It is expected that applicants to the program should be able to demonstrate a knowledge of the following topics:
- Basics of combinatorics (combinations, permutations) and
probability theory (independence, conditional probability).
Elements of group theory:
groups, subgroups, cosets, homomorphisms, quotient groups, structure of finitely generated abelian groups, examples of groups including symmetric, alternating groups, groups of symmetries, matrix groups, deduction groups.
vector spaces and linear maps, basis, dimension, systems of linear equations, Jordan normal form, characteristic and minimal polynomials, quadratic form, positivity.
open and closed subsets in Rn, compactness, connectedness, interior and closure, everywhere dense set and nowhere dense set, continuous maps, uniform continuity, uniform convergence. Intermediate value theorem.
Limits of sequences and functions, series
and their convergencе.
Derivatives and differentials, derivative of complicated function, Taylor series, methods for finding extrema of functions, Lagrange multipliers.
Riemann and Lebesgue integral, Lebesgue measure, Fubini’s theorem. Calculation of lengths and areas using definite integrals.
affine and projective spaces, affine and projective maps, second-order curves (conics).
complex derivative, holomorphic functions, Cauchy integral, residue theorem, Schwarz lemma.
Ordinary differential equations:
existence and uniqueness theorem, separation of variables solution, first and second order linear equations, homogeneous equations. First order partial differential equations: method of characteristic.
Newton laws, principle of minimal action, Euler-Lagrange equations, conservation laws, relativistic mechanics.
- Classical electrodynamics:
vector potential and its curvature, Maxwell equations, electromagnetic stress–energy tensor, charged particles in constant electric and magnetic fields.