I define what it means for a quantum
field theory to be asymptotically safe and
discuss possible applications to theories
of gravity and matter.
This will be a discussion on Stochastic Parameterisation, led by Hannah.
I will explain how to embed arbitrary RAAGs (Right Angled
Artin Groups) in Ham (the group of hamiltonian symplectomorphisms of
the 2-sphere). The proof is combination of topology, geometry and
analysis: We will start with embeddings of RAAGs in the mapping class
groups of hyperbolic surfaces (topology), then will promote these
embeddings to faithful hamiltonian actions on the 2-sphere (hyperbolic
geometry and analysis).
A basic example of shear flow wasintroduced by DiPerna and Majda to study the weaklimit of oscillatory solutions of the Eulerequations of incompressible ideal fluids. Inparticular, they proved by means of this examplethat weak limit of solutions of Euler equationsmay, in some cases, fail to be a solution of Eulerequations. We use this shear flow example toprovide non-generic, yet nontrivial, examplesconcerning the immediate loss of smoothness andill-posedness of solutions of the three-dimensionalEuler equations, for initial data that do notbelong to $C^{1,\alpha}$. Moreover, we show bymeans of this shear flow example the existence ofweak solutions for the three-dimensional Eulerequations with vorticity that is having anontrivial density concentrated on non-smoothsurface. This is very different from what has beenproven for the two-dimensional Kelvin-Helmholtzproblem where a minimal regularity implies the realanalyticity of the interface. Eventually, we usethis shear flow to provide explicit examples ofnon-regular solutions of the three-dimensionalEuler equations that conserve the energy, an issuewhich is related to the Onsager conjecture.
This is a joint work with Claude Bardos.
The main aim of this talk will be to present a proof of the Tsen-Lang theorem on the existence of points on complete intersections and sketch a proof of the Grabber-Harris-Starr theorem giving the existence of a section to a fibration of a rationally connected variety over a curve. Time permitting, recent work of de Jong and Starr on rationally simply connected varieties will be discussed with applications to the number theory of hypersurfaces.
Surfaces of large genus are intriguing objects. Their geometry
has been studied by finding geometric properties that hold for all
surfaces of the same genus, and by finding families of surfaces with
unexpected or extreme geometric behavior. A classical example of this is
the size of systoles where on the one hand Gromov showed that there exists
a universal constant $C$ such that any (orientable) surface of genus $g$
with area normalized to $g$ has a homotopically non-trivial loop (a
systole) of length less than $C log(g)$. On the other hand, Buser and
Sarnak constructed a family of hyperbolic surfaces where the systole
roughly grows like $log(g)$. Another important example, in particular for
the study of hyperbolic surfaces and the related study of Teichmüller
spaces, is the study of short pants decompositions, first studied by Bers.
The talk will discuss two ideas on how to further the understanding of
surfaces of large genus. The first part will be about joint results with
F. Balacheff and S. Sabourau on upper bounds on the sums of lengths of
pants decompositions and related questions. In particular we investigate
how to find short pants decompositions on punctured spheres, and how to
find families of homologically independent short curves. The second part,
joint with L. Guth and R. Young, will be about how to construct surfaces
with large pants decompositions using random constructions.
Abstract: A random polytope $K_n$ is, by definition, the convex hull of $n$ random independent, uniform points from a convex body $K subset R^d$. The investigation of random polytopes started with Sylvester in 1864. Hundred years later R\'enyi and Sulanke began studying the expectation of various functionals of $K_n$, for instance number of vertices, volume, surface area, etc. Since then many papers have been devoted to deriving precise asymptotic formulae for the expectation of the volume of $K \setminus K_n$, for instance. But with few notable exceptions, very little has been known about the distribution of this functional. In the last couple of years, however, two breakthrough results have been proved: Van Vu has given tail estimates for the random variables in question, and M. Reitzner has obtained a central limit theorem in the case when $K$ is a smooth convex body. In this talk I will explain these new results and some of the subsequent development: upper and lower bounds for the variance, central limit theorems when $K$ is a polytope. Time permitting, I will indicate some connections lattice polytopes.
Abstract: We consider the inverse problem of finding the diffusion coefficient of a linear uniformly elliptic partial differential equation in divergence form, from noisy measurements of the forward solution in the interior. We adopt a Bayesian approach to the problem. We consider the prior measure on the diffusion coefficient to be either a Besov or Gaussian measure. We show that if the functions drawn from the prior are regular enough, the posterior measure is well-defined and Lipschitz continuous with respect to the data in the Hellinger metric. We also quantify the errors incurred by approximating the posterior measure in a finite dimensional space. This is joint work with Stephen Harris and Andrew Stuart.
