Counter-examples to the Hasse principle are known for many families of geometrically rational varieties. We discuss how often such failures arise for Chatelet surfaces and certain higher-dimensional hypersurfaces. This is joint work with Regis de la Breteche.
The bounded coherent dg-category on (suitable versions of) the Steinberg stack of a reductive group G is a categorification of the affine Hecke algebra in representation theory. We discuss how to describe the center and universal trace of this monoidal dg-category. Many of the techniques involved are very general, and the description makes use of the notion of "odd micro-support" of coherent complexes. This is joint work with Ben-Zvi and Nadler.
Morse theory gives an estimate of the dimensions of the cohomology groups of a manifold in terms of the critical points of a function.
One can do better and compute the cohomology in terms of this function using the so-called Witten complex.
Already implicit in work of Smale in the fifties, it was rediscovered by Witten in the eighties using techniques from (supersymmetric) quantum field theories.
I will explain Witten's (heuristic) arguments and describe the Witten complex.
The symmetric group S_{mn} acts naturally on the collection of set partitions of a set of size mn into n sets each of size m, and the resulting permutation character is the Foulkes character. These characters are the subject of the longstanding Foulkes Conjecture. In this talk, we define a deflation map which sends a character of the symmetric group S_{mn} to a character of S_n. The values of the images of the irreducible characters under this map are described combinatorially in a rule which generalises two well-known combinatorial rules in the representation theory of symmetric groups, the Murnaghan-Nakayama formula and Young's rule. We use this in a new algorithm for computing irreducible constituents of Foulkes characters and verify Foulkes’ Conjecture in some new cases. This is joint work with Anton Evseev (Birmingham) and Mark Wildon (Royal Holloway).
Many methods for solving high-dimensional statistical inverse problems are based on convex optimization problems formed by the weighted sum of a loss function with a norm-based regularizer.
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Particular examples include $\ell_1$-based methods for sparse vectors and matrices, nuclear norm for low-rank matrices, and various combinations thereof for matrix decomposition and robust PCA. In this talk, we describe an interesting connection between computational and statistical efficiency, in particular showing that the same conditions that guarantee that an estimator has good statistical error can also be used to certify fast convergence of first-order optimization methods up to statistical precision.
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Joint work with Alekh Agarwahl and Sahand Negahban Pre-print (to appear in Annals of Statistics)
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http://www.eecs.berkeley.edu/~wainwrig/Papers/AgaNegWai12b_SparseOptFul…
We consider the problem of optimizing a portfolio of medium to low frequency
quant strategies under heavy tailed distributions. Approaching this problem by modelling
returns through mixture distributions, we derive robust and relative robust methodologies
and discuss conic optimization approaches to solving these models.
The Kolmogorov 1941 theory (K41) is, in a way, the starting point for all
models of turbulence. In particular, K41 and corrections to it provide
estimates of small-scale quantities such as increments and energy spectrum
for a 3D turbulent flow. However, because of the well-known difficulties
involved in studying 3D turbulent flow, there are no rigorous results
confirming or infirming those predictions. Here, we consider a well-known
simplified model for 3D turbulence: Burgulence, or turbulence for the 1D
Burgers equation. In the space-periodic case with a stochastic white in
time and smooth in space forcing term, we give sharp estimates for
small-scale quantities such as increments and energy spectrum.
A topological space is called rigid if its only autohomeomorphism is the identity map. Using the Axiom of Choice it is easy to construct rigid subsets of the real line R, but sets constructed in this way always have size continuum. I will explore the question of whether it is possible to have rigid subsets of R that are small, meaning that their cardinality is smaller than that of the continuum. On the one hand, we will see that forcing can be used to produce models of ZFC in which such small rigid sets abound. On the other hand, I will introduce a combinatorial axiom that can be used to show the consistency with ZFC of the statement "CH fails but every rigid subset of R has size continuum". Only a working knowledge of basic set theory (roughly what one might remember from C1.2b) and topology will be assumed.
