Smooth cubic fourfolds are linked to K3 surfaces via their Hodge structures, due to work of Hassett, and via Kuznetsov's K3 category A. The relation between these two viewpoints has recently been elucidated by Addington and Thomas.
We study both of these aspects further and extend them to twisted K3 surfaces, which in particular allows us to determine the group of autoequivalences of A for the general cubic fourfold. Furthermore, we prove finiteness results for cubics with equivalent K3 categories and study periods of cubics in terms of generalized K3 surfaces.
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