Abstracts

Stochastic Processes and Martingales by Mehmet Öz (Özyeğin University)

This class will serve as an introduction to modern probability theory and stochastic processes through martingales.

Ito Calculus and Stochastic Differential Equations by Mine Çağlar (Koç University)

Brownian motion, Itô’s calculus, stochastic differential equations will be introduced and  Itô’s lemma and existence and uniqueness of the solutions will be discussed.

Partial Differential Equations by Havva Yoldaş (TU Delft)

3 main linear partial differential equations will be introduced. Existence and regularity theories for elliptic and parabolic equations will be discussed. As time permits an introduction to kinetic theory and transport equations will be made.

Stochastic Partial Differential Equations by Avi Mayorcas (University of Bath)

Many important physical, economic and biological phenomena are well described by PDE models of observed quantities, such as temperature, velocity, angular momentum, wealth, population density etc. However, in many cases, exact PDE models provide less information than statistical models, against which one can calibrate statistics of the same observed quantities; e.g. averages, variances and higher moments. Mathematically, these statistical models are given by stochastic perturbations of PDE i.e. stochastic PDE (SPDE). The purpose of this course is to give students an insight into the various analytic tools which can be applied to obtain basic solution theories for a large class of these equations. The main focus will be on the so-called variational formulation, which is of particular relevance as it is well-suited to deriving numerical methods, for example those based on Galerkin or neural network approximations. The course assumes some basic background in functional analysis (familiarity with infinite dimensional Banach and Hilbert spaces) and some basics of probability theory. The course will aim to give sufficient additional background, from this base, to understand the solution theory for a basic class of SPDE as well as give insight into the tools necessary to solve more complex examples.

Machine Learning by Umut Şimşekli (INRIA & École Normale Supérieure)

Many problems in machine learning can be seen as obtaining point estimates (e.g., maximum likelihood). While point estimates have proven very useful, unfortunately, they do not convey any uncertainty information, which can be crucial for risk-intolerant application domains (such as self-driving vehicles). In contrast, Markov Chain Monte Carlo (MCMC) methods are indeed able to provide uncertainty estimates along with point estimates. In this short course, we will cover a specific instance of MCMC algorithms, called "Langevin Monte Carlo" (LMC). The LMC algorithm is built on the Langevin diffusion, a particular stochastic differential equation (SDE) that has been widely used in a broad range of mathematical branches. The particularity of LMC is that it is able to scale up to modern, large-scale machine learning problems, and hence has recently attracted a significant amount of interest in both engineering domains as well as applied probability. The course will be based on the following content:

-- Formalization of the sampling problem and its link to uncertainty estimation.

-- Development of the Langevin SDE for solving the sampling problem.

-- The connections between the Langevin SDE and its associated Fokker-Planck Equation, that is a linear PDE.

-- Development of the LMC algorithm as a time-discretization of the Langevin SDE and its error analysis.

Malliavin Calculus by Soukaina Doussisi (Cadi Ayyad University)

Malliavin calculus, also known as the stochastic calculus of variations, is an infinite-dimensional differential calculus on the Wiener space whose operators act on functional of general Gaussian processes. First initiated by Paul Malliavin in 1976 it was further developed by Stroock, Bismut, Watanabe, and others. In the first part of this course, the students will see some elements of the analysis on Wiener space: Gaussian isonormal processes, Wiener chaos, multiple Wiener integrals and their properties. We will define the basic differential operators and discuss the relationship between them. In the second part of this course, we will see how Stein’s method combined with Malliavin Calculus has permitted to prove CLTs along with their rates of convergence for sequences of functionals of Gaussian fields, we will see some examples from statistical inference for SDEs and SPDEs. This course requires some background in functional analysis and probability theory.

Colloquium talk by Ali Süleyman Üstünel (Bilkent University)

Abstract and Title: TBA