Self-similarity is a fundamental idea in many areas of mathematics. In this talk I will explain how it has entered group theory and the links between self-similar groups and other areas of research. There will also be pretty pictures.
I will give an introduction to The McKay Correspondence, relating the irreducible representations of a finite subgroup Γ ≤ SL2 (C), minimal resolutions of the orbit space C2 /Γ, and affine Dynkin diagrams.
Complex systems emerging from many biochemical applications often exhibit multiscale and multiphysics (MSMP) features: The systems incorporate a variety of physical processes or subsystems across a broad range of scales. A typical MSMP system may come across scales with macroscopic, mesoscopic and microscopic kinetics,
deterministic and stochastic dynamics, continuous and discrete state space, fastscale and slow-scale reactions, and species of both large and small populations. These complex features present great challenges in the modeling and simulation practice. The goal of our research is to develop innovative computational methods and rigorous fundamental theories to answer these challenges. In this talk we will start with introduction of basic stochastic simulation algorithms for biochemical systems and multiscale
features in the stochastic cell cycle model of budding yeast. With detailed analysis of these multiscale features, we will introduce recent progress on simulation algorithms and computational theories for multiscale stochastic systems, including tau-leaping methods, slow-scale SSA, and the hybrid method.
While usual percolation concerns the study of the connected components of
random subgraphs of an infinite graph, bootstrap percolation is a type of
cellular automaton, acting on the vertices of a graph which are in one of
two states: `healthy' or `infected'. For any positive integer $r$, the
$r$-neighbour bootstrap process is the following update rule for the
states of vertices: infected vertices remain infected forever and each
healthy vertex with at least $r$ infected neighbours becomes itself
infected. These updates occur simultaneously and are repeated at discrete
time intervals. Percolation is said to occur if all vertices are
eventually infected.
As it is often difficult to determine precisely which configurations of
initially infected vertices percolate, one often considers a random case,
with each vertex infected independently with a fixed probability $p$. For
an infinite graph, of interest are the values of $p$ for which the
probability of percolation is positive. I will give some of the history
of this problem for regular trees and present some new results for
bootstrap percolation on certain classes of randomly generated trees:
Galton--Watson trees.
***** PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON TUESDAY 19TH FEBRUARY *****
With "fully anisotropic" I describe diffusion models of the form u_t=\nabla \nabla (D(x) u), where the diffusion tensor appears inside both derivatives. This model arises naturally in the modeling of brain tumor spread and wolf movement and other applications. Since this model does not satisfy a maximum principle, it can lead to interesting spatial pattern formation, even in the linear case. I will present a detailed derivation of this model and discuss its application to brain tumors and wolf movement. Furthermore, I will present an example where, in the linear case, the solution blows-up in infinite time; a quite surprising result for a linear parabolic equation (joint work with K.J. Painter and M. Winkler).
In this talk we focus on the incompressible Navier–Stokes equations with variable
density. The aim is to prove existence and uniqueness results in the case of a discontinuous
initial density (typically we are interested in discontinuity along an interface).
In the first part of the talk, by making use of Fourier analysis techniques, we establish the existence of global-in-time unique solutions in a critical
functional framework, under some smallness condition over the initial data,
In the second part, we use another approach to avoid the smallness condition over the nonhomogeneity : as a matter of fact, one may consider any density bounded
and bounded away from zero and still get a unique solution. The velocity is required to have subcritical regularity, though.
In all the talk, the Lagrangian formulation for describing the flow plays a key role in the analysis.
Abstract: A central question in the theory of random walks on groups is how symmetries of the underlying space gives rise to structure and rigidity of the random walks. For example, for nilpotent groups, it is known that random walks have diffusive behavior, namely that the rate of escape, defined as the expected distance of the walk from the identity satisfies E|Xn|~=n^{1/2}. On nonamenable groups, on the other hand we have E|Xn| ~= n. (~= meaning upto (multiplicative) constants )
In this work, for every 3/4 <= \beta< 1 we construct a finitely generated group so that the expected distance of the simple random walk from its starting point after n steps is n^\beta (up to constants). This answers a question of Vershik, Naor and Peres. In other examples, the speed exponent can fluctuate between any two values in this interval.
