Past

Let E be an elliptic curve over the rationals and p a prime such that E admits a rational p-isogeny satisfying some assumptions. In a joint work with J. Lee and C. Skinner, we prove the anticyclotomic Iwasawa main conjecture for E/K for some suitable quadratic imaginary field K. I will explain our strategy and how this, combined with complex and p-adic Gross-Zagier formulae, allows us to prove that if E has rank one, then the p-part of the Birch and Swinnerton-Dyer formula for E/Q holds true.
 

The flow of a thin film down an inclined plane is an important physical phenomenon appearing in many industrial applications, such as coating (where it is desirable to maintain the fluid interface flat) or heat transfer (where a larger interfacial area is beneficial). These applications lead to the need of reliably manipulating the flow in order to obtain a desired interfacial shape. The interface of such thin films can be described by a number of models, each of them exhibiting instabilities for certain parameter regimes. In this talk, I will propose a feedback control methodology based on same-fluid blowing and suction. I use the Kuramoto–Sivashinsky (KS) equation to model interface perturbations and to derive the controls. I will show that one can use a finite number of point-actuated controls based on observations of the interface to stabilise both the flat solution and any chosen nontrivial solution of the KS equation. Furthermore, I will investigate the robustness of the designed controls to uncertain observations and parameter values, and study the effect of the controls across a hierarchy of models for the interface, which include the KS equation, (nonlinear) long-wave models and the full Navier–Stokes equations.

This paper studies the spread of losses and defaults in financial networks with two important features: collateral requirements and alternative contract termination rules in bankruptcy. When collateral is committed to a firm’s counterparties, a solvent firm may default if it lacks sufficient liquid assets to meet its payment obligations. Collateral requirements can thus increase defaults and payment shortfalls. Moreover, one firm may benefit from the failure of another if the failure frees collateral committed by the surviving firm, giving it additional resources to make other payments. Contract termination at default may also improve the ability of other firms to meet their obligations. As a consequence of these features, the timing of payments and collateral liquidation must be carefully specified, and establishing the existence of payments that clear the network becomes more complex. Using this framework, we study the consequences of illiquid collateral for the spread of losses through fire sales; we compare networks with and without selective contract termination; and we analyze the impact of alternative bankruptcy stay rules that limit the seizure of collateral at default. Under an upper bound on derivatives leverage, full termination reduces payment shortfalls compared with selective termination.

We discuss the design of algorithms and codes for the solution of large sparse systems of linear equations on extreme scale computers that are characterized by having many nodes with multi-core CPUs or GPUs. We first use two approaches to get good single node performance. For symmetric systems we use task-based algorithms based on an assembly tree representation of the factorization. We then use runtime systems for scheduling the computation on both multicore CPU nodes and GPU nodes [6]. In this work, we are also concerned with the efficient parallel implementation of the solve phase using the computed sparse factors, and we show impressive results relative to other state-of-the-art codes [3]. Our second approach was to design a new parallel threshold Markowitz algorithm [4] based on Luby’s method [7] for obtaining a maximal independent set in an undirected graph. This is a significant extension since our graph model is a directed graph. We then extend the scope of both these approaches to exploit distributed memory parallelism. In the first case, we base our work on the block Cimmino algorithm [1] using the ABCD software package coded by Zenadi in Toulouse [5, 8]. The kernel for this algorithm is the direct factorization of a symmetric indefinite submatrix for which we use the above symmetric code. To extend the unsymmetric code to distributed memory, we use the Zoltan code from Sandia [2] to partition the matrix to singly bordered block diagonal form and then use the above unsymmetric code on the blocks on the diagonal. In both cases, we illustrate the added parallelism obtained from combining the distributed memory parallelism with the high single-node performance and show that our codes out-perform other state-of-the-art codes. This work is joint with a number of people. We developed the algorithms and codes in an EU Horizon 2020 Project, called NLAFET, that finished on 30 April 2019. Coworkers in this were: Sebastien Cayrols, Jonathan Hogg, Florent Lopez, and Stojce ´ ∗@email 1 Nakov. Collaborators in the block Cimmino part of the project were: Philippe Leleux, Daniel Ruiz, and Sukru Torun. Our codes available on the github repository https://github.com/NLAFET.

