1 semester

1

Name of course

Modern problems of mathematics

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Year

1, speciality Mathematics

3

Semester

1

4

Credits

4

5

Lecturer’s scientific and academic degree, First Name, Patronic Name, Surname

Doctor of Sciences (habilitated), Professor,

Veniyamin Ruhoravič Krotaū

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Goals of course

Acquaintance of undergraduate students with some important achievements of modern mathematics and their new methods and results.

At the end of the course a student will be able:

  • to use main results of mathematics in his scientific activity;
  • to be familiar with modern mathematical literature,
  • to apply mathematical knowledge in practice.

7

Background

Mathematical analysis, algebra and numbers theory, differential geometry and topology, differential equations, probability theory and mathematical statistics, functional analysis, mathematical physics equations, numerical analysis.

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Contents

Ideals on polynomial rings. Monomial ideals and Dickson’s lemma. Hilbert basis theorem. Hröbner’s bases. Concept of affine varieties. Ideals and affine varieties. Radical of an ideal.  Hilbert’s zeros theorem. Irreducible varieties and prime ideals. Decomposition onto irreducible components. Ring of formal power series. Noetherian of the ring of power series. Free abelian groups. Theorem about coherent bases. Structural theorem of finitely generated abelian groups.

          Algebraic manifolds. Bi-rational invariants. Schema, group manifolds. Homotopy theory and typical invariants of topology spaces. Classification of topological manifolds. Homogeneous spaces of Lee groups and invariant structures on.     Analytic continuation. Riemann’s theorem of conformal maps. Schwartz-Christoffel formula. Entire and meromorphic.

          Basic definitions and concepts. Basic theorems for stability of steady moving. Principal problems of a  local theory of differential equations. Bendixon and Poincare spheres. Investigation of behavior of trajectories in whole.  Centre focus and isochronisity problems and their methods of solving. Cauchy theorem of existence and uniqueness of a solution of an initial value problem for a system of differential equations in a complex domain. Characterization and classification of singularities of solutions. Movable and no movable singularities. Painlevé property. Characterization of solutions of linear systems in a neighborhood of a singularities. Regular or irregular singularities. Fuchs class equations. Poicaré problem. Monodromie. Riemann problem. Deformation of linear systems. Schlessinger’s equation and isomonodromic deformation. Nonlinear equations of isomonodromic deformation. Painlevé equations.

          Mixing problem for a string oscillation equation with time-dependant boundary conditions. Formulation of the problem. Construction of a solution of a mixed problem. Solution analysis and some generalizations. Ill-posed problems for heating equation and their regularizations. Formulation of problems. Quasi-inverse method in a initial condition control problem. Regularization by conditions in a initial condition control problem. Inverse problems conceptions.

          Definition of a test functions space and a space of distributions. Regular and singular distributions. Operations with distributions. Differentiations of distributions. Multiplication of distributions onto infinitely differentiable function. Application of distributions theory to mathematical physics. Solving of differential equations by distribution function theory.

          -theory. Definition of Fourier transform, its relation with analysis operations. Lebesgue-Riemann theorem. Approximative identities. Gauss Weierstrass and Poisson kernels and its Fourier integrals. Approximative properties. Semi-group property. -theory. Fourier transform continuation onto . Plancherel theorem. Inverse formula. Relations with convolutions. -inequalities. Distributions of slow growth. Fourier transform on  and its properties. Operators commuting with translations. Multiplicators.

          Stochastic processes with independent  increments. Correlation theory for stochastic processes. A concept of generalized stochastic processes. Elements of Stochastic analysis.

          Old and modern concepts for numeric methods of solving of differential equations in partial derivatives. Compact difference schemes of a spectrum resolution. Iteration method for solving of big systems of linear equations with sparse matrices. Multi-grids methods. Matrix exponent decomposition and splitting methods.

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Bibliography

Algebra

  1. Cox D., Little J., O’Shea D. Ideals, varieties, and algorithms 2ed., Springer, 1997.
  2. Prasolov V.V. Polynomials, Springer, 2004.
  3. Zariski O., Samuel P. Commutative algebra. Vol.2 Van Nostrand, 1960.