Previous examples were only of exponents of the form 1-1/2^k or 1 , and were based on lamplighter (wreath product) constructions.(Other than the standard beta=1/2 and beta=1 known for a wide variety of groups) In this lecture we will describe how a variation of the lamplighter construction, namely the permutational wreath product, can be used to get precise bounds on the rate of escape in terms of return probabilities of the random walk on some Schreier graphs. We will then show how groups of automorphisms of rooted trees, related to automata groups , can be constructed and analyzed to get the desired rate of escape. This is joint work with Balint Virag of the University of Toronto. (Paper available at http://arxiv.org/abs/1203.6226)
No previous knowledge of random walks,automaton groups or wreath products is assumed.
Abstract: Hairer recently had the remarkable insight that Lyons' theory of rough paths can be used to make sense of nonlinear SPDEs that were previously ill-defined due to spatial irregularities. Since rough path theory deals with the integration of functions defined on the real line, the SPDEs studied by Hairer have a one-dimensional spatial index variable. I will show how to combine paraproducts, a notion from functional analysis, with ideas from the theory of controlled rough paths, in order to develop a formulation of rough path theory that works in any index dimension. As an application, I will present existence and uniqueness results for an SPDE with multidimensional spatial index set, for which previously it was not known how to describe solutions. No prior knowledge of rough paths or paraproducts is required for understanding the talk. This is joint work with Massimiliano Gubinelli and Peter Imkeller.
In this talk I will discuss recent developments in information theoretical approaches to fundamental
molecular processes that affect the cellular decision making processes. One of the challenges of applying
concepts from information theory to biological systems is that information is considered independently from
meaning. This means that a noisy signal carries quantifiably more information than a unperturbed signal.
This has, however, led us to consider and develop new approaches that allow us to quantify the level of noise
contributed by any molecular reactions in a reaction network. Surprisingly this analysis reveals an important and hitherto
often overlooked role of degradation reactions on the noisiness of biological systems. Following on from this I will outline
how such ideas can be used in order to understand some aspects of cell-fate decision making, which I will discuss with
reference to the haematopoietic system in health and disease.
InSAR (Interferometric Synthetic Aperture Radar) is an important space geodetic technique (i.e. a technique that uses satellite data to obtain measurements of the Earth) of great interest to geophysicists monitoring slip along fault lines and other changes to shape of the Earth. InSAR works by using the difference in radar phase returns acquired at two different times to measure displacements of the Earth’s surface. Unfortunately, atmospheric noise and other problems mean that it can be difficult to use the InSAR data to obtain clear measurements of displacement.
Persistent Scatterer (PS) InSAR is a later adaptation of InSAR that uses statistical techniques to identify pixels within an InSAR image that are dominated by a single back scatterer, producing high amplitude and stable phase returns (Feretti et al. 2001, Hooper et al. 2004). PS InSAR has the advantage that it (hopefully) chooses the ‘better’ datapoints, but it has the disadvantage that it throws away a lot of the data that might have been available in the original InSAR signal.
InSAR and PS InSAR have typically been used in isolation to obtain slip-rates across faults, to understand the roles that faults play in regional tectonics, and to test models of continental deformation. But could they perhaps be combined? Or could PS InSAR be refined so that it doesn’t throw away as much of the original data? Or, perhaps, could the criteria used to determine what data are signal and what are noise be improved?
The key aim of this workshop is to describe and discuss the techniques and challenges associated with InSAR and PS InSAR (particularly the problem of atmospheric noise), and to look at possible methods for improvement, by combining InSAR and PS InSAR or by methods for making the choice of thresholds.
Motivated by industrial and biological applications, the Waves
Group at Manchester has in recent years been interested in
developing methods for obtaining the effective properties of
complex composite materials. As time allows we shall discuss a
number of issues, such as differences between composites
with periodic and aperiodic distributions of inclusions, and
modelling of nonlinear composites.
