14:00
4D Einstein equations as a gauge theory
Abstract
I will explain a new formulation of Einstein’s equations in 4-dimensions using the language of gauge theory. This was also discovered independently, and with advances, by Kirill Krasnov. I will discuss the advantages and disadvantages of this new point of view over the traditional "Einstein-Hilbert" description of Einstein manifolds. In particular, it leads to natural "sphere conjectures" and also suggests ways to find new Einstein 4-manifolds. I will describe some first steps in these directions. Time permitting, I will explain how this set-up can also be seen via 6-dimensional symplectic topology and the additional benefits that brings.
Global Properties of Supergravity Solutions
Abstract
Design principles and dynamics in clocks, cell cycles and signals
Abstract
I will discuss two topics. Firstly, coupling of the circadian clock and cell cycle in mammalian cells. Together with the labs of Franck Delaunay (Nice) and Bert van der Horst (Rotterdam) we have developed a pipeline involving experimental and mathematical tools that enables us to track through time the phase of the circadian clock and cell cycle in the same single cell and to extend this to whole lineages. We show that for mouse fibroblast cell cultures under natural conditions, the clock and cell cycle phase-lock in a 1:1 fashion. We show that certain perturbations knock this coupled system onto another periodic state, phase-locked but with a different winding number. We use this understanding to explain previous results. Thus our study unravels novel phase dynamics of 2 key mammalian biological oscillators. Secondly, I present a radical revision of the Nrf2 signalling system. Stress responsive signalling coordinated by Nrf2 provides an adaptive response for protection against toxic insults, oxidative stress and metabolic dysfunction. We discover that the system is an autonomous oscillator that regulates its target genes in a novel way.
Positive characteristic version of Ax's theorem
Abstract
Ax's theorem on the dimension of the intersection of an algebraic subvariety and a formal subgroup (Theorem 1F in "Some topics in differential algebraic geometry I...") implies Schanuel type transcendence results for a vast class of formal maps (including exp on a semi-abelian variety). Ax stated and proved this theorem in the characteristic 0 case, but the statement is meaningful for arbitrary characteristic and still implies positive characteristic transcendence results. I will discuss my work on positive characteristic version of Ax's theorem.
D-modules: PDEs, flat connections, and crystals
Abstract
Motivated by the study of PDEs, we introduce the notion of a D-module on a variety X and give the basics of three perspectives on the theory: modules over the sheaf of differential operators on X; quasi-coherent modules with flat connection; and crystals on X. This talk will assume basic knowledge of algebraic geometry (such as rudimentary sheaf theory).
Coherent Lagrangian vortices: The black holes of turbulence
Abstract
We discuss a simple variational principle for coherent material vortices
in two-dimensional turbulence. Vortex boundaries are sought as closed
stationary curves of the averaged Lagrangian strain. We find that
solutions to this problem are mathematically equivalent to photon spheres
around black holes in cosmology. The fluidic photon spheres satisfy
explicit differential equations whose outermost limit cycles are optimal
Lagrangian vortex boundaries. As an application, we uncover super-coherent
material eddies in the South Atlantic, which yield specific Lagrangian
transport estimates for Agulhas rings. We also describe briefly coherent
Lagrangian vortex detection to three-dimensional flows.
Cluster combinatorics and geometrical models (part I)
Abstract
In this talk I will give a definition of cluster algebra and state some main results.
Moreover, I will explain how the combinatorics of certain cluster algebras can be modeled in geometric terms.
Don't be afraid of the 1001st (numerical) derivative
Abstract
The accurate and stable numerical calculation of higher-order
derivatives of holomorphic functions (as required, e.g., in random matrix
theory to extract probabilities from a generating function) turns out to
be a surprisingly rich topic: there are connections to asymptotic analysis,
the theory of entire functions, and to algorithmic graph theory.
see below
Abstract
\textbf{James Newbury} \newline
Title: Heavy traffic diffusion approximation of the limit order book in a one-sided reduced-form model. \newline
Abstract: Motivated by a zero-intelligence approach, we try to capture the
dynamics of the best bid (or best ask) queue in a heavy traffic setting,
i.e when orders and cancellations are submitted at very high frequency.
We first prove the weak convergence of the discrete-space best bid/ask
queue to a jump-diffusion process. We then identify the limiting process
as a regenerative elastic Brownian motion with drift and random jumps to
the origin.
\newline
\textbf{Zhaoxu Hou} \newline
Title: Robust Framework In Finance: Martingale Optimal Transport and
Robust Hedging For Multiple Marginals In Continuous Time
\newline
Abstract: It is proved by Dolinsky and Soner that there is no duality
gap between the robust hedging of path-dependent European Options and a
martingale optimal problem for one marginal case. Motivated by their
work and Mykland's idea of adding a prediction set of paths (i.e.
super-replication of a contingent claim only required for paths falling
in the prediction set), we try to achieve the same type of duality
result in the setting of multiple marginals and a path constraint.
