendly swap. This provides a common generalization for the famous 15-puzzle, transposition Cayley graphs of symmetric groups, and early work of Wilson and of Stanley.
I will describe several recent results and open problems addressing the extremal and typical aspects of the notion, focusing on the result that the threshold probability for connectedness of $FS(X,Y)$ for two independent binomial random graphs $X$ and $Y$ in $G(n,p)$ is $p=p(n)=n-1/2+o(1)$.
Joint work with Colin Defant and Noah Kravitz.

We study the coefficients of the characteristic polynomial (also called secular coefficients) of random unitary matrices drawn from the Circular Beta Ensemble (i.e. the joint probability density of the eigenvalues is proportional to the product of the power beta of the mutual distances between the points). We study the behavior of the secular coefficients when the degree of the coefficient and the dimension of the matrix tend to infinity. The order of magnitude of this coefficient depends on the value of the parameter beta, in particular, for beta = 2, we show that the middle coefficient of the characteristic polynomial of the Circular Unitary Ensemble converges to zero in probability when the dimension goes to infinity, which solves an open problem of Diaconis and Gamburd. We also find a limiting distribution for some renormalized coefficients in the case where beta > 4. In order to prove our results, we introduce a holomorphic version of the Gaussian Multiplicative Chaos, and we also make a connection with random permutations following the Ewens measure.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.