I will explain the beautiful generalization recently discovered by Y. Tian of Heegner's original proof of the existence of infinitely many primes of the form 8n+5, which are congruent numbers. At the end, I hope to mention some possible generalizations of his work to other elliptic curves defined over the field of rational numbers.
Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$. In the third talk we will see that $\Lambda$ admits a canonical quantization if it is a "conification" of a compact exact Lagrangian submanifold of a cotangent bundle. We will see how to use this quantization to recover results of Fukaya-Seidel-Smith and Abouzaid on the topology of $\Lambda$.
Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$.
In the third talk we will see that $\Lambda$ admits a canonical quantization if it is a "conification" of a compact exact Lagrangian submanifold of a
cotangent bundle. We will see how to use this quantization to recover results of Fukaya-Seidel-Smith and Abouzaid on the topology of $\Lambda$.
We address, in the study of acoustic scattering by 2D and 3D planar screens, three inter-related and challenging questions. Each of these questions focuses particularly on the formulation of these problems as boundary integral equations. The first question is, roughly, does it make sense to consider scattering by screens which occupy arbitrary open sets in the plane, and do different screens cause the same scattering if the open sets they occupy have the same closure? This question is intimately related to rather deep properties of fractional Sobolev spaces on general open sets, and the capacity and Haussdorf dimension of their boundary. The second question is, roughly, that, in answering the first question, can we understand explicitly and quantitatively the influence of the frequency of the incident time harmonic wave? It turns out that we can, that the problems have variational formations with sesquilinear forms which are bounded and coercive on fractional Sobolev spaces, and that we can determine explicitly how continuity and coercivity constants depend on the frequency. The third question is: can we design computational methods, adapted to the highly oscillatory solution behaviour at high frequency, which have computational cost essentially independent of the frequency? The answer here is that in 2D we can provably achieve solutions to any desired accuracy using a number of degrees of freedom which provably grows only logarithmically with the frequency, and that it looks promising that some extension to 3D is possible.
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This is joint work with Dave Hewett, Steve Langdon, and Ashley Twigger, all at Reading.
The second order sensitivity of a trading position, the so
called gamma, has a very real and intuitive meaning to the traders.
People think that convex payoffs must generate convex prices. Being long
or short of gamma is a strategy used to balance risks in options books.
While the simples models, like Black Scholes, are consistent with this
intuition other popular models used in the industry are not. I will give
examples of simple and popular models which do not always convert a
convex payoff into a convex price. I will also give the necessary and
sufficient conditions under which the convexity is propagated.
First of all, I will give an overview of what the
phenomenon of homological stability is and why it's useful, with plenty
of examples. I will then introduce configuration spaces -- of various
different kinds -- and give an overview of what is known about their
homological stability properties. A "configuration" here can be more
than just a finite collection of points in a background space: in
particular, the points may be equipped with a certain non-local
structure (an "orientation"), or one can consider unlinked
embedded copies of a fixed manifold instead of just points. If by some
miracle time permits, I may also say something about homological
stability with local coefficients, in general and in particular for
configuration spaces.
We consider the emergence of classical correlations in macroscopic quantum systems, and its connection to monogamy relations for violation of Bell-type inequalities. We work within the framework of Abramsky and Brandenburger [1], which provides a unified treatment of non-locality and contextuality in the general setting of no-signalling empirical models. General measurement scenarios are represented by simplicial complexes that capture the notion of compatibility of measurements. Monogamy and locality/noncontextuality of macroscopic correlations are revealed by our analysis as two sides of the same coin: macroscopic correlations are obtained by averaging along a symmetry (group action) on the simplicial complex of measurements, while monogamy relations are exactly the inequalities that are invariant with respect to that symmetry. Our results exhibit a structural reason for monogamy relations and consequently for the classicality of macroscopic correlations in the case of multipartite scenarios, shedding light on and generalising the results in [2,3].More specifically, we show that, however entangled the microscopic state of the system, and provided the number of particles in each site is large enough (with respect to the number of allowed measurements), only classical (local realistic) correlations will be observed macroscopically. The result depends only on the compatibility structure of the measurements (the simplicial complex), hence it applies generally to any no-signalling empirical model. The macroscopic correlations can be defined on the quotient of the simplicial complex by the symmetry that lumps together like microscopic measurements into macroscopic measurements. Given enough microscopic particles, the resulting complex satisfies a structural condition due to Vorob'ev [4] that is necessary and sufficient for any probabilistic model to be classical. The generality of our scheme suggests a number of promising directions. In particular, they can be applied in more general scenarios to yield monogamy relations for contextuality inequalities and to study macroscopic non-contextuality.