References [1] M. ARIOLI, I. S. DUFF, J. NOAILLES, AND D. RUIZ, A block projection method for sparse matrices, SIAM J. Scientific and Statistical Computing, 13 (1992), pp. 47–70. [2] E. BOMAN, K. DEVINE, L. A. FISK, R. HEAPHY, B. HENDRICKSON, C. VAUGHAN, U. CATALYUREK, D. BOZDAG, W. MITCHELL, AND J. TERESCO, Zoltan 3.0: Parallel Partitioning, Load-balancing, and Data Management Services; User’s Guide, Sandia National Laboratories, Albuquerque, NM, 2007. Tech. Report SAND2007-4748W http://www.cs.sandia. gov/Zoltan/ug_html/ug.html. [3] S. CAYROLS, I. S. DUFF, AND F. LOPEZ, Parallelization of the solve phase in a task-based Cholesky solver using a sequential task flow model, Int. J. of High Performance Computing Applications, To appear (2019). NLAFET Working Note 20. RAL-TR-2018-008. [4] T. A. DAVIS, I. S. DUFF, AND S. NAKOV, Design and implementation of a parallel Markowitz threshold algorithm, Technical Report RAL-TR-2019-003, Rutherford Appleton Laboratory, Oxfordshire, England, 2019. NLAFET Working Note 22. Submitted to SIMAX. [5] I. S. DUFF, R. GUIVARCH, D. RUIZ, AND M. ZENADI, The augmented block Cimmino distributed method, SIAM J. Scientific Computing, 37 (2015), pp. A1248–A1269. [6] I. S. DUFF, J. HOGG, AND F. LOPEZ, A new sparse symmetric indefinite solver using a posteriori threshold pivoting, SIAM J. Scientific Computing, To appear (2019). NLAFET Working Note 21. RAL-TR-2018-012. [7] M. LUBY, A simple parallel algorithm for the maximal independent set problem, SIAM J. Computing, 15 (1986), pp. 1036–1053. [8] M. ZENADI, The solution of large sparse linear systems on parallel computers using a hybrid implementation of the block Cimmino method., These de Doctorat, ´ Institut National Polytechnique de Toulouse, Toulouse, France, decembre 2013.

How would we get a powerful AI to align itself with human preferences? What are human preferences anyway? And how can you code all this?
It turns out that maths give you the grounding to answer these fascinating and vital questions.
 

The Camassa-Holm (CH) equation is a nonlinear nonlocal dispersive equation which arises as a model for the propagation of unidirectional shallow water waves over a flat bottom. One of the most important features of the CH equation is the existence of peaked travelling waves, also called peakons. The aim of this talk is to review some asymptotic stability result for peakon solutions for CH-type equations as well as to present some new result for higher-order generalization of the CH equation.

The automorphism group of a d-regular tree is a topological group with many interesting features. A nice thing about this group is that while some of its features are highly non-trivial (e.g., the existence of infinitely many pairwise non-conjugate simple subgroups), often the ideas involved in the proofs are fairly intuitive and geometric. 
I will present a proof for the fact that the outer automorphism group of (Aut(T)) is trivial. This is original joint work with Gil Goffer, but as is often the case in this area, was already proven by Bass-Lubotzky 20 years ago. I will mainly use this talk to hint at how algebra, topology and geometry all play a role when working with Aut(T).
 

Scrolls play a central role in the construction of varieties; they are ambient spaces for K3 surfaces and Fano 3-folds. In this talk, I will say in two ways what scrolls are and give examples of some embedded varieties in them.

We will be discussing a Fourier-analytic approach
to optimal matching between independent samples, with
an elementary proof of the Ajtai-Komlos-Tusnady theorem.
The talk is based on a joint work with Michel Ledoux.

It is known that random matrix distributions such as those that describe the largest eignevalue of the Gaussian Orthogonal and Symplectic ensembles (GOE, GSE) admit two types of representations: one in terms of a Fredholm Pfaffian and one in terms of a Fredholm determinant. The equality of the two sets of expressions has so far been established via involved computations of linear algebraic nature. We provide a structural explanation of this duality via links (old and new) between the model of last passage percolation and the irreducible characters of classical groups, in particular the general linear, symplectic and orthogonal groups, and by studying, combinatorially, how their representations decompose when restricted to certain subgroups. Based on joint work with Elia Bisi.

