By imposing symmetry on manifolds of exceptional holonomy we get
a variety of differential geometric questions in lower dimensions.
Related to that, one can consider “adiabatic limits”, where the
manifold has a fibration and the fibre size is scaled...
We will discuss the constructions of compact manifolds with
exceptional holonomy (in fact, holonomy $G_{2}$), due to Joyce and
Kovalev. These both use “gluing constructions”. The first involves
de-singularising quotient spaces and the second...
In this lecture we will review the notion of the holonomy group
of a Riemannian manifold and the Berger classification. We will
discuss special algebraic structures in dimensions 6, 7 and 8,
emphasising exterior algebra, and then go on to...
Three fundamental factors determine the quality of a statistical
learning algorithm: expressiveness, optimization and
generalization. The classic strategy for handling these factors is
relatively well understood. In contrast, the radically
different...
Braverman and Kazhdan have conjectured the existence of
summation formulae that are essentially equivalent to the analytic
continuation and functional equation of Langlands L-functions in
great generality. Motivated by their conjectures and related...
(Joint work with Chris Woodward) Consider a Lagrangian
submanifold $\bar L$ in a GIT quotient $\bar X = X//G$. Besides the
usual Fukaya $A_\infty$ algebra $Fuk(\bar L)$ defined by counting
holomorphic disks, another version, called the quasimap...
We prove that if every problem in NP has n^k-size circuits for a
fixed constant k, then for every NP-verifier and every yes-instance
x of length n for that verifier, the verifier's search space has a
witness for x that can be encoded with a circuit...
I will discuss an approach to the subconvexity problem for
L-functions based on period integral formulae and amplification. In
particular, I will describe work in progress with Ruixiang Zhang
that attempts to prove a subconvex bound for the central...
We propose a new second-order method for geodesically convex
optimization on the natural hyperbolic metric over positive
definite matrices. We apply it to solve the operator scaling
problem in time polynomial in the input size and logarithmic in
the...
Complexity of the geometry, randomness of the potential, and
many other irregularities of the system can cause powerful, albeit
quite different, manifestations of localization: a phenomenon of
confinement of waves, or eigenfunctions, to a small...
