Past

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

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.

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 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.

Geometric complexity theory is an approach towards solving computational complexity lower bounds questions using algebraic geometry and representation theory. This talk contains an introduction to geometric complexity theory and a presentation of some recent results. Along the way connections to the study of secant varieties and to classical combinatorial and representation theoretic conjectures will be pointed out.