Geometry

  1. Шафаревич И.Р. Основы алгебраической геометрии, тома 1 и 2 М.: «Наука»,1968.
  2. Hartshorne R. Algebraic geometry, Springer, 1997.
  3. Новиков С.П. Топология, М.: «Наука»,1989.
  4. Постников М.М. Лекции по алгебраической топологии, М.: «Наука»,1986.
  5. Hu S.-T. Homotopy theory, 5pr., AP, 1971.
  6. Daverman R.J. Decompositions of manifolds, «Springer», 1993.
  7. Матвеев С.В. Классификация 3-мерных многообразий, М.: «МЦНМО», 2007.
  8. Kobayashi S., Nomizu K., Foundations of differential geometry, Vol.1. Vol.2, Wiley, 1996.
  9. Warner F.W. Foundations of differentiable manifolds and Lie groups, Springer, 1983.

Complex analysis

  1. Сидоров Ю.В., Федорюк М.Ф., ШабунинМ.И. Лекции по ТФКП. М.,Наука, 1989.
  2. Маркушевич А.И. Теория аналитических функицй. Т. 1, 2. М.: Наука, 1977.
  3. Лаврентьев М.А., Шабат Б.В. Методы теории функций комплексного переменного. М.: Наука, 1973.
  4. Зверович Э.И. Вещественный и комплексный анализ, Ч. 6. Минск, Вышейшая школа, 2008.
  5. Bak J., Newman D.J. Complex analysis, Springer, 2010.

Differential equations

  1. Бибиков Ю.Н. Курс обыкновенных дифференциальных уравнений. М.: Высшая школа, 1991.
  2. Андронов А.А., Леонтович Е.А., Гордон И.И., Майер А.Г. Качественная теория динамических систем второго порядка. М.: Наука, 1966. – 568 с.
  3. Амелькин В.В., Лукашевич Н.А., Садовский А.П. Нелинейные колебания в системах второго порядка. Мн.: Изд-во БГУ, 1982. – 208 с.
  4. Немыцкий В.В., Степанов В.В. Качественная теория дифференциальных уравнений. М.; Л.: Гостехтеориздат, 1949. – 552 с.
  5. Голубев В.В. Лекции по аналитической теории дифференциальных уравнений. Л.: ГИТТЛ, 1941. – 400 с.

Mathematical physics equations

  1. Михлин С.Г. Курс математической физики. М., 1968.
  2. Тихонов А.Н., Самарский А.А. Уравнения математической физики. М., 1977.

Theory of distributions

  1. Антоневич А.Б. , Радыно Я.В. . Функциональный анализ и интегральные уравнения. – Минск: БГУ, 2006. – 430с.
  2. Владимиров В.С. Обобщенные функции в математической физике. – Москва: Наука, 1979. – 320с.
  3. Duistermaat J.J., Kolk J.A.C. Distributions: theory and applications, Springer, 2006.
  4. Grubb G. Distributions and operators, Springer, 2008.

Harmonic analysis

  1. E.Stein, G.Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton, New Jersey, Princeton University Press, 1971.
  2. Strichartz R. A guide to distribution theory and Fourier transforms, CRC Press, 1996.
  3. Grafakos L. Classical Fourier analysis (2ed.). Springer, 2008.

Elements of stochastic processes theory

  1. Вентцель А.Д. Курс теории случайных процессов. – М.: Наука, 1978.
  2. Лазакович Н.В., Сташулёнок С.П., Яблонский О.Л.  Теория вероятностей : учебник. – 2-е изд.,перераб. и доп. – Минск : БГУ, 2007.

Modern numeric calculus of computer modeling

  1. Ольшанский М.А. Лекции и упражнения по многосеточным методам М. 2003.
  2. Lele S.K. Compact finite Difference Schemes with Spectral-like Resolutions. J. of Computational Physics. V. 103 No. 1 1992, pp. 16-42.

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Methods of teaching

Comparative method, problematic method, dialog and heuristic method, method of forming knowledge of a personal significance.

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Language

English

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Requirements, monitoring

  • sample interview.

Final grade on the discipline is a cumulative characteristic which is formed on a base of a undergraduate student’s knowledge during the total period of monitoring and on the semester grade(s).

Monitoring grade component can attain 50% of a final grade.

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Form of assessment

Exam