Tautological bundles on Hilbert schemes of points often enter into enumerative and physical computations. I will explain how to use the Donaldson-Thomas theory of toric threefolds to produce combinatorial identities that are expressed geometrically using tautological bundles on the Hilbert scheme of points on a surface. I'll also explain how these identities can be used to study Euler characteristics of tautological bundles over Hilbert schemes of points on general surfaces.

We show that the scalability challenges of Global Optimisation algorithms can be overcome for functions with low effective dimensionality, which are constant along certain linear subspaces. Such functions can often be found in applications, for example, in hyper-parameter optimization for neural networks, heuristic algorithms for combinatorial optimization problems and complex engineering simulations. We propose the use of random subspace embeddings within a(ny) global minimisation algorithm, extending the approach in Wang et al. (2016). Using tools from random matrix theory and conic integral geometry, we investigate the efficacy and convergence of our random subspace embeddings approach, in a static and/or adaptive formulation. We illustrate our algorithmic proposals and theoretical findings numerically, using state of the art global solvers. This work is joint with Coralia Cartis.
 

Modularity is a function on graphs which is used in algorithms for community detection. For a given graph G, each partition of the vertices has a modularity score, with higher values indicating that the partition better captures community structure in $G$. The (max) modularity $q^\ast(G)$ of the graph $G$ is defined to be the maximum over all vertex partitions of the modularity score, and satisfies $0 \leq q^\ast(G) \leq 1$.

We analyse when community structure of an underlying graph can be determined from an observed subset of the graph. In a natural model where we suppose edges in an underlying graph $G$ appear with some probability in our observed graph $G'$ we describe how high a sampling probability we need to infer the community structure of the underlying graph.

Joint work with Colin McDiarmid.

The QDWH algorithm can compute the polar decomposition of a matrix in a stable and efficient way. We generalize this method in order to compute generalized polar decompositions with respect to signature matrices. Here, the role of the QR decomposition is played by the hyperbolic QR decomposition. However, it doesn't show the same favorable properties concerning stability as its orthogonal counterpart. Remedies are found by exploiting connections to the LDL^T factorization and by employing well-conditioned permuted graph bases. The computed polar decomposition is used to formulate a structure-preserving spectral divide-and-conquer method for pseudosymmetric matrices. Applications of this method are found in computational quantum physics, where eigenvalues and eigenvectors describe optical properties of condensed matter or molecules. Additional properties guarantee fast convergence and a reduction to symmetric definite eigenvalue problems after just one step of spectral divide-and-conquer.

Numerous mathematical models have been proposed for modelling cancerous tumour invasion (Gatenby and Gawlinski 1996), angiogenesis (Owen et al 2008), growth kinetics (Wang et al 2009), response to irradiation (Gao et al 2013) and metastasis (Qiam and Akcay 2018). In this study, we attempt to model the qualitative behavior of growth, invasion, angiogenesis and fragmentation of tumours at the tissue level in an explicitly spatial and continuous manner in two dimensions. We simulate the effectiveness of radiation therapy on a growing tumour in comparison with immunotherapy and propose a novel framework based on vector fields for modelling the impact of interstitial flow on tumour morphology. The results of this model demonstrate the effectiveness of employing a system of partial differential equations along with vector fields for simulating tumour fragmentation and that immunotherapy, when applicable, is substantially more effective than radiation therapy.

A polynomial form is established for the off-shell CHY scattering equations proposed by Lam and Yao. Re-expressing this in terms of independent Mandelstam invariants provides a new expression for the polynomial scattering equations, immediately valid off shell, which makes it evident that they yield the off-shell amplitudes given by massless ϕ3 Feynman graphs. A CHY expression for individual Feynman graphs, valid even off shell, is established through a recurrence relation.

