Past

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.
 

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.

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.

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.

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.

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.

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.

A major challenge in the study of biological systems is that of model discovery: turning data into reduced order models that are not just predictive, but provide insight into the nature of the underlying system that generated the data. We introduce a number of data-driven strategies for discovering nonlinear multiscale dynamical systems and their embeddings from data.  Such data-driven methods can be used in the biological sciences where rich data streams are affording new possibilities for the understanding and characterization of complex, networked systems.  In neuroscience, for instance, the integration of these various concepts (reduced-order modeling, equation-free, machine learning, sparsity, networks, multi-scale physics and adaptive control) are critical to formulating successful modeling strategies that perhaps can say something meaningful about experiments.   These methods will be demonstrated on a number of neural systems.  I will also highlight how such methods can be used to quantify cognitive and decision-making deficits arising from neurodegenerative diseases and/or traumatic brain injuries (concussions).

Whether one uses the sales, the number of employees or any other proxy for firm "size", it is well known that this quantity is power-law distributed, with important consequences to aggregate macroeconomic fluctuations. The Gibrat model explained this by proposing that firms grow multiplicatively, and much work has been done to study the statistics of their growth rates. Inspired by past work in the statistics of financial returns, I present a new framework to study these growth rates. In particular, I will show that they follow approximately Gaussian statistics, provided their heteroskedastic nature is taken into account. I will also elucidate the size/volatility scaling relation, and show that volatility may have a strong sectoral dependence. Finally, I will show how this framework can be used to study intra-firm and supply chain dynamics.

Joint work with JP Bouchaud and Angelo Secchi.

Specialized languages for describing convex optimization problems, and associated parsers that automatically transform them to canonical form, have greatly increased the use of convex optimization in applications. These systems allow users to rapidly prototype applications based on solving convex optimization problems, as well as generate code suitable for embedded applications. In this talk I will describe the general methods used in such systems.

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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.

A topology optimization problem for Stokes flow finds the optimal material distribution of a fluid in Stokes flow that minimizes the fluid’s power dissipation under a volume constraint. In 2003, T. Borrvall and J. Petersson [1] formulated a nonconvex optimization problem for this objective. They proved the existence of minimizers in the infinite-dimensional setting and showed that a suitably chosen finite element method will converge in a weak(-*) sense to an unspecified solution. In this talk, we will extend and refine their numerical analysis. In particular, we will show that there exist finite element functions, satisfying the necessary first-order conditions of optimality, that converge strongly to each isolated local minimizer of the problem.

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Fully nonlinear PDEs involving the eigenvalues of matrix-valued differential operators (such as the Hessian) have been the subject of intensive study over the last few decades, since the seminal work of Caffarelli, Kohn, Nirenberg and Spruck. In this talk I will discuss some recent joint work with Luc Nguyen on the regularity theory for a large class of these equations, with a particular emphasis on a special case known as the sigma_k-Yamabe equation, which arises in conformal geometry. 

[1] T. Borrvall, J. Petersson, Topology optimization of fluids in Stokes flow, International Journal for Numerical Methods in Fluids 41 (1) (2003) 77–107. doi:10.1002/fld.426.

Contagion models on networks can be used to describe the spread of information, rumours, opinions, and (more topically) diseases through a population. In the simplest contagion models, each node represents an individual that can be in one of a number of states (e.g. Susceptible, Infected, or Recovered), and the states of the nodes evolve according to specified rules. Even with simple Markovian models of transmission and recovery, it can be difficult to compute the dynamics of contagion on large networks: running simulations can be slow, and the system of master equations is typically too large to be tractable.

 One approach to approximating contagion dynamics is to assume that each node state is independent of the neighbouring node states; this leads to a system of ODEs for the node state probabilities (the “first-order approximation”) that always overestimates the speed of infection spread. This approach can be made more sophisticated by introducing pair approximations or higher-order moment closures, but this dramatically increases the size of the system and slows computations. In this talk, I will present some alternative node-based approximations for contagion dynamics. The first of these is exact on trees but will always underestimate the speed of infection spread on a network with loops. I will show how this can be combined with the classic first-order node-based approximation to obtain a node-based approximation that has similar accuracy to the pair approximation, but which is considerably faster to solve.

Two classical results of Magidor are: 

(1) from large cardinals it is consistent to have reflection at $\aleph_{\omega+1}$, and 

(2) from large cardinals it is consistent to have the failure of SCH at $\aleph_\omega$.

These principles are at odds with each other. The former is a compactness type principle. (Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object.) In contrast, failure of SCH is an instance of incompactness. The natural question is whether we can have both of these simultaneously. We show the answer is yes.

We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH.  Then we obtain this situation at $\aleph_\omega$. This is joint work with Alejandro Poveda and Assaf Rinot.

