We'll present a proof of the basic Burgess bound for short character sums, following the simplified presentation of Gallagher and Montgomery.
Orthogonal calculus is a calculus of functors, inspired by Goodwillie calculus. It takes as input a functor from finite dimensional inner product spaces to topological spaces and as output gives a tower of approximations by well-behaved functors. The output captures a lot of important homotopical information and is an important tool for calculations.
In this talk I will report on joint work with Peter Oman in which we use model categories to improve the foundations of orthogonal calculus. This provides a cleaner set of results and makes the role of O(n)-equivariance clearer. The classification of n-homogeneous functors in terms of spectra with O(n)-action can then be phrased as a zig-zag of Quillen equivalences.
If one starts with a uniformly ergodic Markov chain on countable states, what sort of perturbation can one make to the transition rates and still retain uniform ergodicity? In this talk, we will consider a class of perturbations, that can be simply described, where a uniform estimate on convergence to an ergodic distribution can be obtained. We shall see how this is related to Ergodic BSDEs in this setting and outline some novel applications of this approach.
The star-triangle transformation is used to obtain an equivalence extending over a set bond percolation models on isoradial graphs. Amongst the consequences are box-crossing (RSW) inequalities and the universality of alternating arms exponents (assuming they exist) for such models, under some conditions. In particular this implies criticality for these models.
(joint with Geoffrey Grimmett)
The banking industry lost a trillion dollars during the global financial crisis. Some of these losses, if not most of them, were attributable to complex derivatives or securities being incorrectly priced and hedged. We introduce a new methodology which provides a better way of trying to hedge and mark-to-market complex derivatives and other illiquid securities which recognise the fundamental incompleteness of markets and the presence of model uncertainty. Our methodology combines elements of the No Good Deals methodology of Cochrane and Saa-Requejo with the Robustness methodology of Hansen and Sargent. We give some numerical examples for a range of both simple and complex problems encompassing not only financial derivatives but also “real options”occurring in commodity-related businesses.
An overview will be given for several examples of fluid mechanical problems in developing household appliances, we discuss some examples of e.g. baby bottles, water treatment, irons, fruit juicers and focus on oral health care where a new air floss product will be discussed.
Masser recently proved a bound on the number of rational points of bounded height on the graph of the zeta function restricted to the interval [2,3]. Masser's bound substantially improves on bounds obtained by Bombieri-Pila-Wilkie. I'll discuss some results obtained in joint work with Gareth Boxall in which we prove bounds only slightly weaker than Masser's for several more natural analytic functions.
Van der Waerden has shown that `almost' all monic integer
polynomials of degree n have the full symmetric group S_n as Galois group.
The strongest quantitative form of this statement known so far is due to
Gallagher, who made use of the Large Sieve.
In this talk we want to explain how one can use recent
advances on bounding the number of integral points on curves and surfaces
instead of the Large Sieve to go beyond Gallagher's result.
Abstract available upon request - Ruth Preston @email
In this first of two talks, I shall introduce the Virtual Haken Conjecture and the major players involved in the proof announced by Ian Agol last year. These are the special cube complexes studied by Dani Wise and his collaborators, with a large supporting cast including the not-inconsiderable presence of Perelman’s Geometrization Theorem and the Surface Subgroup Theorem of Kahn and Markovic. I shall sketch how the VHC follows from Agol’s result that, in spite of the name, specialness is entirely generic among non-positively curved cube complexes.
Tensor numerical methods provide the efficient separable representation of multivariate functions and operators discretized on large $n^{\otimes d}$-grids, providing a base for the solution of $d$-dimensional PDEs with linear complexity scaling in the dimension, $O(d n)$. Modern methods of separable approximation combine the canonical, Tucker, matrix product states (MPS) and tensor train (TT) low-parametric data formats.
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The recent quantized-TT (QTT) approximation method is proven to provide the logarithmic data-compression on a wide class of functions and operators. Furthermore, QTT-approximation makes it possible to represent multi-dimensional steady-state and dynamical equations in quantized tensor spaces with the log-volume complexity scaling in the full-grid size, $O(d \log n)$, instead of $O(n^d)$.
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We show how the grid-based tensor approximation in quantized tensor spaces applies to super-compressed representation of functions and operators (super-fast convolution and FFT, spectrally close preconditioners) as well to hard problems arising in electronic structure calculations, such as multi-dimensional convolution, and two-electron integrals factorization in the framework of Hartree-Fock calculations. The QTT method also applies to the time-dependent molecular Schr{\"o}dinger, Fokker-Planck and chemical master equations.
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Numerical tests are presented indicating the efficiency of tensor methods in approximation of functions, operators and PDEs in many dimensions.
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In this talk, We show that both reflected BSDE and its associated penalized BSDE admit both optimal stopping representation and optimal control
representation. We also show that both multidimensional reflected BSDE and its associated multidimensional penalized BSDE admit optimal switching representation. The corresponding optimal stopping problems for penalized BSDE have the feature that one is only allowed to stop at Poisson arrival times.
The proof of several properties of solutions of hyperbolic systems of conservation laws in one space dimension (existence, stability, regularity) depends on the existence of a decreasing functional, controlling the nonlinear interactions of waves. In a special case (genuinely nonlinear systems) the interaction functional is quadratic, while in the general case it is cubic. Several attempts to prove the existence of a a quadratic functional also in the most general case have been done. I will present the approach we follow in order to prove this result, an some of its implication we hope to exploit.
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Work in collaboration with Stefano Modena.
Complex dynamical systems have been very well studied in recent years, in particular since computer graphics now enable us to peer deep into structures such as the Mandlebrot set and Julia sets, which beautifully illustrate the intricate dynamical behaviour of these systems. Using new techniques from Symbolic Dynamics, we demonstrate previously unknown properties of a class of quadratic maps on their Julia sets.
Cell polarity in the rod-shaped bacterium Myxococcus xanthus is crucial for the direction of movement of individual cells. Polarity is governed by a regulatory system characterized by a dynamic spatiotemporal oscillation of proteins between the opposite cell poles. A mathematical framework for a minimal macroscopic model is presented which produces self-sustained regular oscillations of the protein concentrations. The mathematical model is based on a reaction-diffusion PDE system and is independent of external triggers. Necessary conditions on the reaction terms leading to oscillating solutions are derived theoretically. Possible scenarios for protein interaction are numerically tested for robustness against parameter variation. Finally, possible extensions of the model will be addressed.
I will discuss similarities and differences between the geometry of
relatively hyperbolic groups and that of mapping class groups.
I will then discuss results about random walks on such groups that can
be proven using their common geometric features, namely the facts that
generic elements of (non-trivial) relatively hyperbolic groups are
hyperbolic, generic elements in mapping class groups are pseudo-Anosovs
and random paths of length $n$ stay $O(\log(n))$-close to geodesics in
(non-trivial) relatively hyperbolic groups and
$O(\sqrt{n}\log(n))$-close to geodesics in mapping class groups.
Razborov's flag algebras provide a formal system
for operating with asymptotic inequalities between subgraph densities,
allowing to do extensive "book-keeping" by a computer. This novel use
of computers led to progress on many old problems of extremal
combinatorics. In some cases, finer structural information can be
derived from a flag algebra proof by by using the Removal Lemma or
graph limits. This talk will overview this approach.