The main goal of this course is to present mathematical methods used in solid states physics. The main tools here are the $C^*$-algebras and $K$-theory. Each of these subjects is considered in a separate chapter of the course.

In the chapter, devoted to the theory of $C^*$-algebras, we pay special attention to graded and real $C^*$-algebras. An important role in the sequel is play by the $C^*$-modules and invariant traces. The theory of $C^*$-algebras is closely related to the theory of $C^*$-dynamic systems and crossed products. We also briefly recall basics of the theory of von Neumann algebras and $W^*$-dynamic systems.

The $K$-theory is a natural mathematical language for the description of solid state physics. In the chapter, devoted to this theory, we consider in detail the topological $K$-groups amd algebraic ones as well.

The topological invariants of solid bodies are defined on the base of the index theory of $\mathcal T$-Fredholm operators. The main invariants are given by the Chern cocecles defined in terls of cyclic cohomologies.

The developed mathematical methods are applied to the theory of solid states. Of special interest is the so called BB-correspondence connecting the topological invariants of the solid body with topological invariants of its boundary.

An important role in the solid state physics is played by graphene. This is a solid body with sexagonal crystal lattice having carbon atoms in its sites. Graphene and uts topological properties are considered in the kast chapter of the course.

PROGRAM

I. INTRODUCTION
II. BLOCH THEORY
2.1. One-particle Schroedinger operator
2.2. Physical interpretation
2.3. Fermion Fock space
2.4. Fermion Fock space of the solid body
III. $C^*$-ALGEBRAS
3.1. $C^*$-algebras. Basic definitions
3.2. Graded and real $C^*$-algebras
3.3. Clifford algebras
3.4. $C^*$-modules
3.5. $C^*$-dynamical systems and crossed products
3.6. $W^*$-dynamical systems
3.7. Invariant traces
IV. $K$-THEORY
4.1. $K_0$-group
4.2. Grothendieck construction
4.3. Algebraic $K_0$-group
4.4. $K_1$-group
4.5. Algebraic $K_1$-group
4.6. Higher $K$-groups
V. INDEC THEORY
5.1. Spaces of smooth elements
5.2. $\mathcal T$-Fredholm operators
5.3 Cyclic cohomologies
5.4. Chern cocycles and Toeplitz operators
5.5. Toeplitz extensions
VI. APPICATIONS TO THE SOLID STATE THEORY
6.1. Algebraic interpretation of the solid state theory
6.2. Halfspace and boundary algebras
6.3. Topological invariants of the solid state
6.4. Smooth $BB$-correspondence
VII. GRAPHENE
7.1. Structure of graphene
7.2. Topological invariants of graphene

Lecturer
Sergeev Armen Glebovich

Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, agreement no. 075-15-2025-303).

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center