Virtual

We will discuss recent developments of the theory of a-contraction with shifts to study the stability of discontinuous solutions of systems of equations modeling inviscid compressible flows, like the compressible Euler equation.

In the one dimensional configuration, the Bressan theory shows that small BV solutions are stable under small BV perturbations (together with a technical condition known as bounded variations on space-like curve).

The theory of a-contraction allows to extend the Bressan theory to a weak/BV stability result allowing wild perturbations fulfilling only the so-called strong trace property. Especially, it shows that the technical condition of BV on space-like curve is not needed. (joint work with Sam Krupa and Geng Chen). 

We will show several applications of the theory of a-contraction with shifts on the barotropic Navier-Stokes equation. Together with Moon-Jin Kang and Yi wang, we proved the conjecture of Matsumura (first mentioned in 1986). It consists in proving the time asymptotic stability of composite waves of viscous shocks and rarefactions. 

Together with Moon-Jin Kang, we proved also that inviscid shocks of the Euler equation, are stable among the family of inviscid limits of Navier-Stokes equation (Inventiones 2021). This stability result holds in the class of wild perturbations of inviscid limits, without any regularity restriction (not even strong trace property). This shows that the class of inviscid limits of Navier-Stokes equations is better behaved that the class of weak solutions to the inviscid limit problem.

This is obtained thanks to a stability result at the level of Navier-Stokes, which is uniform with respect to the viscosity, allowing asymptotically infinitely large perturbations (JEMS 2021).

A first multi D result of stability of contact discontinuities without shear, in the class of inviscid limit of Fourier-Navier-Stokes, shows that the same property is true for some situations even in multi D (joint work with Moon-jin Kang and Yi Wang). 

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.

We study the minimization of functionals of the form $$ u  \mapsto \int_\Omega  f(\nabla u) \, dx $$

with a convex integrand $f$ of linear growth (such as the area integrand), among all functions in the Sobolev space W$^{1,1}$ with prescribed boundary values. Due to insufficient compactness properties of these Dirichlet classes, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations and might in fact fail, as it is well-known already for the non-parametric minimal surface problem. In such cases, the functional is extended suitably to the space BV of functions of bounded variation via relaxation, and for the relaxed functional one can in turn guarantee the existence of minimizers. However, in contrast to the original minimization problem, these BV minimizers might in principle have interior jump discontinuities or not attain the prescribed boundary values.

After a short introduction to the problem I want to focus on the question of regularity of BV minimizers. In past years, Sobolev regularity was established provided that the lack of ellipticity -- which is always inherent for such linear growth integrands -- is mild, while, in general, only some structure results seems to be within reach. In this regard, I will review several results which were obtained in cooperation with Miroslav Bulíček (Prague), Franz Gmeineder (Bonn), Erika Maringová (Vienna), and Thomas Schmidt (Hamburg).

Numerical relativity allows us to simulate the behaviour of regions of space and time where gravity is strong and dynamical. For example, it allows us to calculate precisely the gravitational waveform that should be generated by the merger of two inspiralling black holes. Since the first detection of gravitational waves from such an event in 2015, banks of numerical relativity “templates” have been used to extract further information from noisy data streams. In this talk I will give an overview of the field - what are we simulating, why, and what are the main challenges, past and future.

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Counting coherent sheaves on Calabi--Yau fourfolds is a subject in its infancy. An evidence of this is given by how little is known about perhaps the simplest case - counting ideal sheaves of length $n$. On the other hand, the parallel story for surfaces while with many open questions has seen many new results, especially in the direction of understanding virtual integrals over Quot-schemes. Motivated by the conjectures of Cao--Kool and Nekrasov, we study virtual integrals over Hilbert schemes of points of top Chern classes $c_n(L^{[n]})$ and their K-theoretic refinements. Unlike lower-dimensional sheaf-counting theories, one also needs to pay attention to orientations. In this, we rely on the conjectural wall-crossing framework of Joyce. The same methods can be used for Quot-schemes of surfaces and we obtain a generalization of the work of Arbesfeld--Johnson--Lim--Oprea--Pandharipande for a trivial curve class. As a result, there is a correspondence between invariants for surfaces and fourfolds in terms of a universal transformation.

We introduce a new method for high-dimensional, online changepoint detection in settings where a $p$-variate Gaussian data stream may undergo a change in mean. The procedure works by performing likelihood ratio tests against simple alternatives of different scales in each coordinate, and then aggregating test statistics across scales and coordinates. The algorithm is online in the sense that both its storage requirements and worst-case computational complexity per new observation are independent of the number of previous observations. We prove that the patience, or average run length under the null, of our procedure is at least at the desired nominal level, and provide guarantees on its response delay under the alternative that depend on the sparsity of the vector of mean change. Simulations confirm the practical effectiveness of our proposal, which is implemented in the R package 'ocd', and we also demonstrate its utility on a seismology data set.

In its simplest instance, kinetic Brownian in R^d is a C¹ random path (m_t, v_t) with unit velocity v_t a Brownian motion on the unit sphere run at speed a > 0. Properly time rescaled as a function of the parameter a, its position process converges to a Brownian motion in R^d as a tends to infinity. On the other side the motion converges to the straight line motion (= geodesic motion) when a goes to 0. Kinetic Brownian motion provides thus an interpolation between geodesic and Brownian flows in this setting. Think now about changing R^d for the diffeomorphism group of a fluid domain, with a velocity vector now a vector field on the domain. I will explain how one can prove in this setting an interpolation result similar to the previous one, giving an interpolation between Euler’s equations of incompressible flows and a Brownian-like flow on the diffeomorphism group.

Path signature has unique advantages on extracting high-order differential features of sequential data. Our team has been studying the path signature theory and actively applied it to various applications, including infant cognitive score prediction, human motion recognition, hand-written character recognition, hand-written text line recognition and writer identification etc. In this talk, I will share our most recent works on infant cognitive score prediction using deep path signature. The cognitive score can reveal individual’s abilities on intelligence, motion, language abilities. Recent research discovered that the cognitive ability is closely related with individual’s cortical structure and its development. We have proposed two frameworks to predict the cognitive score with different path signature features. For the first framework, we construct the temporal path signature along the age growth and extract signature features of developmental infant cortical features. By incorporating the cortical path signature into the multi-stream deep learning model, the individual cognitive score can be predicted with missing data issues. For the second framework, we propose deep path signature algorithm to compute the developmental feature and obtain the developmental connectivity matrix. Then we have designed the graph convolutional network for the score prediction. These two frameworks have been tested on two in-house cognitive data sets and reached the state-of-the-art results.

The Temperley-Lieb algebra is a diagrammatic algebra - defined on a basis of "string diagrams" with multiplication given by "joining the diagrams together".  It first arose as an algebra of operators in statistical mechanics but quickly found application in knot theory (where Jones used it to define his famed polynomial) and the representation theory of $sl_2$.  From the outset, the representation theory of the Temperley-Lieb algebra itself has been of interest from a physics viewpoint and in characteristic zero it is well understood.  In this talk we will explore the representation theory over mixed characteristic (i.e. over positive characteristic fields and specialised at a root of unity).  This gentle introduction will take the listener through the beautifully fractal-like structure of the algebras and their cell modules with plenty of examples.

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