At the heart of the interior point method in optimization is a linear system solve, but how accurate must this solve be? The behaviour of such methods is well-understood when a direct solver is used, but the scale of problems being tackled today means that users increasingly turn to iterative methods to approximate its solution. Current suggestions of the accuracy required can be seen to be too stringent, leading to inefficiency.
In this talk I will give conditions on the accuracy of the solution in order to guarantee the inexact interior point method converges at the same rate as if there was an exact solve. These conditions can be shown numerically to be tight, in that performance degrades rapidly if a weaker condition is used. Finally, I will describe how the norms that appear in these condition are related to the natural norms that are minimized in several popular Krylov subspace methods. This, in turn, could help in the development of new preconditioners in this important field.
There is an obvious product-free subset of the symmetric group of density 1/2, but what about the alternating group? An argument of Gowers shows that a product-free subset of the alternating group can have density at most n^(-1/3), but we only know examples of density n^(-1/2 + o(1)). We'll talk about why in fact n^(-1/2 + o(1)) is the right answer, why
Gowers's argument can't prove that, and how this all fits in with a more general 'product mixing' phenomenon. Our tools include some nonabelian Fourier analysis, a version of entropy subadditivity adapted to the symmetric group, and a concentration-of-measure result for rearrangements of inner products.
The formal degree is a fundamental invariant of a discrete series representation which generalizes the notion of dimension from finite dimensional representations. For discrete series with unipotent cuspidal support, a formula for formal degrees, conjectured by Hiraga-Ichino-Ikeda, was verified by Opdam (2015). For split exceptional groups, this formula was previously known from the work of Reeder (2000). I will present a different interpretation of the formal degrees of unipotent discrete series in terms of the nonabelian Fourier transform (introduced by Lusztig in the character theory of finite groups of Lie type) and certain invariants arising in the elliptic theory of the affine Weyl group. This interpretation relates to recent conjectures of Lusztig about `almost characters' of p-adic groups. The talk is based on joint work with Eric Opdam.
Operators, functions, and functionals are combined in many problems of computational science in a fashion that has the same logical structure as is familiar for block matrices and vectors. It is proposed that the explicit consideration of such block structures at the continuous as opposed to discrete level can be a useful tool. In particular, block operator diagrams provide templates for spectral discretization by the rectangular differentiation, integration, and identity matrices introduced by Driscoll and Hale. The notion of the rectangular shape of a linear operator can be made rigorous by the theory of Fredholm operators and their indices, and the block operator formulations apply to nonlinear problems too, where the convention is proposed of representing nonlinear blocks as shaded. At each step of a Newton iteration, the structure is linearized and the blocks become unshaded, representing Fréchet derivative operators, square or rectangular. The use of block operator diagrams makes it possible to precisely specify discretizations of even rather complicated problems with just a few lines of pseudocode.
[Joint work with Nick Trefethen]
In this talk, I will review an inverse scattering construction of interacting integrable quantum field theories on two-dimensional Minkowski space and its ramifications. The construction starts from a given two-body S-matrix instead of a classical Lagrangean, and defines corresponding quantum field theories in a non-perturbative manner in two steps: First certain semi-local fields are constructed explicitly, and then the analysis of the local observable content is carried out with operator-algebraic methods (Tomita-Takesaki modular theory, split subfactor inclusions). I will explain how this construction solves the inverse scattering problem for a large family of interactions, and also discuss perspectives on extensions of this program to higher dimensions and/or non-integrable theories.
Consider the following question. Given a $k$-colouring of the positive integers, must there exist a solution to $x+y=z^2$ with $x,y,z$ all the same colour (and not all equal to 2)? Using $10$ colours a counterexample can be given to show that the answer is "no". If one instead asks the same question over $\mathbb{Z}/p\mathbb{Z}$ for some prime $p$, the answer turns out to be "yes", provided $p$ is large enough in terms of the number of colours used. I will talk about how to prove this using techniques developed by Ben Green and Tom Sanders. The main ingredients are a regularity lemma, a counting lemma and a Ramsey lemma.
For maps from surfaces there is a close connection between the area of the surface parametrised by the map and its Dirichlet energy and this translates also into a relation for the corresponding critical points. As such, when trying to find minimal surfaces, one route to take is to follow a suitable gradient flow of the Dirichlet energy. In this talk I will introduce such a flow which evolves both a map and a metric on the domain in a way that is designed to change the initial data into a minimal immersions and discuss some question concerning the existence of solutions and their asymptotic behaviour. This is joint work with Peter Topping.
