This talk will be a gentle introduction to the main ideas behind some of the obstructions to the Hasse principle. In particular, I'll focus on the Brauer-Manin obstruction and on the descent obstruction, and explain briefly how other types of obstructions could be constructed.
A new nematic phase has recently been discovered and characterized experimentally. It embodies a theoretical prediction made by Robert B. Meyer in 1973 on the basis of mere symmetry considerations to the effect that a nematic phase might also exist which in its ground state would acquire a 'heliconical' configuration, similar to the chiral molecular arrangement of cholesterics, but with the nematic director precessing around a cone about the optic axis. Experiments with newly synthetized materials have shown chiral heliconical equilibrium structures with characteristic pitch in the range of 1o nanometres and cone semi-amplitude of about 20 degrees. In 2001, Ivan Dozov proposed an elastic theory for such (then still speculative) phase which features a negative bend elastic constant along with a quartic correction to the nematic energy density that makes it positive definite. This lecture will present some thoughts about the possibility of describing the elastic response of twist-bend nematics within a purely quadratic gradient theory.
Abstract:In this talk we consider a continuous time random
walk $X$ on $\mathbb{Z}^d$ in an environment of random conductances taking
values in $[0, \infty)$. Assuming that the law of the conductances is
ergodic with respect to space shifts, we present a quenched invariance
principle for $X$ under some moment conditions on the environment. The key
result on the sublinearity of the corrector is obtained by Moser's iteration
scheme. Under the same conditions we also present a local limit theorem. For
the proof some Hölder regularity of the transition density is needed, which
follows from a parabolic Harnack inequality. This is joint work with J.-D.
Deuschel and M. Slowik.
There are various Floer-theoretical invariants of links and 3-manifolds
which take the form of homology groups which are the E_infinity page of
spectral sequences starting from Khovanov homology. We shall discuss recent
work, joint with Raphael Zentner, and work in progress, joint with John
Baldwin and Matthew Hedden, in investigating and exploiting these spectral
sequences.
When cast in a Bayesian setting, the solution to an inverse problem is given as a distribution over the space where the quantity of interest lives. When the quantity of interest is in principle a field then the discretization is very high-dimensional. Formulating algorithms which are defined in function space yields dimension-independent algorithms, which overcome the so-called curse of dimensionality. These algorithms are still often too expensive to implement in practice but can be effectively used offline and on toy-models in order to benchmark the ability of inexpensive approximate alternatives to quantify uncertainty in very high-dimensional problems. Inspired by the recent development of pCN and other function-space samplers [1], and also the recent independent development of Riemann manifold methods [2] and stochastic Newton methods [3], we propose a class of algorithms [4,5] which combine the benefits of both, yielding various dimension-independent and likelihood-informed (DILI) sampling algorithms. These algorithms can be effective at sampling from very high-dimensional posterior distributions.
[1] S.L. Cotter, G.O. Roberts, A.M. Stuart, D. White. "MCMC methods for functions: modifying old algorithms to make them faster," Statistical Science (2013).
[2] M. Girolami, B. Calderhead. "Riemann manifold Langevin and Hamiltonian Monte Carlo methods," Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73 (2), 123–214 (2011).
[3] J. Martin, L. Wilcox, C. Burstedde, O. Ghattas. "A stochastic newton mcmc method for large-scale statistical inverse problems with application to seismic inversion," SIAM Journal on Scientific Computing 34(3), 1460–1487 (2012).
[4] K. J. H. Law. "Proposals Which Speed Up Function-Space MCMC," Journal of Computational and Applied Mathematics, in press (2013). http://dx.doi.org/10.1016/j.cam.2013.07.026
[5] T. Cui, K.J.H. Law, Y. Marzouk. Dimension-independent, likelihood- informed samplers for Bayesian inverse problems. In preparation.
I will define and describe in some details a large class of gauge theories in four dimensions. These theories admit a variational principle with the action a functional of only the gauge field. In particular, no metric appears in the Lagrangian or is used in the construction of the theory. The Euler-Lagrange equations are second order PDE's on the gauge field. When the gauge group is taken to be SO(3), a particular theory from this class can be seen to be (classically) equivalent to Einstein's General Relativity. All other points in the SO(3) theory space can be seen to describe "deformations" of General Relativity. These keep many of GR's properties intact, and may be important for quantum gravity. For larger gauge groups containing SO(3) as a subgroup, these theories can be seen to describe gravity plus Yang-Mills gauge fields, even though the associated geometry is much less understood in this case.
In this talk, we discuss an analogue of the Hermitian-Einstein equations for generalized Kaehler manifolds proposed by N. Hitchin. We explain in particular how these equations are equivalent to a notion of stability, and that there is a Kobayahsi-Hitchin-type of correspondence between solutions of these equations and stable objects. The correspondence holds even for non-Kaehler manifolds, as long as they are endowed with Gauduchon metrics (which is always the case for generalized Kaehler structures on 4-manifolds).
This is joint work with Shengda Hu and Reza Seyyedali.
We extend Kyle's (1985) model of insider trading to the case where liquidity provided
by noise traders follows a general stochastic process. Even though the level of noise
trading volatility is observable, in equilibrium, measured price impact is stochastic.
