International Undergraduate Mathematics Summer School

This mini-course gives an introduction to complex spaces and the resolution of singularities. We begin with the foundations of complex spaces, the Weierstrass preparation theorem, and the Oka coherence theorem, followed by Riemann's second extension theorem, the proper mapping theorem, Stein factorization, and Zariski's main theorem.

Building on these foundations, we formulate resolution of singularities and its applications to the elimination of points of indeterminacy and the Chow lemma. We then examine resolution in several concrete settings: curves via Newton's method, ADE surface singularities via double covers, and determinantal singularities. The course concludes with an introduction to foliations and the reduction of singularities in dimension two.

This course will focus on methods for working with functions of noncommuting operators. These methods find application in a wide range of areas of mathematical physics. For example, in the adiabatic approximation, the method of operator separation of variables can be used, which will be demonstrated in the Cauchy problem for a wave equation with rapidly oscillating velocity. Furthermore, problems of constructing asymptotic solutions in shallow water theory and in electromagnetic wave propagation problems will be considered.

This lecture series gives a first introduction to K-theory for C*-algebras. Starting from basic examples, we explain how projections and unitaries give rise to the groups K₀ and K₁. We then discuss the six-term exact sequence, one of the main computational tools in the subject, and illustrate it through concrete examples including the Toeplitz extension, Bott periodicity, UHF algebras, irrational rotation algebras, and Cuntz algebras.

The automorphism group of an affine algebraic variety is a complex and in many ways mysterious object that has captivated mathematicians for decades. Remarkably, even for the simplest varieties, the structure of this group is far from being completely understood.

In this three-lecture mini-course, we will discuss both classical results and the significant breakthroughs of the last two decades. We will begin by explaining why the automorphism group of an affine variety is, in general, not an algebraic group. We will then move on to the classical theorem of Jung–van der Kulk and the resulting distinction between tame and wild automorphisms. The central example here will be the famous Nagata automorphism. In the final part, we will examine classes of varieties for which the automorphism group can be described explicitly and formulate several open problems.

Coxeter groups are a special class of groups generated by involutions. They play important roles in various areas of mathematics. These lectures will focus in particular on how Coxeter groups can be used to construct interesting examples of discrete subgroups of Lie groups.

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