1. Arithmetic of elliptic curves
Lecturer: Denis Osipov
Description： An elliptic curve over a field is a smooth projective curve of genus 1 which has a point defined over the basic field. In other words, a genus 1 curve is a curve which has a rational differential form without poles or zeros, or, equivalently, it is a smooth cubic curve in the two-dimensional projective space. On the set of points of an elliptic curve which are defined over the basic field, one can introduce the structure of an abelian group. If the ground field is the field of complex numbers, then this group is not very interesting, because it is isomorphic to a two-dimensional torus. If the ground field is finite, then one obtains a finite group which has a lot of applications in coding theory. If the ground field is the field Q of rational numbers, then one can prove that the resulting abelian group is finitely generated. A lot of famous conjectures in arithmetic algebraic geometry are connected with invariants of this group.
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Meeting ID: 849 6500 9261
Passcode: 842085
Assistant's Email：lingwang@stu.pku.edu.cn
Time：Thursdays and Fridays 18:40-20:30 Beijing time
2. Topology of Integrable Hamiltonian Systems
Lecturer: Andrey Oshemkov
Description: Integrable Hamiltonian systems are a very popular object of research in mathematics. They were studied from different points of view including the search of exact analytical solutions, the study of algebraic constructions underlying integrability, topological analysis of specific integrable systems arising in geometry, mechanics and mathematical physics. This course is devoted to the presentation of the basic concepts, ideas and methods necessary to study the topology of integrable systems, in particular singularities of integrable Hamiltonian systems. By Liouville theorem, it is the structure of singularities of an integrable Hamiltonian system that determines its topological properties and makes it possible to investigate its topological invariants. We will discuss the local classification of singularities of integrable Hamiltonian systems, the semi-local structure of non-degenerate singularities (i.e., their structure in a neighborhood of a singular fiber), as well as some modern results on the global properties of integrable systems reflecting their behavior in the large (i.e., on the whole phase space). The initial part of the course will be devoted to discussing the basic definitions and facts from symplectic and Poisson geometry, which are the basic tools for presenting the theory of Hamiltonian systems and investigating their properties.
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Meeting ID: 419 450 5642
Passcode: 654321
Assistant's Email：zhuling@stu.pku.edu.cn
Time: Thursdays 15:10-17:00 and 18:40-20:30 Beijing time
3. Diophantine approximation
Lecturer: Oleg German
Description: For the first time, the existence of transcendental numbers was proved in the middle of the 19th century by Joseph Liouville. He showed that algebraic numbers cannot be “too well” approximated by rationals. This gave rise to the theory of Diophantine approximation. Since then, this theory has been developing with the help of various tools: geometry, algebra, and analysis. We shall start with rather elementary geometry of numbers, provide a geometric interpretation of continued fractions, and discuss different types of Diophantine exponents. Then, we shall describe methods of finding good rational approximations for such fundamental constants as e and π, which involve analytic observations, such as Hermite’s integral identity. This approach will lead us to prove both the irrationality of π and the transcendence of e. We shall also prove the transcendence of π. To this end we shall study finite extensions of Q. Finally, if time allows, we shall address the problem of approximating real numbers with algebraic ones and muse on Wirsing’s famous conjecture and related problems.
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Meeting ID: 864 8584 5041
Passcode: 990240
Assistant's Email：lingwang@stu.pku.edu.cn
Time：Mondays and Tuesdays 18:40-20:30 Beijing time
4. Triangulated and Derived Categories in Algebra and Geometry
Lecturer: Anton Fonarev
Description：Derived categories were originally introduced by Grothendieck and Verdier as a nice language to deal with derived functors. Some decades later it became clear that derived categories are often fascinating invariants themselves and carry lots of algebraic/geometric structures wherever they appear. The goal of this course is to present the basics of the theory of derived and triangulated categories beginning with abelian categories and classical derived functors all the way up to semiorthogonal decompositions and DG-categories. Along the way, we will also deal with some categories of sheaves appearing in topology and algebraic geometry.
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Meeting ID: 827 1402 6695
Pasacode: 825287（Tuesdays）
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Meeting ID: 894 1647 0147
Passcode: 749059（Wednesdays）
Assistant's Email：zhuling@stu.pku.edu.cn
Time：Tuesdays 13:00-14:50 Beijing time; Wednesdays 18:40-20:30 Beijing time