The Cartan model computes the equivariant cohomology of a smooth manifold X with
differentiable action of a compact Lie group G, from the invariant functions on
the Lie algebra with values in differential forms and a deformation of the de Rham
differential. Before extracting invariants, the Cartan differential does not square
to zero. Unrecognised was the fact that the full complex is a curved algebra,
computing the quotient by G of the algebra of differential forms on X. This
generates, for example, a gauged version of string topology. Another instance of
the construction, applied to deformation quantisation of symplectic manifolds,
gives the BRST construction of the symplectic quotient. Finally, the theory for a
X point with an additional quadratic curving computes the representation category
of the compact group G.
We derive a Stratonovich-to-Skorohod integral conversion formula for a class of integrands which are path-level solutions to RDEs driven by Gaussian rough paths. This is done firstly by showing that this class lies in the domain of the Skorohod integral, and secondly, by appending the Riemann-sum approximants of the Skorohod integral with a suitable compensation term. To show the convergence of the Riemann-sum approximants, we utilize a novel characterization of the Cameron-Martin norm using higher dimensional Young-Stieltjes integrals. Moreover, in the case where complementary regularity is absent, i.e. when the integrand has finite p-variation and the integrator has finite q-variation but 1/p + 1/q <= 1, we give new and sufficient conditions for the convergence these Young integrals.
After reviewing the main results relating holomorphic Poisson geometry to generalized Kahler structures, I will explain some recent progress in deforming generalized Kahler structures. I will also describe a new way to view generalized kahler geometry purely in terms of Poisson structures.
The Hilbert transform is a central operator in harmonic analysis as it gives access to the harmonic conjugate function. The link between pairs of martingales (X,Y) under differential subordination and the pair (f,Hf) of a function and its Hilbert transform have been known at least since the work of Burkholder and Bourgain in the UMD setting.
During the last 20 years, new and more exact probabilistic interpretations of operators such as the Hilbert transform have been studied extensively. The motivation for this was in part the study of optimal weighted estimates in harmonic analysis. It has been known since the 70s that H:L^2(w dx) to L^2(w dx) if and only if w is a Muckenhoupt weight with its finite Muckenhoupt characteristic. By a sharp estimate we mean the correct growth of the weighted norm in terms of this characteristic. In one particular case, such an estimate solved a long standing borderline regularity problem in complex PDE.
In this lecture, we present the historic development of the probabilistic interpretation in this area, as well as recent results and open questions.
A large number of examples of compact G2 manifolds, relevant to supersymmetric compactifications of M-Theory to four dimensions, can be constructed by forming a twisted connected sum of two appropriate building blocks times a circle. These building blocks, which are appropriate K3-fibred threefolds, are shown to have a natural and elegant construction in terms of tops, which parallels the construction of Calabi-Yau manifolds via reflexive polytopes.
The question of deriving Fluid Mechanics equations from deterministic
systems of interacting particles obeying Newton's laws, in the limit
when the number of particles goes to infinity, is a longstanding open
problem suggested by Hilbert in his 6th problem. In this talk we shall
present a few attempts in this program, by explaining how to derive some
linear models such as the Heat, acoustic and Stokes-Fourier equations.
This corresponds to joint works with Thierry Bodineau and Laure Saint
Raymond.
In this talk I will review mathematical models used to describe the dynamics of ice sheets, and highlight some current areas of active research. Melting of glaciers and ice sheets causes an increase in global sea level, and provides many other feedbacks on isostatic adjustment, the dynamics of the ocean, and broader climate patterns. The rate of melting has increased in recent years, but there is still considerable uncertainty over why this is, and whether the increase will continue. Central to these questions is understanding the physics of how the ice intereacts with the atmosphere, the ground on which it rests, and with the ocean at its margins. I will given an overview of the fluid mechanical problems involved and the current state of mathematical/computational modelling. I will focus particularly on the issue of changing lubrication due to water flowing underneath the ice, and discuss how we can use models to rationalise observations of ice speed-up and slow-down.
TBA
Partial differential equations with more than three coordinates arise naturally if the model features certain kinds of stochasticity. Typical examples are the Schroedinger, Fokker-Planck and Master equations in quantum mechanics or cell biology, as well as quantification of uncertainty.
