Past

In this talk I will introduce the study of lattices in locally compact groups through their actions CAT(0) spaces. This is an extremely rich class of groups including S-arithmetic groups acting on products of symmetric spaces and buildings, right angled Artin and Coxeter groups acting on polyhedral complexes, Burger-Mozes simple groups acting on products of trees, and the recent CAT(0) but non biautomatic groups of Leary and Minasyan. If time permits I will discuss some of my recent work related to the Leary-Minasyan groups.

There are many open conjectures about the algebraic behaviour of transcendental functions in arithmetic geometry, one of which is the Existential Closedness problem. In this talk I will review recent developments made on this question: the cases where we have unconditional existence of solutions, the conditional existence of generic solutions (depending on the conjecture of periods and Zilber-Pink), and even a few cases of unconditional existence of generic solutions. Many of the results I will mention are joint work with (different subsets of) Vahagn Aslanyan, Jonathan Kibry, Sebastián Herrero, and Roy Zhao. 

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

I will introduce the notion of moduli spaces of curves and specifically genus 0 curves. They are in general not compact, and we will discuss the most common way to compactify them. In particular, I will try to explain the construction of Mbar_{0,5}, together with how to classify the boundary, how it is related to a moduli space of tropical curves, and how to do intersection theory on this space.

In this talk, I will discuss AdS super gluon scattering amplitudes in various spacetime dimensions. These amplitudes are dual to correlation functions in a variety of non-maximally supersymmetric CFTs, such as the 6d E-string theory, 5d Seiberg exceptional theories, etc. I will introduce a powerful method based on symmetries and consistency conditions, and show that it fixes all the infinitely many four-point amplitudes at tree level. I will also point out many interesting properties and structures of these amplitudes, which include the flat space limit, Parisi-Sourlas-like dimensional reduction, hidden conformal symmetry, and a color-kinematic duality in AdS. Along the way, I will also review some earlier progress and the relation with this work. I will conclude with a brief discussion of various open problems. 

We discuss the behavior of geodesics in the continuous models of random geometry known as the Brownian map and the Brownian plane. We say that a point $x$ is a geodesic star with $m$ arms if $x$ is the endpoint of $m$ disjoint geodesics. We prove that the set of all geodesic stars with $m$ arms has dimension $5-m$, for $m=1,2,3,4$. This complements recents results of Miller and Qian, who derived upper bounds for these dimensions.

A group of projection based approaches for solving large-scale linear systems is known for its speed and simplicity. For example, Kaczmarz algorithm iteratively projects the previous approximation x_k onto the solution spaces of the next equation in the system. An elegant proof of the exponential convergence of this method, using correct randomization of the process, was given in 2009 by Strohmer and Vershynin, and succeeded by many extensions and generalizations. I will discuss our newly developed variants of these methods that successfully avoid large and potentially adversarial corruptions in the linear system. I specifically focus on the random matrix and high-dimensional probability results that play a crucial role in proving convergence of such methods. Based on the joint work with Jamie Haddock, Deanna Needell, and Will Swartworth.

Riemannian optimization is a powerful and active area of research that studies the optimization of functions defined on manifolds with structure. A class of functions of interest is the set of geodesically convex functions, which are functions that are convex when restricted to every geodesic. In this talk, we will present an accelerated first-order method, nearly achieving the same rates as accelerated gradient descent in the Euclidean space, for the optimization of smooth and g-convex or strongly g-convex functions defined on the hyperbolic space or a subset of the sphere. We will talk about accelerated optimization of another non-convex problem, defined in the Euclidean space, that we solve as a proxy. Additionally, for any Riemannian manifold of bounded sectional curvature, we will present reductions from optimization methods for smooth and g-convex functions to methods for smooth and strongly g-convex functions and vice versa.

This talk is based on the paper https://arxiv.org/abs/2012.03618.

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How much does the chromatic number of the random graph $G(n, 1/2)$ vary? Shamir and Spencer proved that it is contained in some sequence of intervals of length about $n^{1/2}$. Alon improved this slightly to $n^{1/2} / \log n$. Until recently, however, no lower bounds on the fluctuations of the chromatic number of $G(n, 1/2)$ were known, even though the question was raised by Bollobás many years ago. I will talk about the main ideas needed to prove that, at least for infinitely many $n$, the chromatic number of $G(n, 1/2)$ is not concentrated on fewer than $n^{1/2-o(1)}$ consecutive values.
I will also discuss the Zigzag Conjecture, made recently by Bollobás, Heckel, Morris, Panagiotou, Riordan and Smith: this proposes that the correct concentration interval length 'zigzags' between $n^{1/4+o(1)}$ and $n^{1/2+o(1)}$, depending on $n$.
Joint work with Oliver Riordan.