There are nontrivial solutions of the incompressible Euler equations which are compactly supported in space and time. If they were to model the motion of a real fluid, we would see it suddenly start moving after staying at rest for a while, without any action by an external force. There are C1 isometric embeddings of a fixed flat rectangle in arbitrarily small balls of the three dimensional space. You should therefore be able to put a fairly large piece of paper in a pocket of your jacket without folding it or crumpling it. I will discuss the corresponding mathematical theorems, point out some surprising relations and give evidences that, maybe, they are not merely a mathematical game.
The static two price economy of conic finance is first employed to
define capital, profit, and subsequently return and leverage. Examples
illustrate how profits are negative on claims taking exposure to loss
and positive on claims taking gain exposure. It is argued that though
markets do not have preferences or objectives of their own, competitive
pressures lead markets to become capital minimizers or leverage
maximizers. Yet within a static context one observes that hedging
strategies must then depart from delta hedging and incorporate gamma
adjustments. Finally these ideas are generalized to a dynamic context
where for dynamic conic finance, the bid and ask price sequences are
seen as nonlinear expectation operators associated with the solution of
particular backward stochastic difference equations (BSDE) solved in
discrete time at particular tenors leading to tenor specific or
equivalently liquidity contingent pricing. The drivers of the associated
BSDEs are exhibited in complete detail.
I will present a decidability result for theories of large fields of algebraic numbers, for example certain subfields of the field of totally real algebraic numbers. This result has as special cases classical theorems of Jarden-Kiehne, Fried-Haran-Völklein, and Ershov.
The theories in question are axiomatized by Galois theoretic properties and geometric local-global principles, and I will point out the connections with the seminal work of Ax on the theory of finite fields.
Several stochastic simulation algorithms (SSAs) have been recently proposed for modelling reaction-diffusion processes in cellular and molecular biology. In this talk, two commonly used SSAs will be studied. The first SSA is an on-lattice model described by the reaction-diffusion master equation. The second SSA is an off-lattice model based on the simulation of Brownian motion of individual molecules and their reactive collisions. The connections between SSAs and the deterministic models (based on reaction-diffusion PDEs) will be presented. I will consider chemical reactions both at a surface and in the bulk. I will show how the "microscopic" parameters should be chosen to achieve the correct "macroscopic" reaction rate. This choice is found to depend on which SSA is used. I will also present multiscale algorithms which use models with a different level of detail in different parts of the computational domain.
We consider the iterative solution of large sparse linear least squares (LS) problems. Specifically, we focus on the design and implementation of reliable stopping criteria for the widely-used algorithm LSQR of Paige and Saunders. First we perform a backward perturbation analysis of the LS problem. We show why certain projections of the residual vector are good measures of convergence, and we propose stopping criteria that use these quantities. These projections are too expensive to compute to be used directly in practice. We show how to estimate them efficiently at every iteration of the algorithm LSQR. Our proposed stopping criteria can therefore be used in practice.
This talk is based on joint work with Xiao-Wen Chang, Chris Paige, Pavel Jiranek, and Serge Gratton.
I will first introduce and motivate the notion of 'homological stability' for a sequence of spaces and maps. I will then describe a method of proving homological stability for configuration spaces of n unordered points in a manifold as n goes to infinity (and applications of this to sequences of braid groups). This method also generalises to the situation where the configuration has some additional local data: a continuous parameter attached to each point.
However, the method says nothing about the case of adding global data to the configurations, and indeed such configuration spaces rarely do have homological stability. I will sketch a proof -- using an entirely different method -- which shows that in some cases, spaces of configurations with additional global data do have homological stability. Specifically, this holds for the simplest possible global datum for a configuration: an ordering of the points up to even permutations. As a corollary, for example, this proves homological stability for the sequence of alternating groups.
Many problems in computer science can be modelled as metric spaces, whereas for mathematicians they are more likely to appear as the opening question of a second year examination. However, recent interesting results on the geometry of finite metric spaces have led to a rethink of this position. I will describe some of the work done and some (hopefully) interesting and difficult open questions in the area.
We first present the localized virtual cycles by cosections of obstruction sheaves constructed by Kiem and Li. This construction has two kinds of applications: one is define invariants for non-proper moduli spaces; the other is to reduce the obstruction classes. We will present two recent applications of this construction: one is the Gromov-Witten invariants of stable maps with fields (joint work with Chang); the other is studying Donaldson-Thomas invariants of Calabi-Yau threefolds (joint work with Kiem).