The principal difficulty of a straightforward numerical solution of such equations is the `curse of dimensionality': the storage cost of the discrete solution grows exponentially with the number of coordinates (dimensions).
One way to reduce the complexity is the low-rank separation of variables. One can see all discrete data (such as the solution) as multi-index arrays, or tensors. These large tensors are never stored directly.
We approximate them by a sum of products of smaller factors, each carrying only one of the original variables. I will present one of the simplest but powerful of such representations, the Tensor Train (TT) decomposition. The TT decomposition generalizes the approximation of a given matrix by a low-rank matrix to the tensor case. It was found that many interesting models allow such approximations with a significant reduction of storage demands.
A workhorse approach to computations with the TT and other tensor product decompositions is the alternating optimization of factors. The simple realization is however prone to convergence issues.
I will show some of the recent improvements that are indispensable for really many dimensions, or solution of linear systems with non-symmetric or indefinite matrices.
I will discuss a couple of techniques often useful to prove quasi-isometric rigidity results for isometry groups. I will then sketch how these were used by B. Kleiner and B. Leeb to obtain quasi-isometric rigidity for the class of fundamental groups of closed locally symmetric spaces of noncompact type.
In this talk, I'll discuss some personal experiences - good, bad, and
ugly - of disclosing vulnerabilities in a range of different cryptographic
standards and implementations. I'll try to draw some general lessons about
what works well and what does not.
Given an Artin stack $X$, there is growing evidence that there should be an associated `category of B-branes', which is some subcategory of the derived category of coherent sheaves on $X$. The simplest case is when $X$ is just a vector space modulo a linear action of a reductive group, or `gauged linear sigma model' in physicists' terminology. In this case we know some examples of what the category B-branes should be. Hori has conjectured a physical duality between certain families of GLSMs, which would imply that their B-brane categories are equivalent. We prove this equivalence of categories. As an application, we construct Homological Projective Duality for (non-commutative resolutions of) Pfaffian varieties.
The chromatic number of the Erdős–Rényi random graph G(n,p) has been an intensely studied subject since at least the 1970s. A celebrated breakthrough by Bollobás in 1987 first established the asymptotic value of the chromatic number of G(n,1/2), and a considerable amount of effort has since been spent on refining Bollobás' approach, resulting in increasingly accurate bounds. Despite this, up until now there has been a gap of size O(1) in the denominator between the best known upper and lower bounds for the chromatic number of dense random graphs G(n,p) where p is constant. In contrast, much more is known in the sparse case.
In this talk, new upper and lower bounds for the chromatic number of G(n,p) where p is constant will be presented which match each other up to a term of size o(1) in the denominator. In particular, they narrow down the optimal colouring rate, defined as the average colour class size in a colouring with the minimum number of colours, to an interval of length o(1). These bounds were obtained through a careful application of the second moment method rather than a variant of Bollobás' method. Somewhat surprisingly, the behaviour of the chromatic number changes around p=1-1/e^2, with a different limiting effect being dominant below and above this value.
I will describe 5 definitions of Euler characteristic for a space with perfect obstruction theory (i.e. a well-behaved moduli space), and their inter-relations. This is joint work with Yunfeng Jiang. Then I will describe work of Yuuji Tanaka on how to this can be used to give two possible definitions of Vafa-Witten invariants of projective surfaces in the stable=semistable case.
Adaptive cubic regularization methods have recently emerged as a credible alternative to line search and trust-region for smooth nonconvex optimization, with optimal complexity amongst second-order methods. Here we consider a general class of adaptive regularization methods, that use first- or higher-order local Taylor models of the objective regularized by a(ny) power of the step size. We investigate the worst-case complexity/global rate of convergence of these algorithms, in the presence of varying (unknown) smoothness of the objective. We find that some methods automatically adapt their complexity to the degree of smoothness of the objective; while others take advantage of the power of the regularization step to satisfy increasingly better bounds with the order of the models. This work is joint with Nick Gould (RAL) and Philippe Toint (Namur).
tba
Modular invariance is ubiquitous in string theory. This is the symmetry of genus-one amplitudes, as well as the non-perturbative duality symmetry of type IIb superstring in ten dimensions. The alpha’ expansion of string theory amplitudes leads to interesting new modular forms. In this talk we will describe the properties of the new modular forms. We will explain that the modular forms entering the alpha’ expansion of genus one type-II superstring amplitude are naturally expressed as particular values of single valued elliptic multiple polylogarithm. They are natural modular generalization of the single valued elliptic multiple-zeta introduced by Francis Brown.
