Кто читает:: Факультет математики

Статус:: Курс обязательный

Когда читается:: 1-й курс, 1, 2 модуль

Abstract

Research Seminar of Master’s Programme “Open Problems of Modern Mathematics” is compulsory and accessible to any first year student of the master’s program in mathematics, no special pre-requisite required. Each participant of the seminar give a talk about open problems in the area of his/her own research.

Learning Objectives

The seminar is intended to introduce most popular open mathematical problems and known approaches to solve them. Also it offers the students an opportunity to prepare and give a talk.

Expected Learning Outcomes

knows the current state of various branches of mathematics, which problems are open now and what is already done, improves presentation skills and ability to understand mathematics from each other

Course Contents

Number theory-1
Grand Riemann hypothesis. Generalized Riemann hypothesis. Hilbert's ninth problem. Hilbert's eleventh problem. Hilbert's twelfth problem. Lehmer's totient problem: if φ(n) divides n − 1, must n be prime? Are there infinitely many amicable numbers? Are there any pairs of amicable numbers which have opposite parity? Are there any pairs of relatively prime amicable numbers? Are there infinitely many betrothed numbers? Are there any pairs of betrothed numbers which have same parity? Are there infinitely many perfect numbers? Do quasiperfect numbers exist? Do any odd weird numbers exist? Do any Lychrel numbers exist? Exponent pair conjecture. Is π a normal number (its digits are "random")? Which integers can be written as the sum of three perfect cubes?
Group theory
Is every finitely presented periodic group finite? For which positive integers m, n is the free. Burnside group B(m,n) finite? Is every group surjunctive? Andrews–Curtis conjecture. Herzog–Schönheim conjecture. Are there an infinite number of Leinster Groups?
Partial differential equations
Regularity of solutions of Euler equations. Existence and regularity of solutions of Navier-Stokes equation. Regularity of solutions of Vlasov–Maxwell equations.

Assessment Elements

active participation at the seminar
own talk at the seminar

Interim Assessment

Interim assessment (2 module)
0.4 * active participation at the seminar + 0.6 * own talk at the seminar

Bibliography

Recommended Core Bibliography

Connes, A., & Kouneiher, J. (2019). Sir Michael Atiyah, a Knight Mathematician A tribute to Michael Atiyah, an inspiration and a friend. https://doi.org/10.1090/noti1981

Recommended Additional Bibliography

De Lellis, C. (2016). The masterpieces of John Forbes Nash Jr. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1606.02551

Магистерская программа «Математика»

Research Seminar of Master’s Programme 1

Course Syllabus