Abstract: The signature of a path characterizes the non-commutative evolvements along the path trajectory. Nevertheless, one can extract local commutativities from the signature, thus leading to an inversion scheme.
We consider a large class of three dimensional continuous dynamic fluctuation models, and show that they all rescale and converge to the stochastic Allen-Cahn equation, whose solution should be interpreted after a suitable renormalization procedure. The interesting feature is that, the coefficient of the limiting equation is different from one's naive guess, and the renormalization required to get the correct limit is also different from what one would naturally expect. I will also briefly explain how the recent theory of regularity structures enables one to prove such results. Joint work with Martin Hairer.
For many questions of scientific interest, all-atom molecular simulations are still out of reach, in particular in materials engineering where large numbers of atoms, and often expensive force fields, are required. A long standing challenge has been to construct concurrent atomistic/continuum coupling schemes that use atomistic models in regions of space where high accuracy is required, with computationally efficient continuum models (e.g., FEM) of the elastic far-fields.
Many different mechanisms for achieving such atomistic/continuum couplings have been developed. To assess their relative merits, in particular accuracy/cost ratio, I will present a numerical analysis framework. I will use this framework to analyse in more detail a particularly popular scheme (the BQCE scheme), identifying key approximation parameters which can then be balanced (in a surprising way) to obtain an optimised formulation.
Finally, I will demonstrate that this analysis shows how to remove a severe bottlenecks in the BQCE scheme, leading to a new scheme with optimal convergence rate.
In elasticity theory, one naturally requires that the Jacobian determinant of the deformation is positive or even a-priori prescribed (for example incompressibility). However, such strongly non-linear and non-convex constraints are difficult to deal with in mathematical models. In this talk, which is based on joint work with K. Koumatos (Oxford) and E. Wiedemann (UBC/PIMS), I will present various recent results on how this constraint can be manipulated in subcritical Sobolev spaces, where the integrability exponent is less than the dimension.
In particular, I will give a characterization theorem for Young measures under this side constraint, which are widely used in the Calculus of Variations to model limits of nonlinear functions of weakly converging "generating" sequences. This is in the spirit of the celebrated Kinderlehrer--Pedregal Theorem and based on convex integration and "geometry" in matrix space.
Finally, applications to the minimization of integral functionals, the theory of semiconvex hulls, incompressible extensions, and approximation of weakly orientation-preserving maps by strictly orientation-preserving ones in Sobolev spaces are given.
The'signature', from the theory of differential equations driven by rough paths,
provides a very efficient way of characterizing curves. From a machine learning
perspective, the elements of the signature can be used as a set of features for
consumption by a classification algorithm.
Using datasets of letters, digits, Indian characters and Chinese characters, we
see that this improves the accuracy of online character recognition---that is
the task of reading characters represented as a collection of pen strokes.
We combine recent breakthroughs in modularity lifting with a
3-5-7 modularity switching argument to deduce modularity of elliptic curves over real
quadratic fields. We
discuss the implications for the Fermat equation. In particular we
show that if d is congruent
to 3 modulo 8, or congruent to 6 or 10 modulo 16, and $K=Q(\sqrt{d})$
then there is an
effectively computable constant B depending on K, such that if p>B is prime,
and $a^p+b^p+c^p=0$ with a,b,c in K, then abc=0. This is based on joint work with Nuno Freitas (Bayreuth) and Bao Le Hung (Harvard).
Many problems in the physical sciences
require the determination of an unknown
function from a finite set of indirect measurements.
Examples include oceanography, oil recovery,
water resource management and weather forecasting.
The Bayesian approach to these problems
is natural for many reasons, including the
under-determined and ill-posed nature of the inversion,
the noise in the data and the uncertainty in
the differential equation models used to describe
complex mutiscale physics. The object of interest
in the Bayesian approach is the posterior
probability distribution on the unknown field [1].
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However the Bayesian approach presents a
computationally formidable task as it
results in the need to probe a probability
measure on separable Banach space. Monte
Carlo Markov Chain methods (MCMC) may be
used to achieve this [2], but can be
prohibitively expensive. In this talk I
will discuss approximation of probability measures
by a Gaussian measure, looking for the closest
approximation with respect to the Kullback-Leibler
divergence. This methodology is widely
used in machine-learning [3]. In the context of
target measures on separable Banach space
which themselves have density with respect to
a Gaussian, I will show how to make sense of the
resulting problem in the calculus of variations [4].
Furthermore I will show how the approximate
Gaussians can be used to speed-up MCMC
sampling of the posterior distribution [5].
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[1] A.M. Stuart. "Inverse problems: a Bayesian
perspective." Acta Numerica 19(2010) and
http://arxiv.org/abs/1302.6989
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[2] S.L.Cotter, G.O.Roberts, A.M. Stuart and D. White,
"MCMC methods for functions: modifying old algorithms
to make them faster". Statistical Science 28(2013).
http://arxiv.org/abs/1202.0709
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[3] C.M. Bishop, "Pattern recognition and machine learning".
Springer, 2006.
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[4] F.J. Pinski G. Simpson A.M. Stuart H. Weber, "Kullback-Leibler
Approximations for measures on infinite dimensional spaces."
http://arxiv.org/abs/1310.7845
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[5] F.J. Pinski G. Simpson A.M. Stuart H. Weber, "Algorithms
for Kullback-Leibler approximation of probability measures in
infinite dimensions." In preparation.
Nesetril and Ossona de Mendez introduced a new notion of convergence of graphs called FO convergence. This notion can be viewed as a unified notion of convergence of dense and sparse graphs. In particular, every FO convergent sequence of graphs is convergent in the sense of left convergence of dense graphs as studied by Borgs, Chayes, Lovasz, Sos, Szegedy, Vesztergombi and others, and every FO convergent sequence of graphs with bounded maximum degree is convergent in the Benjamini-Schramm sense.
FO convergent sequences of graphs can be associated with a limit object called modeling. Nesetril and Ossona de Mendez showed that every FO convergent sequence of trees with bounded depth has a modeling. We extend this result
to all FO convergent sequences of trees and discuss possibilities for further extensions.
The talk is based on a joint work with Martin Kupec and Vojtech Tuma.
We consider a class of CR manifold which are defined as asymptotically
Heisenberg,
and for these we give a notion of mass. From the solvability of the
$\Box_b$ equation
in a certain functional class ([Hsiao-Yung]), we prove positivity of the
mass under the
condition that the Webster curvature is positive and that the manifold
is embeddable.
We apply this result to the Yamabe problem for compact CR manifolds,
assuming positivity
of the Webster class and non-negativity of the Paneitz operator. This is
joint work with
J.H.Cheng and P.Yang.
I will discuss two topics. Firstly, coupling of the circadian clock and cell cycle in mammalian cells. Together with the labs of Franck Delaunay (Nice) and Bert van der Horst (Rotterdam) we have developed a pipeline involving experimental and mathematical tools that enables us to track through time the phase of the circadian clock and cell cycle in the same single cell and to extend this to whole lineages. We show that for mouse fibroblast cell cultures under natural conditions, the clock and cell cycle phase-lock in a 1:1 fashion. We show that certain perturbations knock this coupled system onto another periodic state, phase-locked but with a different winding number. We use this understanding to explain previous results. Thus our study unravels novel phase dynamics of 2 key mammalian biological oscillators. Secondly, I present a radical revision of the Nrf2 signalling system. Stress responsive signalling coordinated by Nrf2 provides an adaptive response for protection against toxic insults, oxidative stress and metabolic dysfunction. We discover that the system is an autonomous oscillator that regulates its target genes in a novel way.