[1] Samson Abramsky and Adam Brandenburger, The sheaf-theoretic structure of non-locality and contextuality, New Journal of Physics 13 (2011), no. 113036.
[2] MarcinPawłowski and Caslav Brukner, Monogamy of Bell’s inequality violations in nonsignaling theories, Phys. Rev. Lett. 102 (2009), no. 3, 030403.
[3] R. Ramanathan, T. Paterek, A. Kay, P. Kurzynski, and D. Kaszlikowski, Local realism of macroscopic correlations, Phys. Rev. Lett. 107 (2011), no. 6, 060405.
[4] N.N.Vorob’ev, Consistent families of measures and their extensions, Theory of Probability and its Applications VII (1962), no. 2, 147–163, (translated by N. Greenleaf, Russian original published in Teoriya Veroyatnostei i ee Primeneniya).
Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$.
In the second talk we will introduce a stack on $\Lambda$ by localization of the category of sheaves on $M$. We deduce topological obstructions on $\Lambda$ for the existence of a quantization.
A number is called transcendental if it is not algebraic, that is it does not satisfy a polynomial equation with rational coefficients. It is easy to see that the algebraic numbers are countable, hence the transcendental numbers are uncountable. Despite this fact, it turns out to be very difficult to determine whether a given number is transcendental. In this talk I will discuss some famous examples and the theorems which allow one to construct many different transcendental numbers. I will also give an outline of some of the many open problems in the field.
I propose to present modeling and experimental data about the organization of telomeres (ends of the chromosomes): the 32 telomeres in Yeast form few local aggregates. We built a model of diffusion-aggregation-dissociation for a finite number of particles to estimate the number of cluster and the number of telomere/cluster and other quantities. We will further present based on eingenvalue expansion for the Fokker-Planck operator, asymptotic estimation for the mean time a piece of a polymer (DNA) finds a small target in the nucleus.
We discuss the problem to what extend a group action determines geometry of the space.
More precisely, we show that for a large class of actions measurable isomorphisms must preserve
the geometric structure as well. This is a joint work with Bader, Furman, and Weiss.
Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$. In the first talk we will see that the graph of a Hamiltonian isotopy admits a canonical quantization and we deduce a new proof of Arnold's non-displaceability conjecture.
Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$.
In the first talk we will see that the graph of a Hamiltonian isotopy admits a canonical quantization and we deduce a new proof of Arnold's non-displaceability conjecture.
I will describe how a search for the answer to an old question about the existence of monotone arithmetic progressions in permutations of positive integers led to the study of infinite words with bounded additive complexity. The additive complexity of a word on a finite subset of integers is defined as the function that, for a positive integer $n$, counts the maximum number of factors of length $n$, no two of which have the same sum.
Defect measures have successfully been used, in a variety of
contexts, as a tool to quantify the lack of compactness of bounded
sequences of square-integrable functions due to concentration and
oscillation effects. In this talk we shall present some results on the
structure of the set of possible defect measures arising from sequences
of solutions to the linear Schrödinger equation on a compact manifold.
This is motivated by questions related to understanding the effect of
geometry on dynamical aspects of the Schrödinger flow, such as
dispersive effects and unique continuation.
It turns out that the answer to these questions depends strongly on
global properties of the geodesic flow on the manifold under
consideration: this will be illustrated by discussing with a certain
detail the examples of the the sphere and the (flat) torus.