Last year, G. Kings and D. Rossler related the degree zero part of the polylogarithm
on abelian schemes pol^0 with another object previously defined by V. Maillot and D.
Rossler. More precisely, they proved that the canonical class of currents constructed
by Maillot and Rossler provides us with the realization of pol^0 in analytic Deligne
cohomology.
I will show that, adding some properness conditions, it is possible to give a
refinement of Kings and Rossler’s result involving Deligne-Beilinson cohomology
instead of analytic Deligne cohomology.
I will consider higher derivative corrections to the graviton 3-point coupling within a weakly coupled theory of gravity. Lorentz invariance allows further structures beyond that of Einstein’s theory. I will argue that these structures are constrained by causality, and show that the problem cannot be fixed by adding conventional particles with spins J ≤ 2, but adding an infinite tower of massive particles with higher spins. Implications of this result in the context of AdS/CFT, quantum gravity in asymptotically flat space-times, and non-Gaussianity features of primordial gravitational waves are discussed.
If the abundances of the constituent molecules of a biochemical reaction system are sufficiently high then their concentrations are typically modelled by a coupled set of ordinary differential equations (ODEs). If, however, the abundances are low then the standard deterministic models do not provide a good representation of the behaviour of the system and stochastic models are used. In this talk, I will first introduce both the stochastic and deterministic models. I will then provide theorems that allow us to determine the qualitative behaviour of the underlying mathematical models from easily checked properties of the associated reaction network. I will present results pertaining to so-called ``complex-balanced'' models and those satisfying ``absolute concentration robustness'' (ACR). In particular, I will show how ACR models, which are stable when modelled deterministically, necessarily undergo an extinction event in the stochastic setting. I will then characterise the behaviour of these models prior to extinction.
We examine the Foreign Exchange (FX) spot price spreads with and without Last Look on the transaction. We assume that brokers are risk-neutral and they quote spreads so that losses to latency arbitrageurs (LAs) are recovered from other traders in the FX market. These losses are reduced if the broker can reject, ex-post, loss-making trades by enforcing the Last Look option which is a feature of some trading venues in FX markets. For a given rejection threshold the risk-neutral broker quotes a spread to the market so that her expected profits are zero. When there is only one venue, we find that the Last Look option reduces quoted spreads. If there are two venues we show that the market reaches an equilibrium where traders have no incentive to migrate. The equilibrium can be reached with both venues coexisting, or with only one venue surviving. Moreover, when one venue enforces Last Look and the other one does not, counterintuitively, it may be the case that the Last Look venue quotes larger spreads.
a working version of the paper may be found here
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2630662
Higgs bundles have a rich structure and play a role in many different areas including gauge theory, hyperkähler geometry, surface group representations, integrable systems, nonabelian Hodge theory, mirror symmetry and Langlands duality. In this introductory talk I will explain some basic notions of G-Higgs – including the Hitchin fibration and spectral data - and illustrate how this relates to mirror symmetry.
We shall describe a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. Our proof produces a numerical upper-bound for the degree of the finite morphism from an isotrivial variety appearing in the statement of the Mordell-Lang conjecture. This upper-bound is given in terms of the Frobenius-stabilised slopes of the cotangent bundle of the variety.
We shall describe a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety.
Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. Our proof produces a numerical upper-bound for the degree of the finite morphism from an isotrivial variety appearing in the statement of the Mordell-Lang conjecture. This upper-bound is given in terms of the Frobenius-stabilised slopes of the cotangent bundle of the variety.
With few exceptions, optimal stopping assumes that the underlying system is stopped immediately after the decision is made.
In fact, most stoppings take time. This has been variously referred to as "time-to-build", "investment lag" and "gestation period",
which is often non negligible.
In this talk, we consider a class of optimal stopping/switching problems with delivery lags, or equivalently, delayed information,
by using reflected BSDE method. As an example, we study American put option with delayed exercise, and show that it can be decomposed
as a European put option and a premium, the latter of which involves a new optimal stopping problem where the investor decides when to stop
to collect the Greek theta of such a European option. We also give a complete characterization of the optimal exercise boundary by resorting to free boundary analysis.
Joint work with Zhou Yang and Mihail Zervos.