I will talk about Jochen’s theorem about the existence of some non-trivial Henselian valuation given by investigating the absolute Galois group.
The size Ramsey number r'(H) of a graph H is the smallest number of edges in a graph G which is Ramsey with respect to H, that is, such that any 2-colouring of the edges of G contains a monochromatic copy of H. A famous result of Beck states that the size Ramsey number of the path with n vertices is at most bn for some fixed constant b > 0. An extension of this result to graphs of maximum degree ∆ was recently given by Kohayakawa, Rödl, Schacht and Szemerédi, who showed that there is a constant b > 0 depending only on ∆ such that if H is a graph with n vertices and maximum degree ∆ then r'(H) < bn^{2 - 1/∆} (log n)^{1/∆}. On the other hand, the only known lower-bound on the size Ramsey numbers of bounded-degree graphs is of order n (log n)^c for some constant c > 0, due to Rödl and Szemerédi.
Together with David Conlon, we make a small step towards improving the upper bound. In particular, we show that if H is a ∆-bounded-degree triangle-free graph then r'(H) < s(∆) n^{2 - 1/(∆ - 1/2)} polylog n. In this talk we discuss why 1/∆ is the natural "barrier" in the exponent and how we go around it, why we need the triangle-free condition and what are the limits of our approach.
In this talk I will present some recent work on the amplituhedron formulation of scattering amplitudes. Very recently it has been conjectured that amplitudes in planar N=4 sYM are nothing else but the volume of a completely new mathematical object, called amplituhedron, which generalises the positive Grassmannian. After a review of the main ingredients which will be used, I will discuss some of the questions which remain open in this framework. I will then describe a new direction which promises to solve these issues and compute the volume of the amplituhedron at tree level.
The Ginzburg-Landau functional serves as a model for the formation of vortices in many physical contexts. The natural gradient flow, the parabolic Ginzburg-Landau equation, converges in the limit of small vortex size and finite number of vortices to a system of ODEs. Passing to the limit of many vortices in this ODE, one can derive a mean field PDE, similar to the passage from point vortex systems to the 2D Euler equations. In the talk, I will present quantitative estimates that allow us to directly connect the parabolic GL equation to the limiting mean field PDE. In contrast to recent work by Serfaty, our work is restricted to a fairly low number of vortices, but can handle vortex sheet initial data in bounded domains. This is joint work with Daniel Spirn (University of Minnesota).
In this talk we want to present a detailed study of the algebraic objects appearing in the theory of regularity structures. In particular we aim at introducing a class of co-algebras on labelled forests and trees and show that these allow to describe in an unified setting the structure group and the renormalisation group. Based on joint work with Yvain Bruned and Martin Hairer
We aim to study the long time behaviour of the solution to a rough differential equation (in the sense of Lyons) driven by a random rough path. To do so, we use the theory of random dynamical systems. In a first step, we show that rough differential equations naturally induce random dynamical systems, provided the driving rough path has stationary increments. If the equation satisfies a strong form of stability, we can show that the solution admits an invariant measure.
This is joint work with I. Bailleul (Rennes) and M. Scheutzow (Berlin).
For the programme see
Computing distinct solutions of differential equations -- Patrick Farrell
Abstract: TBA
Triangles and equations -- Yufei Zhao
Abstract: I will explain how tools in graph theory can be useful for understanding certain problems in additive combinatorics, in particular the existence of arithmetic progressions in sets of integers.
I will discuss how \zeta(3) occurs in quantum corrections to the Einstein action, and how this causes \zeta(3) to be seen in the moduli space of CY manifolds. I will also draw attention to the fact that the dependence of the moduli space on \zeta(3) has a p-adic analogue.
Topological Quantum Field Theories are functors from a category of bordisms of manifolds to (usually) some categorification of the notion of vector spaces. In this talk we will first discuss why mathematicians are interested in these in general and an overview of the relevant notions. After this we will have a closer look at the example of functors from the bordism category of 1-, 2- and 3-dimensional manifolds equipped with principal G-bundles, for G a finite group, to nice categorifications of vector spaces.
I will describe some diophantine problems and results motivated by the analogy between powers of the modular curve and powers of the multiplicative group in the context of the Zilber-Pink conjecture.
In this talk, we present a pathwise method to construct confidence
intervals on the value of some discrete time stochastic dynamic
programming equations, which arise, e.g., in nonlinear option pricing
problems such as credit value adjustment and pricing under model
uncertainty. Our method generalizes the primal-dual approach, which is
popular and well-studied for Bermudan option pricing problems. In a
nutshell, the idea is to derive a maximization problem and a
minimization problem such that the value processes of both problems
coincide with the solution of the dynamic program and such that
optimizers can be represented in terms of the solution of the dynamic
program. Applying an approximate solution to the dynamic program, which
can be precomputed by any algorithm, then leads to `close-to-optimal'
controls for these optimization problems and to `tight' lower and upper
bounds for the value of the dynamic program, provided that the algorithm
for constructing the approximate solution was `successful'. We
illustrate the method numerically in the context of credit value
adjustment and pricing under uncertain volatility.
The talk is based on joint work with C. Gärtner, N. Schweizer, and J.
Zhuo.
While there have been recent advances for analyzing the complex deterministic
behavior of systems with discontinuous dynamics, there are many open questions about
understanding and predicting noise-driven and noise-sensitive phenomena in the
non-smooth context. Stochastic effects can often change the picture dramatically,
particularly if multiple time scales are present. We demonstrate novel approaches
for exploring and explaining surprising phenomena driven by the interplay of
nonlinearities, delays, randomness, in specific applications with piecewise smooth
dynamics - nonlinear models of balance, relay control, and impacting dynamics.
Effective techniques typically depend on the combination of mathematical techniques,
multiple scales techniques, and phenomenological intuition from seemingly unrelated
canonical models of biophysics, mechanics, and chemical dynamics. The appropriate
strategy is not always immediately obvious from the area of application or model
type. This gap may follow from the limited attention that stochastic models with
discontinuous dynamics have received in the past, or it may be the reason for this
limited attention. Combining the geometrical perspective with asymptotic approaches
in physical and phase space appears to be a critical part of developing effective
approaches.
Quadrature is the term for the numerical evaluation of integrals. It's a beautiful subject because it's so accessible, yet full of conceptual surprises and challenges. This talk will review ten of these, with plenty of history and numerical demonstrations. Some are old if not well known, some are new, and two are subjects of my current research.
For a positive measure set of Klein-Gordon masses mu^2 > 0, we construct one-parameter families of solutions to the Einstein-Klein-Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-periodic. An immediate corollary is that for these Klein-Gordon masses, the Kerr family is not asymptotically stable as a solution to the Einstein-Klein-Gordon equations. This is joint work with Otis Chodosh.