Week 09.10.2023 – 15.10.2023

second lecture by Prof Leonid A. Pastur

See https://dsadvancedlectures.weebly.com/

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.

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).

Mean Curvature Flow, the negative gradient flow for the volume functional of submanifolds of Riemannian manifolds, is a well-studied field of modern geometric analysis. Of particular interest are classifications of self-similar solutions (shrinkers, expanders, and translators) and finite-time singularities\DSEMIC projects which when completed will hopefully allow one to apply the flow to prove results in Riemannian geometry and differential topology. Moreover, in a Calabi-Yau manifold the class of Lagrangian submanifolds is preserved by mean curvature flow, a fact which inspired Thomas and Yau to make influential conjectures about existence of special Lagrangians in Calabi-Yau manifolds.

In this talk, we aim to make progress towards an understanding of self-similar solutions and singularities of Lagrangian mean curvature flow, by focusing on Lagrangians in C^n that are cohomogeneity-one under the action of a compact Lie group. Interestingly, each such Lagrangian lies in a level set \mu^{-1}(c) of the moment map \mu, and mean curvature flow preserves this containment. Using this, we classify all shrinking, expanding, and translating solitons, and in the zero level set \mu^{-1}(0), we classify the Type I and Type II blowup models of LMCF singularities. Finally, given any special Lagrangian in \mu^{-1}(0), we’ll show that it arises as a Type II blowup, thereby yielding infinitely many new singularity models of Lagrangian mean curvature flow.

The results presented in this talk are contained in the preprint ‘Cohomogeneity-One Lagrangian Mean Curvature Flow’, which is jointly written with Jesse Madnick, University of Oregon.

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.

Plants undergo several key developmental transitions, such as the decision to flower, that farmers would like to synchronise to maximise their yields. In this talk I will describe (i) a novel experimental design to understand how these transitions happen and (ii) a novel application of functional data analysis to help farmers breed more synchronised crops. To understand the biological regulation that leads to these transition points, a high temporal resolution of sampling would be required\DSEMIC however, the degree of developmental asynchrony makes such an experiment difficult to design. Instead, we sample a large collection of individual plants at the transition point and then estimate their age retroactively with a bootstrapping strategy, enabling us to order the plants along a pseudotime, giving us an unprecedented level of detail of the cascade of biological events that lead to the initiation of flowering. We then hypothesised that plants that are more sensitive to changes in day length (as occur in the spring and autumn) would have more synchronised development. Using functional data analysis approaches, we developed a predictive model of flowering synchrony on the basis of how the circadian rhythms of plants respond to changes in day length, in a population of plants with parents adapted from different latitudes. We are further adapting FDA methods to identify genetic loci that are significantly associated with these clock-related traits, which can be used to direct crop breeding for synchronised development.