We study solutions of the irregular Stratonovich SDE $dX = X|^\alpha \circ dB$, $\alpha\in (0, 1)$. In particular we construct solutions spending positive time in 0, describe solutions spending zero time in 0, and show how a particular physically natural solution can be singled out by means of an additional external "ambient" noise.

This talk is based on the joint works with G. Shevchenko (Kiev).

The Random Walk Metropolis (RWM) is a simple and enduring Markov chain-based algorithm for approximate simulation from an intractable ���target��� probability distribution. In this work, we study quantitatively the convergence of this algorithm, providing non-asymptotic estimates on mixing times, with explicit dependence on dimension and other problem parameters. The results hold at a reasonable level of generality, and are often sharp in a suitable sense.

The focus of the talk will be conceptual rather than technical, with an eye towards enabling intuition for i) which high-level aspects of the target distribution influence the convergence behaviour of RWM, and ii) which concrete properties must be verified in order to obtain a rigorous proof. No prior knowledge of the RWM is required from the audience.

second lecture by Prof Leonid A. Pastur

See https://dsadvancedlectures.weebly.com/

The non-minimal coupling of scalar fields to gravity has been claimed to violate energy conditions,
leading to exotic phenomena such as traversable wormholes, even in classical theories. In this work
we adopt the view that the non-minimal coupling can be viewed as part of an effective field
theory (EFT) in which the field value is controlled by the theory���s cutoff. Under this assumption,
the average null energy condition, whose violation is necessary to allow traversable wormholes, is
obeyed both classically and in the context of quantum field theory. In addition, we establish a type
of ���smeared��� null energy condition in the non-minimally coupled theory, showing that the null
energy averaged over a region of spacetime obeys a state dependent bound, in that it depends on
the allowed field range. We finally motivate our EFT assumption by considering when the gravity
plus matter path integral remains semi-classically controlled.

In the Euclidean space setting, symmetrization inequalities is a classical theory that has been quite useful in solving problems coming from various parts of analysis: spectral geometry, variational problems, mathematical physics, spectral theory, to name a few. In my talk, I will discuss a possible extensions of this theory to the setting of graphs. It is a fairly new topic and most of the results in the area are proved in the last two years. I will talk about these developments, connections of this theory with discrete isoperimetric inequalities, and its possible applications to problems concerning 'analysis on graphs'. The talk will be at the interface of discrete math and analysis, and will be based on a joint work with Stefan Steinerberger.

We consider the renormalization group flow of a quantum field theory (QFT) in Anti-de Sitter (AdS) space. We derive sum rules that express UV data and the energy of a chosen eigenstate in terms of the spectral densities and certain correlation functions of the theory. In two dimensions, this leads to a bootstrap setup that involves the UV central charge and may allow us to follow a Renormalization Group (RG) flow non-perturbatively by continuously varying the AdS radius. Along the way, we establish the convergence properties of the newly discovered local block decomposition, which applies to three-point functions involving one bulk and two boundary operators.

We will review some general concepts of deformation theory. Then we will apply these ideas in order to explore the geometry of the moduli space Inv of foliations on a given variety $X$ around the points corresponding to foliations induced by Lie group actions. More precisely, let $X$ be a smooth projective variety over the complex numbers and $S(d)$ the scheme parametrizing $d$-dimensional Lie subalgebras of $H^0(X,\mathcal{T} X)$. For every $\mathfrak{g} \in S(d)$ one can consider the corresponding element $\mathcal{F}(\mathfrak{g})\in Inv$, whose generic leaf coincides with an orbit of the action of $\exp(\mathfrak{g})$ on $X$. We will show that under mild hypotheses, after taking a stratification $\coprod_i S(d)_i\to S(d)$ this assignment yields an isomorphism $\coprod_i S(d)_i\to Inv$ locally around $\mathfrak{g}$ and $\mathcal{F}(\mathfrak{g})$.

Cylindrical L��vy processes are a natural extension of cylindrical Brownian motion which has been the standard model of random perturbations of partial differential equations and other models in infinite dimensions for the last 50 years. Here, the attribute cylindrical refers to the fact that cylindrical Brownian motions are not classical stochastic processes attaining values in the underlying space but are generalised objects. The reasons for the choice of cylindrical but not classical Brownian motion can be found in the facts that there does not exist a classical Brownian motion with independent components in an infinite dimensional Hilbert space, and that cylindrical processes enable a very flexible modelling of random noise in time and space.

This talk is a very introductory presentation to cylindrical L��vy processes. We explain the difficulty to define random noises in infinite dimensions and explain the approach by cylindrical measures and cylindrical random variables, which are strongly related to other areas such as harmonic analysis and operator theory. We present some specific examples of cylindrical L��vy processes in detail and discuss their relations to other models of random perturbations in the literature. We explain how a theory of stochastic integration for cylindrical L��vy processes can be developed although standard approaches to stochastic integration cannot be applied, and how this theory can be used to derive a theory of stochastic partial differential equations driven by Cylindrical L��vy processes.