If noise trading volatility is mean-reverting, then the equilibrium price follows a
multivariate stochastic volatility `bridge' process. More private information is revealed
when volatility is higher. This is because insiders choose to optimally wait to trade
more aggressively when noise trading volatility is higher. In equilibrium, market makers
anticipate this, and adjust prices accordingly. In time series, insiders trade more
aggressively, when measured price impact is lower. Therefore, aggregate execution costs
to uninformed traders can be higher when price impact is lower
Clouds play a key role in the climate system. Driven by radiation, clouds power the hydrological cycle and global atmospheric dynamics. In addition, clouds fundamentally affect the global radiation balance by reflecting solar radiation back to space and trapping longwave radiation. The response of clouds to global warming remains poorly understood and is strongly regime dependent. In addition, anthropogenic aerosols influence clouds, altering cloud microphysics, dynamics and radiative properties. In this presentation I will review progress and limitations of our current understanding of the role of clouds in climate change and discuss the state of the art of the representation of clouds and aerosol-cloud interactions in global climate models, from (slightly) better constrained stratiform clouds to new frontiers: the investigation of anthropogenic effects on convective clouds.
I am interested in integer solutions to equations of the form $f(x)=0$ where $f$ is a transcendental, globally analytic function defined in a neighbourhood of $\infty$ in $\mathbb{R}^n \cup \{\infty\}$. These notions will be defined precisely, and clarified in the wider context of globally semi-analytic and globally subanalytic sets.
The case $n=1$ is trivial (the global assumption forces there to be only finitely many (real) zeros of $f$) and the case $n=2$, which I shall briefly discuss, is completely understood: the number of such integer zeros of modulus at most $H$ is of order $\log\log H$. I shall then go on to consider the situation in higher dimensions.
We motivate and dicuss the Beilinson Theorem for sheaves on projective spaces. Hopefully we see some examples along the way.
After an MISG there is time to reflect. I will report briefly on the follow up to two problems that we have worked on.
Crack Repair:
It has been found that thin elastically weak spray on liners stabilise walls and reduce rock blast in mining tunnels. Why? The explanation seems to be that the stress field singularity at a crack tip is strongly altered by a weak elastic filler, so cracks in the walls are less likely to extend.
Boundary Tracing:
Using known exact solutions to partial differential equations new domains can be constructed along which prescribed boundary conditions are satisfied. Most notably this technique has been used to extract a large class of new exact solutions to the non-linear Laplace Young equation (of importance in capillarity) including domains with corners and rough boundaries. The technique has also been used on Poisson's, Helmholtz, and constant curvature equation examples. The technique is one that may be useful for handling modelling problems with awkward/interesting geometry.
A popular approach within the signal processing and machine learning communities consists in modelling signals as sparse linear combinations of atoms selected from a learned dictionary. While this paradigm has led to numerous empirical successes in various fields ranging from image to audio processing, there have only been a few theoretical arguments supporting these evidences. In particular, sparse coding, or sparse dictionary learning, relies on a non-convex procedure whose local minima have not been fully analyzed yet. Considering a probabilistic model of sparse signals, we show that, with high probability, sparse coding admits a local minimum around the reference dictionary generating the signals. Our study takes into account the case of over-complete dictionaries and noisy signals, thus extending previous work limited to noiseless settings and/or under-complete dictionaries. The analysis we conduct is non-asymptotic and makes it possible to understand how the key quantities of the problem, such as the coherence or the level of noise, can scale with respect to the dimension of the signals, the number of atoms, the sparsity and the number of observations.
This is joint work with Rodolphe Jenatton & Francis Bach.
We will discuss TQFTs (at a basic level), then higher categorical extensions, and see how these lead naturally to the notion of Segal spaces.
Given a residually finite group, we analyse a growth function measuring the minimal index of a normal subgroup in a group which does not contain a given element. This growth (called residual finiteness growth) attempts to measure how ``efficient'' of a group is at being residually finite. We review known results about this growth, such as the existence of a Gromov-like theorem in a particular case, and explain how it naturally leads to the study of a second related growth (called intersection growth). Intersection growth measures asymptotic behaviour of the index of the intersection of all subgroups of a group that have index at most n. In this talk I will introduce these growths and give an overview of some cases and properties.
This is joint work with Ian Biringer, Khalid Bou-Rabee and Martin Kassabov.
Rabinowitz Floer homology (RFH) is the Floer theory associated to the Rabinowitz action functional. One can think of this functional as a Lagrange multiplier functional of the unperturbed action functional of classical mechanics. Its critical points are closed orbits of arbitrary period but with fixed energy.
This fixed energy problem can be transformed into a fixed period problem on an enlarged phase space. This provides a way to see RFH as a "standard" Hamiltonian Floer theory, and allows one to treat RFH on an equal footing to other related Floer theories. In this talk we explain how this is done and discuss several applications.
Joint work with Alberto Abbondandolo and Alexandru Oancea.
The antitriangular factorisation of real symmetric indefinite matrices recently proposed by Mastronardi and van Dooren has several pleasing properties. It is backward stable, preserves eigenvalues and reveals the inertia, that is, the number of positive, zero and negative eigenvalues.
In this talk we show that the antitriangular factorization simplifies for saddle point matrices, and that solving a saddle point system in antitriangular form is equivalent to applying the well-known nullspace method. We obtain eigenvalue bounds for the saddle point matrix and discuss the role of the factorisation in preconditioning.
We prove a generalization of Helly's theorem concerning intersections of convex sets that has an interesting voting theory interpretation. We then
consider various extensions in which compelling mathematical problems are motivated from very natural questions in the voting context.