It is known that many real-world networks exhibit geometric properties.  Brain networks, social networks, and wireless communication networks are a few examples.  Storage and transmission of the information contained in the topologies and structures of these networks are important tasks, which, given their scale, is often nontrivial.  Although some (but not much) work has been done to characterize and develop compression limits and algorithms for nonspatial graphs, little is known for the spatial case.  In this talk, we will discuss an information theoretic formalism for studying compression limits for a fairly broad class of random geometric graphs.  We will then discuss entropy bounds for these graphs and, time permitting, local (pairwise) connection rules that yield maximum entropy properties in the induced graph distribution.

Given a number field K, it is natural to ask whether it has a finite extension with ideal class number one. This question can be translated into a fundamental question in class field theory, namely the class field tower problem. In this talk, we are going to discuss this problem as well as its solution due to Golod and Shafarevich using methods of group cohomology.
 

 The Steklov eigenvalue problem is an eigenvalue problem whose spectral parameters appear in the boundary condition. On a Riemannian surface with smooth boundary, Steklov eigenvalues have a very sharp asymptotic expansion. Also, a number of interesting sharp bounds for the $k$th Steklov eigenvalues have been known. We extend these results on orbisurfaces and discuss how the structure of orbifold singularities comes into play. This is joint work with Arias-Marco, Dryden, Gordon, Ray and Stanhope.

A broad challenge in the theory of finitely generated groups is to understand their subgroups. A group is commensurably coHopfian if its finite index subgroups are distinct from its infinite index subgroups (that is to say not abstractly isomorphic). We will focus primarily on hyperbolic groups, and give the first examples of one-ended hyperbolic groups that are not commensurably coHopfian.
This is joint work with Emily Stark.
 

"We consider a spatial Lambda-Fleming-Viot process, a model in mathematical biology, with a randomly chosen (rough) selection field. We study the scaling limit of this process in different regimes. This leads to the analysis of semi-discrete approximations of singular SPDEs, in particular the Parabolic Anderson Model and allows to extend previous results to weakly nonlinear cases. The subject presented is based on joint works with Aleksander Klimek and Nicolas Perkowski."

In this talk I will introduce a continuous wetting model consisting of the law of a Brownian meander tilted by its local time at a positive level h, with h small. I will prove that this measure converges, as h tends to 0, to the same weak limit as for discrete critical wetting models. I will also discuss the corresponding gradient dynamics, which is expected to converge to a Bessel SPDE admitting the law of a reflecting Brownian motion as invariant measure. This is based on joint work with Jean-Dominique Deuschel and Tal Orenshtein.

McDuff and Schlenk determined when a four-dimensional symplectic ellipsoid can be symplectically embedded into a four-dimensional ball. They found that if the ellipsoid is close to round, the answer is given by an ``infinite staircase" determined by the odd index Fibonacci numbers, while if the ellipsoid is sufficiently stretched, all obstructions vanish except for the volume obstruction. Infinite staircases have also been found when embedding ellipsoids into polydisks (Frenkel - Muller, Usher) and into the ellipsoid E(2, 3) (Cristofaro-Gardiner - Kleinman). In this talk, we will see how the sharpness of ECH capacities for embedding of ellipsoids implies the existence of infinite staircases for these and three other target spaces.  We will then discuss the relationship with toric varieties, lattice point counting, and the Philadelphia subway system. This is joint work with Dan Cristofaro-Gardiner, Alessia Mandini,
and Ana Rita Pires.

In the process of studying the zeta-function for one parameter families of Calabi-Yau manifolds we have been led to a manifold, for which the quartic numerator of the zeta-function factorises into two quadrics remarkably often. Among these factorisations, we find persistent factorisations; these are determined by a parameter that satisfies an algebraic equation with coefficients in Q, so independent of any particular prime.  We note that these factorisations are due a splitting of Hodge structure and that these special values of the parameter are rank two attractor points in the sense of IIB supergravity. To our knowledge, these points provide the first explicit examples of non-singular, non-rigid rank two attractor points for Calabi-Yau manifolds of full SU(3) holonomy. Modular groups and modular forms arise in relation to these attractor points in a way that, to a physicist, is unexpected. This is a report on joint work with Xenia de la Ossa, Mohamed Elmi and Duco van Straten.