One of the main characterisations of word-hyperbolic groups is that they are the groups with a linear isoperimetric function. That is, for a compact 2-complex X, the hyperbolicity of its fundamental group is equivalent to the existence of a linear isoperimetric function for disc diagrams D -->X.
It is likewise known that hyperbolic groups have a linear annular isoperimetric function and a linear homological isoperimetric function. I will talk about these isoperimetric functions, and about a (previously unexplored)  generalisation to all homotopy types of surface diagrams. This is joint work with Dani Wise.

We generalize the fact that all graphs omitting a fixed finite bipartite graph can be uniformly approximated by rectangles (Alon-Fischer-Newman, Lovász-Szegedy), showing that hypergraphs omitting a fixed finite $(k+1)$-partite $(k+1)$-uniform hypergraph can be approximated by $k$-ary cylinder sets. In particular, in the decomposition given by hypergraph regularity one only needs the first $k$ levels: such hypergraphs can be approximated using sets of vertices, sets of pairs, and so on up to sets of $k$-tuples, and on most of the resulting $k$-ary cylinder sets, the density is either close to 0 or close to 1. Moreover, existence of such approximations uniformly under all measures on the vertices is a characterization. Our proof uses a combination of analytic, combinatorial and model-theoretic methods, and involves a certain higher arity generalization of the epsilon-net theorem from VC-theory.  Joint work with Henry Towsner.

Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson metal-insulator phase transition (crossover) even in 1d. In this talk we will discuss some recent progress in application of the supersymmetric method (SUSY) and transfer matrix approach to the analysis of local spectral characteristics of some specific types of 1d RBM.

The Schmidt subspace theorem is a far-reaching generalization of the Thue-Siegel-Roth theorem in diophantine approximation. In this talk I will give an interpretation of Schmidt's subspace theorem in terms of the dynamics of diagonal flows on homogeneous spaces and describe how the exceptional subspaces arise from certain rational Schubert varieties associated to the family of linear forms through the notion of Harder-Narasimhan filtration and an associated slope formalism. This geometric understanding opens the way to a natural generalization of Schmidt's theorem to the setting of diophantine approximation on submanifolds of $GL_d$, which is our main result. In turn this allows us to recover and generalize the main results of Kleinbock and Margulis regarding diophantine exponents of submanifolds. I will also mention an application to diophantine approximation on Lie groups. Joint work with Nicolas de Saxcé.

Networks are an imperfect representation of a dataset, yet often there is little consideration for how these imperfections may affect network evolution and structure.

In this talk, I want to discuss a simple set of generative network models in which the mechanism of network growth is decomposed into two layers. The first layer represents the “observed” network, corresponding to our conventional understanding of a network. Here I want to consider the scenario in which the network you observe is not self-contained, but is driven by a second hidden network, comprised of the same nodes but different edge structure. I will show how a range of different network growth models can be constructed such that the observed and hidden networks can be causally decoupled, coupled only in one direction, or coupled in both directions.

One consequence of such models is the emergence of abrupt transitions in observed network topology – one example results in scale-free degree distributions which are robust up to an arbitrarily long threshold time, but which naturally break down as the network grows larger. I will argue that such examples illustrate why we should be wary of an overreliance on static networks (measured at only one point in time), and will discuss other possible implications for prediction on networks.

I take the “collapse of the wave-function” to be an objective physical process—OR (the Objective Reduction of the quantum state)—which I argue to be intimately related to a basic conflict between the principles of equivalence and quantum linear superposition, which leads us to a fairly specific formula (in agreement with one found earlier by Diósi) for the timescale for OR to take place. Moreover, we find that for consistency with relativity, OR needs to be “instantaneous” but with curious retro-active features. By extending an argument due to Donadi, for EPR situations, we find a fundamental conflict with “gradualist” models such as CSL, in which OR is taken to be the result of a (stochastic) evolution of quantum amplitudes.

Mean field games (MFG) and mean field control problems (MFC) are frameworks to study Nash equilibria or social optima in games with a continuum of agents. These problems can be used to approximate competitive or cooperative situations with a large finite number of agents. They have found a broad range of applications, from economics to crowd motion, energy production and risk management. Scalable numerical methods are a key step towards concrete applications. In this talk, we propose several numerical methods for MFG and MFC. These methods are based on machine learning tools such as function approximation via neural networks and stochastic optimization. We provide numerical results and we investigate the numerical analysis of these methods by proving bounds on the approximation scheme. If time permits, we will also discuss model-free methods based on extensions of the traditional reinforcement learning setting to the mean-field regime.  

I will discuss joint work with S. Galatius and A. Kupers in which we investigate the homology of general linear groups over a ring $A$ by considering the collection of all their classifying spaces as a graded $E_\infty$-algebra. I will first explain diverse results that we obtained in this investigation, which can be understood without reference to $E_\infty$-algebras but which seem unrelated to each other: I will then explain how the point of view of cellular $E_\infty$-algebras unites them.