Abstracts of the courses

The aim of these lectures is to provide a practical approach to algebra, analysis, geometry and probability. This also leads to a deeper understanding. To this end, we shall focus on various topics and stress their connections. Students who follow the lectures will be able to understand the practical aspects of the abstract theory learned in their undergraduate courses.

Further details here.

Practical Approach to Algebraic Methods for Applications

GHIMENTI Marco (University of Pisa): Introduction to the Lusternik Schnirelmann category and application to nonlinear elliptic PDE

The aim of this mini-course is to give the flavor of the so-called “Topological methods in nonlinear analysis", which combines geometric and algebraic tools with classical variational methods to obtain existence and multiplicity results for solutions of nonlinear elliptic PDE.

We will start with some preliminary notions to give the variational framework (Euler Lagrange equation, the energy functional, etc), and we will sketch the proof of a “Deformation Lemma”, which establishes a link between critical points of the energy functional and the change of topology of the sublevels of the functional. This lemma, paired with a compactness condition, is the base of all topological methods.

At this point we will focus on the definition and the principal properties of the Lusternik Schnirelmann category of a set. Roughly speaking a set A has LS category k if k is the minimum number of closed contractible sets which covers A.

The main point of this course is that, given a nonlinear elliptic PDE, the LS category of the sublevels of the associated energy functional provides a lower bound on the solutions of the PDE.

Some application of these methods to the stationary nonlinear schrodinger equation will be provided.

SHIBUKAWA Youichi (University of Hokkaido): Quantum Yang-Baxter equation and its variants

The quantum Yang-Baxter equation (QYBE for short) has been providing Mathematics with new useful concepts. The quantum group is one of the most famous examples. The main topic of this course is the QYBE, and we will focus on its set-theoretic solutions initiated by Drinfeld. In the construction of the solutions, we will make use of the group; in particular, the group with the unique factorization plays a central role. Through generalizing the construction of these solutions, I will introduce a variant of the QYBE, the QYBE on tensor categories.

Practical approach to analysis

KUBO Hideo (University of Hokkaido): On the theory of self-adjoint operators and its application

The topic of this course is on the theory of self-adjoint operators and its application to PDEs. You can learn basic concepts of the linear operators such as closed operators, adjoint operators, and self-adjoint operators. You learn how to extend symmetric operators to self-adjoint operators. You also learn the functional calculus of a self-adjoint operator. Furthermore, you learn the method of representing the solution to partial differential equations in terms of the functional calculus.

MOLLE Riccardo (Università di Roma Tor Vergata): An introduction to nonlinear analysis

I will first describe the variational approach to Partial Differential Equations (PDE). Then, I will introduce some methods typical of the variational approach to nonlinear problems, such as the minimization method (on suitable manifolds), the “Mountain Pass” Theorem, and other linking methods. Meanwhile, applications to PDEs arising from practical problems will be discussed.

Practical Approach to Differential Geometry and Topology

BUZANO Reto (University of Torino): Introduction to mean curvature flow with applications in the two-convex case

The aim of this course is to give an introduction to the mean curvature flow and explain some applications, in particular in the case of two-convex hypersurfaces. We aim at explaining everything from an intuitive viewpoint that does not assume prior familiarity with the topic and we prefer to illustrate the ideas and results with pictures, often foregoing analytical rigor.

We start with an informal discussion of the basics of mean curvature flow on hypersurfaces (explicit examples, evolution of geometric quantities, finite time singularities, parabolic rescaling, etc.). We then explain Huisken’s monotonicity formula and how it can be used to study singularity models of certain finite time singularities, before restricting to the two-convex case, where all local singularities are essentially modelled on shrinking cylinders.

Next, still assuming two-convexity, we sketch the idea of mean curvature flow with surgery and explain a gluing construction to topologically undo the surgeries again. We then finish the course by discussing applications of this surgery and gluing approach towards the study of path-components of the spaces of two-convex embeddings of spheres and tori into Euclidean space or into three-dimensional lens spaces.

KASUYA Naohiko (University of Hokkaido): Fiber bundles and Vector bundles

The basic idea of this course is to underline the relationship between various fields of mathematics, especially of geometry. You can learn basic concepts of fiber bundles and vector bundles. You learn transition functions, reduction of structure group, pullback bundles and monodromies. You also learn characteristic classes such as Euler class and Chern class from the viewpoint of the classifying spaces. Furthermore, you learn how to classify orientable circle bundles by their Euler classes, and orientable torus bundles over closed orientable surfaces by their monodromies and Euler classes.

Practical Approach to Probability and its Application

LANTERI Alessandro (University of Torino): Retrieve information from high dimensional data

Dealing with high dimensional data, we may face several problems related to the so-called curse of dimensionality phenomenon. A reduction of the data-complexity with the use of a low dimensional representation of the data is often required to obtain meaningful results. Such low dimensional representation can be obtained with robust methods which are implemented with fast and scalable algorithms.

TABATA Koji (University of Hokkaido): Introduction to Multi-Armed Bandit

This course provides a comprehensive understanding of multi-armed bandit problems in machine learning. Multi-armed bandits are a class of problems where an agent must choose between multiple options (arms) with uncertain rewards to maximize the total reward over time. Fundamental concepts, key algorithms such as ε-greedy, UCB, and Thompson Sampling, and the exploration-exploitation dilemma are covered. The course includes theoretical foundations and practical applications of multi-armed bandit algorithms.

Schedule of the talks: