Oxford

A common strategy to solve shape optimization problems is to select an initial domain and to update it iteratively until it satisfies certain optimality crietria. In the presence of PDE-constraints, computing these updates requires solving a boundary value problem on a domain that changes at every iteration. We explain how to use isoparametric finite elements to tackle this issue. We also show how finite elements allow computing these updates without deriving shape derivative formulas by hand.

The E-string theory is usually considered as the simplest among 6D (1,0) superconformal field theories. Nonetheless, we still have little information about its spectrum of operators. In this talk, I'm going to describe our recent geometric approach using F-theory compactification on an elliptic Calabi-Yau threefold. The elliptic fibration is non-flat, which means that there are complex surface components in the fiber direction. From the geometry of non-flat fiber, we read out an infinite tower of particle states in the E-string theory. I will also discuss its relevance to 4D standard model building, which is a main motivation of this work.
 

The zeta-function of a manifold varies with the parameters and may be evaluated in terms of the periods. For a one parameter family of CY manifolds, the periods satisfy a single 4th order differential equation. Thus there is a straight and, it turns out, readily computable path that leads from a differential operator to a zeta-function. Especially interesting are the specialisations to singular manifolds, for which the zeta-function manifests modular behaviour. We are also able to find, from the zeta function, attractor points. These correspond to special values of the parameter for which there exists a 10D spacetime for which the 6D corresponds to a CY manifold and the 4D spacetime corresponds to an extremal supersymmetric black hole. These attractor CY manifolds are believed to have special number theoretic properties. This is joint work with Xenia de la Ossa, Mohamed Elmi and Duco van Straten.

The well known (and notoriously hard) P vs NP problem asks whether every Boolean function with polynomial-size proofs is also computable in
polynomial time.

The standard approach to the P vs NP problem is via circuit complexity. For progressively richer classes of Boolean circuits (networks of AND, OR and NOT
logic gates), one wishes to show super-polynomial lower bounds on the sizes of circuits (as a function of the size of the input) computing some Boolean
function known to be in NP, such as the Satisfiability problem.

However, there is a more logic-oriented approach initiated by Cook and Reckhow, going through proof complexity rather than circuit complexity. For
progressively richer proof systems, one wishes to show super-polynomial lower bounds on the sizes of proofs (as a function of the size of the tautology) of
some sequence of propositional tautologies.

I will give a brief overview on known results along these two directions, and on their limitations. Somewhat surprisingly, similar techniques have been found
to be useful for these seemingly different approaches. I will say something about known connections between the approaches, and pose the question of
whether there are deeper connections.

Finally, I will discuss how the perspective of proof complexity can be used to formalize the difficulty of proving lower bounds on the sizes of computations
(or of proofs).

 

 The talk is about the normal modal logics of elementary classes defined by first-order formulas of the form
 'for all x_0 there exist x_1, ..., x_n phi(x_0, x_1, ... x_n)' with phi being a conjunction of binary atoms.
 I'll show that many properties of these logics, such as finite axiomatisability,
 elementarity,  axiomatisability by a set of canonical formulas or by a single generalised Sahlqvist formula,
 together with modal definability of the initial formula, either simultaneously hold or simultaneously do not hold.
 

Analytic continuation is ill-posed, but becomes merely ill-conditioned (though with an infinite condition number) if it is known that the function in question is bounded in a given region of the complex plane.
This classical, seemingly theoretical subject has many connections with numerical practice.  One argument indicates that if one tracks an analytic function from z=1 around a branch point at z=0 and back to z=1 again by a Weierstrass chain of disks, the number of accurate digits is divided by about exp(2 pi e) ~= 26,000,000.

Formally, the Fourier coefficients of Eisenstein series on Kac-Moody groups contain as yet mysterious automorphic L-functions relevant to open conjectures such as that of Ramanujan and Langlands functoriality. In this talk, we will consider the constant Fourier coefficient, if it even makes sense rigorously, and its relationship to the geometry and combinatorics of a Kac-Moody group. Joint work with Kyu-Hwan Lee.

Following recent papers by Nima Arkani-Hamed.

In this talk, we are going to talk about the Type I singularity on 4-dimensional manifolds foliated by homogeneous S3 evolving under the Ricci
flow. We review the study on rotationally symmetric manifolds done by Angenent and Isenberg as well as by Isenberg, Knopf and Sesum. In the latter, a global frame for the tangent bundle, called the Milnor frame, was used to set up the problem. We shall look at the symmetries of the manifold, derived from Lie groups and its ansatz metrics, and this global tangent bundle frame developed by Milnor and Bianchi. Numerical simulations of the Ricci flow on these manifolds are done, following the work by Garfinkle and Isenberg, providing insight and conjectures for the main problem. Some analytic results will be proven for the manifolds S1×S3 and S4 using maximum principles from parabolic PDE theory and some sufficiency conditions for a neckpinch singularity will be provided. Finally, a problem from general relativity with similar metric symmetries but endowed on a manifold with differenttopology, the Taub-Bolt and Taub-NUT metrics, will be discussed.