The development of high frequency and algorithmic trading allowed to considerably reduce the bid ask spread by increasing liquidity in limit order books. Beyond the problem of optimal placement of market and limit orders, the possibility to cancel orders for free leaves room for price manipulations, one of such being spoofing. Detecting spoofing from a regulatory viewpoint is challenging due to the sheer amount of orders and difficulty to discriminate between legitimate and manipulative flows of orders. However, it is empirical evidence that volume imbalance reflecting offer and demand on both sides of the limit order book has an impact on subsequent price movements. Spoofers use this effect to artificially modify the imbalance by posting limit orders and then execute market orders at subsequent better prices while canceling at a high speed their previous limit orders. In this work we set up a model to determine where a spoofer would place its limit orders to maximize its gains as a function of the imbalance impact on the price movement. We study the solution of this non local optimization problem as a function of the imbalance. With this at hand, we calibrate on real data from TMX the imbalance impact (as a function of its depth) on the resulting price movement. Based on this calibration and theoretical results, we then provide some methods and numerical results as how to detect in real time some eventual spoofing behavior based on Wasserstein distances. Joint work with Tao Xuan (SJTU), Ling Lan (SJTU) and Andrew Day (Western University)
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 In this talk I will discuss possible extensions of the euclidean quantitative isoperimetric inequality to compact Riemannian manifolds. 
This is joint work with O. Chodosh (Stanford) and M. Engelstein (University of Minnesota).

Hardt-Simon proved that every area-minimizing hypercone $C$ having only an isolated singularity fits into a foliation of $R^{n+1}$ by smooth, area-minimizing hypersurfaces asymptotic to $C$. We prove that if a minimal hypersurface $M$ in the unit ball $B_1 \subset R^{n+1}$ lies sufficiently close to a minimizing quadratic cone (for example, the Simons' cone), then $M \cap B_{1/2}$ is a $C^{1,\alpha}$ perturbation of either the cone itself, or some leaf of its associated foliation. In particular, we show that singularities modeled on these cones determine the local structure not only of $M$, but of any nearby minimal surface. Our result also implies the Bernstein-type result of Simon-Solomon, which characterizes area-minimizing hypersurfaces in $R^{n+1}$ asymptotic to a quadratic cone as either the cone itself, or some leaf of the foliation.  This is joint work with Luca Spolaor.

In the talk, we will discuss the connection between quantitative hypoelliptic PDE methods and the long-time dynamics of stochastic differential equations (SDEs). In a recent joint work with Alex Blumenthal and Sam Punshon-Smith, we put forward a new method for obtaining quantitative lower bounds on the top Lyapunov exponent of stochastic differential equations (SDEs). Our method combines (i) an (apparently new) identity connecting the top Lyapunov exponent to a degenerate Fisher information-like functional of the stationary density of the Markov process tracking tangent directions with (ii) a  quantitative version of Hörmander's hypoelliptic regularity theory in an L1 framework which estimates this (degenerate) Fisher information from below by a W^{s,1} Sobolev norm using the associated Kolmogorov equation for the stationary density. As an initial application, we prove the positivity of the top Lyapunov exponent for a class of weakly-dissipative, weakly forced SDE and we prove that this class includes the classical Lorenz 96 model in any dimension greater than 6, provided the additive stochastic driving is applied to any consecutive pair of modes. This is the first mathematically rigorous proof of chaos (in the sense of positive Lyapunov exponents) for Lorenz 96 and, more recently, for finite dimensional truncations of the shell models GOY and SABRA (stochastically driven or otherwise), despite the overwhelming numerical evidence. If time permits, I will also discuss joint work with Kyle Liss, in which we obtain sharp, quantitative estimates on the spectral gap of the Markov semigroups. In both of these works, obtaining various kinds of quantitative hypoelliptic regularity estimates that are uniform in certain parameters plays a pivotal role.  

Cuntz introduced pure infiniteness for simple C*-algebras as a C*-algebraic analogue of type III von Neumann factors. Notable examples include the Calkin algebra B(H)/K(H), the Cuntz algebras O_n, simple Cuntz-Krieger algebras, and other C*-algebras you would encounter in the wild. The separable, nuclear ones were classified in celebrated work by Kirchberg and Phillips in the mid 90s. I will talk about these topics including the non-simple case if time permits.

Unitary solutions of the Yang-Baxter equation ("R-matrices") play a prominent role in several fields, such as quantum field theory and topological quantum computing, but are difficult to find directly and remain somewhat mysterious. In this talk I want to explain how one can use subfactor techniques to learn something about unitary R-matrices, and a research programme aiming at the classification of unitary R-matrices up to a natural equivalence relation. This talk is based on joint work with Roberto Conti, Ulrich Pennig, and Simon Wood.

Identifying discrete patterns in binary data is an important dimensionality reduction tool in machine learning and data mining. In this paper, we consider the problem of low-rank binary matrix factorisation (BMF) under Boolean arithmetic. Due to the NP-hardness of this problem, most previous attempts rely on heuristic techniques. We formulate the problem as a mixed integer linear program and use a large scale optimisation technique of column generation to solve it without the need of heuristic pattern mining. Our approach focuses on accuracy and on the provision of optimality guarantees. Experimental results on real world datasets demonstrate that our proposed method is effective at producing highly accurate factorisations and improves on the previously available best known results for 16 out of 24 problem instances.

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In this talk, we will present an approach to constrained optimisation when the set of constraints is a smooth manifold. This setting is of particular interest in data science applications, as many interesting sets of matrices have a manifold structure. We will show how we may couple classic ideas from differential geometry with modern methods such as autodifferentiation to simplify optimisation problems from spaces with a difficult topology (e.g. problems with orthogonal or fixed-rank constraints) to problems on ℝⁿ where we can use any classical optimisation methods to solve them. We will also show how to use these methods to automatically compute quantities such as the Riemannian gradient and Hessian. We will present the library GeoTorch that allows for putting these kind of constraints within models written in PyTorch by adding just one line to the model. We will also comment on some convergence results if time allows.

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.

We develop a theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes. Our approach generalizes the usual (co)variance to the setting of weighted graphs and retains many of its intuitive and desired properties. As a particular application, we define the maximum-variance problem on graphs with respect to the effective resistance distance, and characterize the solutions to this problem both numerically and theoretically. We show how the maximum-variance distribution can be interpreted as a core-periphery measure, illustrated by the fact that these distributions are supported on the leaf nodes of tree graphs, low-degree nodes in a configuration-like graph and boundary nodes in random geometric graphs. Our theoretical results are supported by a number of experiments on a network of mathematical concepts, where we use the variance and covariance as analytical tools to study the (co-)occurrence of concepts in scientific papers with respect to the (network) relations between these concepts. Finally, I will draw connections to related notion of assortativity on networks, a network analogue of correlation used to describe how the presence and absence of edges covaries with the properties of nodes.

https://arxiv.org/abs/2008.09155

(joint work with T. Kania, Academy of Sciences of the Czech Republic, Prague)
In this talk we discuss our recent result on the inverse eigenvalue problem for symmetric doubly stochastic matrices. 
Namely, we provide a new sufficient condition for a list of real numbers to be the spectrum of a symmetric doubly stochastic matrix. 
In our construction of such matrices, we employ the eigenvectors of the transition probability matrix of a simple symmetric random walk on the circle. 
We also demonstrate a simple algorithm for generating random doubly stochastic matrices based on our construction. Examples will be provided.