The complementarity and substitutability between products are essential concepts in retail and marketing. Qualitatively, two products are said to be substitutable if a customer can replace one product by the other, while they are complementary if they tend to be bought together. In this article, we take a network perspective to help automatically identify complements and substitutes from sales transaction data. Starting from a bipartite product-purchase network representation, with both transaction nodes and product nodes, we develop appropriate null models to infer significant relations, either complements or substitutes, between products, and design measures based on random walks to quantify their importance. The resulting unipartite networks between products are then analysed with community detection methods, in order to find groups of similar products for the different types of relationships. The results are validated by combining observations from a real-world basket dataset with the existing product hierarchy, as well as a large-scale flavour compound and recipe dataset.

arXiv link: https://arxiv.org/abs/2103.02042

We propose a randomized algorithm for solving a linear system $Ax = b$ with a highly numerically rank-deficient coefficient matrix $A$ with nearly consistent right-hand side possessing a small-norm solution. Our algorithm finds a small-norm solution with small residual in $O(N_r + nrlogn + r^3 )$ operations, where $r$ is the numerical rank of $A$ and $N_r$ is the cost of multiplying an $n\times r$ matrix to $A$. 

Joint work with Marcus Webb (Manchester). 

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 index of a saddle point of a smooth function is the number of descending directions of the saddle. While the index can usually be retrieved by counting the number of negative eigenvalues of the Hessian at the critical point, we may not have the luxury of having second derivatives in data deriving from practical applications. To address this problem, we develop a computational pipeline for estimating the index of a non-degenerate saddle point without explicitly computing the Hessian. In our framework, we only require a sufficiently dense sample of level sets of the function near the saddle point. Using techniques in Morse theory and Topological Data Analysis, we show how the shape of saddle points can help us infer the index of the saddle. Furthermore, we derive an explicit upper bound on the density of point samples necessary for inferring the index depending on the curvature of level sets. 

The outstanding issue of a non-singular extension of the Kerr-NUT- (anti) de Sitter solutions to Einstein’s equations is solved completely. The Misner’s method of obtaining the extension for Taub-NUT spacetime is generalized in a non-singular manner. The Killing vectors that define non-singular spaces of non-null orbits are derived and applied. The global structure of spacetime is discussed. The non-singular conformal geometry of theinfinities is derived. The Killing horizons are present.

For every positive integer N and every α ∈ [0,1), let B(N, α) denote the probabilistic model in which a random set A of (1,...,N) is constructed by choosing independently every element of (1,...,N) with probability α. We prove that, as N → +∞, for every A in B(N, α) we have |AA| ~ |A|^2/2 with probability 1-o(1), if and only if (log(α^2(log N)^{log 4-1}))(√loglog N) → ∞. This improves on a theorem of Cilleruelo, Ramana and Ramar\'e, who proved the above asymptotic between |AA| and |A|^2/2 when α =o(1/√log N), and supplies a complete characterization of maximal product sets of random sets.

A fundamental question in fluid dynamics concerns the formation of discontinuous shock waves from smooth initial data. In previous works, we have established stable generic shock formation for the compressible Euler system, showing that at the first singularity the solution has precisely C^{1/3} Holder regularity, a so-called preshock. The focus of this talk is a complete space-time description of the solution after this initial singularity. We show that three surfaces of discontinuity emerge simultaneously and instantaneously from the preshock: the classical shock discontinuity that propagates by the Rankine–Hugoniot conditions, together with two distinct surfaces in space-time, along which C^{3/2} cusp singularities form.

The unknotting number of a knot is the minimum number of crossing changes needed to untie the knot. It is one of the simplest knot invariants to define, yet remains notoriously difficult to compute. We will survey some basic knot theory invariants and constructions, including the satellite knot construction, a straightforward way to build new families of knots. We will give a lower bound on the unknotting number of certain satellites using knot Floer homology. This is joint work in progress with Tye Lidman and JungHwan Park.

The Atiyah-Floer conjecture asserts the instanton Floer homology of a closed three-manifold (constructed via gauge theory) is isomorphic to the Lagrangian Floer homology of a pair of Lagrangian submanifolds associated to a splitting of the three manifold (constructed via symplectic geometry). This conjecture has remained open for more than three decades. In this talk I will explain two compactness results for the SO(3) case of the conjecture in the neck-stretching process. One result is related to the construction of a natural bounding chain in the Lagrangian Floer theory and a conjecture of Fukaya.

We discuss constraints from perturbative unitarity and crossing on the leading contributions of the higher-dimension operators to the four-graviton amplitude in four spacetime dimensions. We focus on the leading order effect due to exchange by massive degrees of freedom which makes the amplitudes of interest IR finite. To test the constraints we obtain nontrivial effective field theory data by computing and taking the large mass expansion of the one-loop minimally-coupled four-graviton amplitude with massive particles up to spin 2 circulating in the loop. Remarkably, the leading EFT corrections to Einstein gravity of physical theories, both string theory and QFT coupled to gravity, end up in minuscule islands which are much smaller than what is suggested by the generic bounds obtained from consistency of the 2-2 graviton scattering amplitude. We discuss the underlying mechanism for this phenomenon.

