Past

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.

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.

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.

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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.

The wreath product of a finite group, or more generally an algebra, with a symmetric group is a familiar and important construction in representation theory and other areas of Mathematics. I shall present some highlights from my work on the representation theory of wreath products. These will include both structural properties (for example, that the wreath product of a cellular algebra with a symmetric group is again a cellular algebra) and cohomological ones (one 
particular point of interest being a generalisation of the result of Hemmer and Nakano on filtration multiplicities to the wreath product of two symmetric groups). I will also give an outline of some potential applications of this and related theory to important open  problems in algebraic combinatorics.

I will discuss how to construct cycles of many different lengths in graphs, in particular answering the following two problems on odd and even cycles. Erdős and Hajnal asked in 1981 whether the sum of the reciprocals of the odd cycle lengths in a graph diverges as the chromatic number increases, while, in 1984, Erdős asked whether there is a constant $C$ such that every graph with average degree at least $C$ contains a cycle whose length is a power of 2.

Topology optimisation finds the optimal material distribution of a fluid or solid in a domain, subject to PDE, volume, and box constraints. The optimisation problem is normally nonconvex and can support multiple local minima. In recent work [1], the authors developed an algorithm for systematically discovering multiple minima of two-dimensional problems through a combination of barrier methods, active-set strategies, and deflation. The bottleneck of the algorithm is solving the Newton systems that arise. In this talk, we will present preconditioning methods for these linear systems as they occur in the topology optimization of Stokes flow. The strategies involve a mix of block preconditioning and specialized multigrid relaxation schemes that reduce the computational work required and allow the application of the algorithm to three-dimensional problems.

[1] “Computing multiple solutions of topology optimization problems”, I. P. A. Papadopoulos, P. E. Farrell, T. M. Surowiec, 2020, https://arxiv.org/abs/2004.11797

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.

Empirical networks often exhibit different meso-scale structures, such as community and core-periphery structure. Core-periphery typically consists of a well-connected core, and a periphery that is well-connected to the core but sparsely connected internally. Most core-periphery studies focus on undirected networks. In this talk we discuss  a generalisation of core-periphery to directed networks which  yields a family of core-periphery blockmodel formulations in which, contrary to many existing approaches, core and periphery sets are edge-direction dependent. Then we shall  focus on a particular structure consisting of two core sets and two periphery sets, and we introduce  two measures to assess the statistical significance and quality of this  structure in empirical data, where one often has no ground truth. The idea will be illustrated on three empirical networks --  faculty hiring, a world trade data-set, and political blogs.

This is based on joint work with Andrew Elliott, Angus Chiu, Marya Bazzi and Mihai Cucuringu, available at https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2019.0783

Diffusion processes are widely used to model the evolution of random values over time. In many applications, the diffusion process is constrained to a finite domain. We consider the estimation problem of a diffusion process constrained by a polytope, i.e. intersection of finitely many (hyper-)planes, given a discretely observed time series data. Since the boundary behaviours of a diffusion process are characterised by its drift and diffusion functions, we derive sufficient conditions on the drift and diffusion functions for the nonattainablity of a polytope. We use deep learning to estimate the drift and diffusion, and ensure that their constraints are satisfied throughout the training.

Precision predictions for binary mergers are essential for the nascent field of gravitational wave astronomy. The initial inspiral part can be treated perturbatively. We present results for the post-Minkowskian expansion of conservative binary dynamics, previously available only at the 2nd order for several decades, at the 3rd and 4th orders in the expansion. Our calculations are based on quantum field theory and use powerful methods developed in the modern scattering amplitudes program, as well as loop integration techniques developed for precision collider physics. Furthermore, we take initial steps in calculating radiative binary dynamics and obtain analytically the total radiated energy in hyperbolic black hole scattering, at the lowest order in G but all orders in velocity.

Local-to-global principles are a key tool in arithmetic geometry. Through a theorem of Siegel on representations of totally positive numbers as sums of squares in number fields we give a concrete introduction to the Hasse principle, and briefly talk about other local-to-global principles. No prerequisites from algebraic number theory are assumed, although some familiarity is helpful for context.

An open market is a subset of a larger equity market, composed of a certain fixed number of top‐capitalization stocks. Though the number of stocks in the open market is fixed, their composition changes over time, as each company's rank by market capitalization fluctuates. When one is allowed to invest also in a money market, an open market resembles the entire “closed” equity market in the sense that the market viability (lack of arbitrage) is equivalent to the existence of a numéraire portfolio (which cannot be outperformed). When access to the money market is prohibited, the class of portfolios shrinks significantly in open markets; in such a setting, we discuss how to construct functionally generated stock portfolios and the concept of the universal portfolio.

This talk is based on joint work with Ioannis Karatzas.

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I will introduce the Friedl-Tillmann polytope for one-relator groups, and then discuss how it can be generalised to the Friedl-Lück polytope, how it connects to the Thurston polytope, and how we can view it as a convenient source of intuition and ideas.

In this talk I describe a holographic perspective to study field spaces that arise in string compactifications. The constructions are motivated by a general description of the asymptotic, near-boundary regions in complex structure moduli spaces of Calabi-Yau manifolds using Hodge theory. For real two-dimensional field spaces, I introduce an auxiliary bulk theory and describe aspects of an associated sl(2) boundary theory. The classical bulk reconstruction is provided by the sl(2)-orbit theorem, which is a famous and general result in Hodge theory. I then apply this correspondence to the flux landscape of Calabi-Yau fourfold compactifications and discuss how this allows us to prove that the number of self-dual flux vacua is finite. I will point out how the finiteness result for supersymmetric fluxes relates to the Hodge conjecture.