Questionary
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).
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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.
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Linear algebra:
vector spaces and linear maps, basis, dimension, systems of linear equations, Jordan normal form, characteristic and minimal polynomials, quadratic form, positivity.
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Topology:
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.
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Limits of sequences and functions, series
and their convergencе.
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Differential calculus:
Derivatives and differentials, derivative of complicated function, Taylor series, methods for finding extrema of functions, Lagrange multipliers.
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Integral:
Riemann and Lebesgue integral, Lebesgue measure, Fubini’s theorem. Calculation of lengths and areas using definite integrals.
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Geometry:
affine and projective spaces, affine and projective maps, second-order curves (conics).
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Complex analysis:
complex derivative, holomorphic functions, Cauchy integral, residue theorem, Schwarz lemma.
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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.
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Classical mechanics:
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.