Given a persistence diagram with n points, we give an algorithm that produces a sequence of n persistence diagrams converging in bottleneck distance to the input diagram, the ith of which has i distinct (weighted) points and is a 2-approximation to the closest persistence diagram with that many distinct points. For each approximation, we precompute the optimal matching between the ith and the (i+1)st. Perhaps surprisingly, the entire sequence of diagrams as well as the sequence of matchings can be represented in O(n) space. The main approach is to use a variation of the greedy permutation of the persistence diagram to give good Hausdorff approximations and assign weights to these subsets. We give a new algorithm to efficiently compute this permutation, despite the high implicit dimension of points in a persistence diagram due to the effect of the diagonal. The sketches are also structured to permit fast (linear time) approximations to the Hausdorff distance between diagrams -- a lower bound on the bottleneck distance. For approximating the bottleneck distance, sketches can also be used to compute a linear-size neighborhood graph directly, obviating the need for geometric data structures used in state-of-the-art methods for bottleneck computation.

The circadian clock generates ~24h rhythms everyday via a transcriptional-translational negative feedback loop. Although this involves the daily entry of repressor molecules into the nucleus after random diffusion through a crowded cytoplasm, the period remains extremely consistent. In this talk, I will describe how we identified a key molecular mechanism for such robustness of the circadian clock against spatio-temporal noise by analyzing spatio-temporal timeseries data of clock molecules. Furthermore, I will illustrate a systemic modeling approach that can identify hidden molecular interactions from oscillatory timeseries with an example of a circadian clock and tumorigenesis system.  Finally, I will talk about a fundamental question underlying the model-based time-series analysis: “Can we always fit a model to given timeseries data as long as the number of parameters is large?”. That is, is Von Neumann's quote “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk” true?

n this talk we consider a p-isometrisability property of discrete groups. If p=2 this property is equivalent to unitarisability. We prove that any group containing a non-abelian free subgroup is not p-isometrisable for any p∈(1,∞). We also discuss some open questions and possible relations of p-isometrisability with the recently introduced Littlewood exponent Lit(Γ).

Abstract: We formulate and solve a multi-player stochastic differential game between financial agents who seek to cost-efficiently liquidate their position in a risky asset in the presence of jointly aggregated transient price impact on the risky asset's execution price along with taking into account a common general price predicting signal. In contrast to an interaction of the agents through purely permanent price impact as it is typically considered in the literature on multi-player price impact games, accrued transient price impact does not persist but decays over time. The unique Nash-equilibrium strategies reveal how each agent's liquidation policy adjusts the predictive trading signal for the accumulated transient price distortion induced by all other agents' price impact; and thus unfolds a direct and natural link in equilibrium between the trading signal and the agents' trading activity. We also formulate and solve the corresponding mean field game in the limit of infinitely many agents and show how the latter provides an approximate Nash-equilibrium for the finite-player game. Specifically we prove the convergence of the N-players game optimal strategy to the optimal strategy of the mean field game.     (Joint work with Moritz Voss)
 

Unsupervised learning, in particular learning general nonlinear representations, is one of the deepest problems in machine learning. Estimating latent quantities in a generative model provides a principled framework, and has been successfully used in the linear case, especially in the form of independent component analysis (ICA). However, extending ICA to the nonlinear case has proven to be extremely difficult: A straight-forward extension is unidentifiable, i.e. it is not possible to recover those latent components that actually generated the data. Recently, we have shown that this problem can be solved by using additional information, in particular in the form of temporal structure or some additional observed variable. Our methods were originally based on "self-supervised" learning increasingly used in deep learning, but in more recent work, we have provided likelihood-based approaches. In particular, we have developed computational methods for efficient maximization of the likelihood for two variants of the model, based on variational inference or Riemannian relative gradients, respectively.

Dissipative Hamiltonian systems are an important class of dynamical systems that arise in all areas of science and engineering. They are a special case of port-Hamiltonian control systems. When the system is linearized arround a stationary solution one gets a linear dissipative Hamiltonian typically differential-algebraic system. Despite the fact that the system looks unstructured at first sight, it has remarkable properties.  Stability and passivity are automatic, spectral structures for purely imaginary eigenvalues, eigenvalues at infinity, and even singular blocks in the Kronecker canonical form are very restricted and furthermore the structure leads to fast and efficient iterative solution methods for asociated linear systems. When port-Hamiltonian systems are subject to (structured) perturbations, then it is important to determine the minimal allowed perturbations so that these properties are not preserved. The computation of these structured distances to instability, non-passivity, or non-regularity, is typically a very hard non-convex optimization problem. However, in the context of dissipative Hamiltonian systems, the computation becomes much easier and can even be implemented efficiently for large scale problems in combination with model reduction techniques. We will discuss these distances and the computational methods and illustrate the results via an industrial problem.

 

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