2022/2023

Research Seminar "Real Algebraic and Toric Geometry"

Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'

Category 'Best Course for New Knowledge and Skills'

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

When: 3, 4 module

Open to: students of all HSE University campuses

Instructors: Alexander Isaakovich Esterov

Language: English

ECTS credits: 3

Contact hours: 36

Course Syllabus

Abstract

Algebraic geometry and convex geometry are among key fields in modern mathematics that should be somewhat familiar to every working mathematician. Fundamental introductory textbooks to these fields are usually set up at the graduate level This text, although presenting particular topics in these fields, is intended to be read before fundamental introductions and to provide exemples and motivation that may subsequently facilitate systematic study of algebraic geometry and convex geometry. More specifically, this course is an introduction to real algebraic geometry (1st module) and toric varieties (2nd module). Although the two topics are formally independent, the first one provides a natural context for the second. The first module will be devoted to Hilbert’s 16th problem for algebraic curves, which was one of the starting points for real algebraic geometry. The problem asks for the topological classification of smooth plane real algebraic curves of a given degree. Hilbert himself solved the problem up to degree 6 modulo one elusive topological type, whose existence was proved only 70 years later. Our aim is Viro’s patchworking theorem, which allows to construct algebraic curves of a given degree with prescribed topology. The second module will be devoted to toric varieties — certain algebraic varieties that can be assigned to integer polytopes. This correspondence between algebraic and geometric objects turns out to be profitable for both fields of study. For instance, on the polyhedral side, it solves the Upper bound conjecture regarding the number of faces of a simple polytope, while, on the algebro-geometric side, it produces the theory of Newton polytopes and provides the technique behind Viro’s patchworking..

Learning Objectives

• To gain basic understanding of toric geometry, patchworking and Newton polytopes

Expected Learning Outcomes

• To be able to apply methods of toric geometry, patchworking and Newton polytopes in their research work and to read scientific papers referring to such methods. and to undertake a broader and more in-depth study of algebraic geometry.

Course Contents

• Hilbert’s 16th problem and Harnack’s inequality
• Viro’s patchworking
• Real and complex projective toric varieties
• Newton polytopes and tropical compactifications
• Kouchnirenko–Bernstein formula

Assessment Elements

• Homework
  I will send you by email two exercises from the lecture notes. You will send me written solutions. Then we meet in Zoom, and I ask some questions on your solutions.
• Homework
  I will send you by email two exercises from the lecture notes. You will send me written solutions. Then we meet in Zoom, and I ask some questions on your solutions.

Interim Assessment

• 2022/2023 4th module
  0.5 * Homework + 0.5 * Homework

Bibliography

Recommended Core Bibliography

• Dequantization of Real Algebraic Geometry on Logarithmic Paper. (2000). Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.CD7F9A4E
• Kaveh, K., & Khovanskii, A. G. (2008). Algebraic equations and convex bodies. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.1DF414A4
• O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov, Dedicated Vladimir, & Abramovich Rokhlin. (n.d.). Elementary Topology Problem Textbook. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.FF37742C
• V. M. Kharlamov, A Korchagin, G. M. Polotovskiĭ, & O Viro. (2016). Topology of real algebraic varieties and related topics. [N.p.]: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1494946
• Viro, J., & Viro, O. (2006). Configurations of skew lines. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.math%2f0611374

Recommended Additional Bibliography

• G. Mayo, & David R. Cox. (n.d.). 2nd Lehmann Symposium- Optimality IMS Lecture Notes- Mongraphs Series (2006) FREQUENTIST STATISTICS AS A THEORY OF INDUCTIVE INFERENCE. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.25E050E1