## Sirince school on mathematical and theoretical physics

27 August - 3 September, Nesin Mathematics Village, SIrince, Turkey

### Basic

Applied methods of mathematical analysis: complex analysis, special functions

D. Diakonov

Gamma function

Zeta function

Fourier transform, Laplace transform and Mellin transform.

Completeness and orthogonality relations, Wronskian

Field quantization in curved space-time

Applied methods of mathematical analysis: integrals, differential equations

K. Bazarov

1. Сomplex integrals

2. Steepest descent method

3. Airy function

4. Spherical harmonics

5. Legandre functions

Introduction to General Relativity: Black Holes and Wormholes

P. Anempodistov

1. Point particle in curved spacetime. Tensor analysis. Equations of motion for a particle in curved spacetime.

2. Riemannian geometry.

3. Einstein-Hilbert action. Einstein equations, their linearization.

4. Solutions of Einstein equations. Black holes. Motion of particles in black hole background.

5. Wormhole solutions.

Gauge theory and solitons

E. Musaev

Dirac monopole

Scalar field theory in 1+1 dimensions, kink

Abrikosov vortex

BPS solitons and supersymmetry

Braneworld models

### Advanced

Quantization methods

K. Gubarev

1. Setup of quantization problem. Canonical quantization.

2. Dirac approach to constrained systems.

3. Path integral quantization.

4. Deformation quantization.

5. Geometrical quantization.

Introduction to Supersymmetry

L. Astrakhantsev

1. Supersymmetry algebra

2. Superspace and Superfields I

3. Superspace and Superfields II

4. Wess-Zumino model

5. Sigma-model and Yang-Mills theories

Quantum effects in external fields

E. Akhmedov, A. Morozov

Klein effect

Schwinger effect from path integral instantons

Functional determinant in external electromagnetic field

4. Particle currents created by external fields (two lectures)

Student talks

V. Lapushkin, Quantum effects in external fields

V. Novoseltsev, Instantons in double well potentials