Moving interface problems in multi-D compressible Euler equations
16:00
Learning spaces
Abstract
Working together with the Blue Brain Project at the EPFL, I'm trying to develop new topological methods for neural modelling. As a mathematician, however, I'm really motivated by how these questions in neuroscience can inspire new mathematics. I will introduce new work that I am doing, together with Kathryn Hess and Ran Levi, on brain plasticity and learning processes, and discuss some of the topological and geometric features that are appearing in our investigations.
11:30
Straight edge and compass to Origami
Abstract
I will look at the classical constructions that can be made using a straight edge and compass, I will then look at the limits of these constructions. I will then show how much further we can get with Origami, explaining how it is possible to trisect an angle or double a cube. Compasses not supplied.
Some thoughts on the Burnside problem
Abstract
Thoughts on the Burnside problem
Quasimaps, wall-crossings, and Mirror Symmetry II
Abstract
Quasimaps provide compactifications, depending on a stability parameter epsilon, for moduli spaces of maps from nonsingular algebraic curves to a large class of GIT quotients. These compactifications enjoy good properties and in particular they carry virtual fundamental classes. As the parameter epsilon varies, the resulting invariants are related by wall-crossing formulas. I will present some of these formulas in genus zero, and will explain why they can be viewed as generalizations (in several directions) of Givental's toric mirror theorems. I will also describe extensions of wall-crossing to higher genus, and (time permitting) to orbifold GIT targets as well.
The talk is based on joint works with Bumsig Kim, and partly also with Daewoong Cheong and with Davesh Maulik.
Structure exploitation in Hessian computations
Abstract
Hessians of functionals of PDE solutions have important applications in PDE-constrained optimisation (Newton methods) and uncertainty quantification (for accelerating high-dimensional Bayesian inference). With current techniques, a typical cost for one Hessian-vector product is 4-11 times the cost of the forward PDE solve: such high costs generally make their use in large-scale computations infeasible, as a Hessian solve or eigendecomposition would have costs of hundreds of PDE solves.
In this talk, we demonstrate that it is possible to exploit the common structure of the adjoint, tangent linear and second-order adjoint equations to greatly accelerate the computation of Hessian-vector products, by trading a large amount of computation for a large amount of storage. In some cases of practical interest, the cost of a Hessian-
vector product is reduced to a small fraction of the forward solve, making it feasible to employ sophisticated algorithms which depend on them.
Hypergraph matchings
Abstract
Perfect matchings are fundamental objects of study in graph theory. There is a substantial classical theory, which cannot be directly generalised to hypergraphs unless P=NP, as it is NP-complete to determine whether a hypergraph has a perfect matching. On the other hand, the generalisation to hypergraphs is well-motivated, as many important problems can be recast in this framework, such as Ryser's conjecture on transversals in latin squares and the Erdos-Hanani conjecture on the existence of designs. We will discuss a characterisation of the perfect matching problem for uniform hypergraphs that satisfy certain density conditions (joint work with Richard Mycroft), and a polynomial time algorithm for determining whether such hypergraphs have a perfect matching (joint work with Fiachra Knox and Richard Mycroft).
14:15
Coupling across scales in geophysical flows
Quantitative sparse signal recovery guarantees of nonconvex nonsmooth first-order methods
Abstract
Finding a sparse signal solution of an underdetermined linear system of measurements is commonly solved in compressed sensing by convexly relaxing the sparsity requirement with the help of the l1 norm. Here, we tackle instead the original nonsmooth nonconvex l0-problem formulation using projected gradient methods. Our interest is motivated by a recent surprising numerical find that despite the perceived global optimization challenge of the l0-formulation, these simple local methods when applied to it can be as effective as first-order methods for the convex l1-problem in terms of the degree of sparsity they can recover from similar levels of undersampled measurements. We attempt here to give an analytical justification in the language of asymptotic phase transitions for this observed behaviour when Gaussian measurement matrices are employed. Our approach involves novel convergence techniques that analyse the fixed points of the algorithm and an asymptotic probabilistic analysis of the convergence conditions that derives asymptotic bounds on the extreme singular values of combinatorially many submatrices of the Gaussian measurement matrix under matrix-signal independence assumptions.
This work is joint with Andrew Thompson (Duke University, USA).
Quasimaps, wall-crossings, and Mirror Symmetry I
Abstract
Quasimaps provide compactifications, depending on a stability parameter epsilon, for moduli spaces of maps from nonsingular algebraic curves to a large class of GIT quotients. These compactifications enjoy good properties and in particular they carry virtual fundamental classes. As the parameter epsilon varies, the resulting invariants are related by wall-crossing formulas. I will present some of these formulas in genus zero, and will explain why they can be viewed as generalizations (in several directions) of Givental's toric mirror theorems. I will also describe extensions of wall-crossing to higher genus, and (time permitting) to orbifold GIT targets as well.
The talk is based on joint works with Bumsig Kim, and partly also with Daewoong Cheong and with Davesh Maulik.
Mixed Motives in Number Theory
Abstract
Mixed motives turn up in number theory in various guises. Rather than discuss the rather deep foundational questions involved, this talk will aim
to give several illustrations of the ubiquity of mixed motives and their realizations. Along the way I hope to mention some of: the Mordell-Weil
theorem, the theory of height pairings, special values of L-functions, the Mahler measure of a polynomial, Galois deformations and the motivic
fundamental group.