Forschungsgruppe Nichtlineare Algebra

Direktor:
Bernd Sturmfels

Kontakt: Email
Telefon:
+49 (0) 341 - 9959 - 750

Anschrift:
Inselstr. 22
04103 Leipzig

Sekretariat:
Saskia Gutzschebauch
Email, Telefon/Fax:
+49 (0) 341 - 9959
- 752
- 658

Winter 2020/2021

Computing with integrals in nonlinear algebra

Abstract

  • How to compute 100 digits of the volume of a semialgebraic (defined by polynomial inequalities)?

  • How to compute the moments \(\int_{[0,1]^n} f(x_1,\dotsc,x_n)^k \mathrm{d}x_1 \dotsc \mathrm{d}x_n\) for large \(k\)? This problem stems from polynomial optimization.

  • How to compute a recurrence relation for the numbers \(\sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2\)? This relation is central in Apéry's proof of the irrationality of \(\zeta(3)\).

  • How to count the number of smooth rational curves of degree \(d\) on a smooth quartic surface in \(\mathbb{P}^3\)?

These questions from diverse area of mathematics all feature period functions of rational integrals: analytic functions defined by integrating multivariate rational functions. In some regards, period functions are the simplest nonalgebraic functions. One of the first instance to be studied was the perimeter of the ellipse, as a function eccentricity.

I will first expose the fundamental material to compute with period function: linear differential equations as a data structure, symbolic integration and numerical analytic continuation. Next, I will show how to apply these technique in practice on many different problems, including the four questions above. As much as possible, I will connect with current research questions.

See the page from Pierre Lairez for detailed content description and references.


Introduction to Enumerative Geometry

  • Lecturer: Fulvio Gesmundo
  • Date: January 11 / 13 / 15 / 18 / 20 / 22, 2021: 16:30 - 18:30 CET
  • Room: Videobroadcast
  • Prerequisites: A first course in algebraic geometry is recommended but not strictly required. Familiarity with the notion of algebraic variety and the Zariski topology in affine and projective space is assumed. Some familiarity with commutative algebra or algebraic topology will be helpful but not necessary.
  • Remarks: Attendance is free but registration is required. To register or for any other information contact Fulvio Gesmundo or Chiara Meroni.

Abstract

This is an intensive short course on enumerative geometry. The course covers an introduction to intersection theory, and applies the acquired techniques to some classical problems. We will introduce the basics of intersection theory: Chow ring, Chern classes, and basics of Schubert calculus. The theoretical tools which are developed will be applied to the enumerative geometry of some Grassmannian problem and to the Thom-Porteous formula for the calculation of the degree of determinantal varieties. If time permits, we will draw connections to the representation theory of the general linear group.
  • Lecture 1, 2: Basics of intersection theory. Chow ring. Grassmannians.
  • Lecture 3, 4: Chern classes. Schubert calculus. Enumerative problems.
  • Lecture 5, 6: Thom-Porteous’s Formula. Representation Theory.

References

  • D. Eisenbud, J. Harris 3264 and All That (Cambridge 2016) [main reference; lecture notes adapted from this reference will be provided (in non-final version)]
  • E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris Geometry of Algebraic Curves, Vol. I (Springer 1985)
  • L. Manivel Symmetric Functions, Schubert Polynomials, and Degeneracy Loci (SMF/AMS 1998)
  • W. Fulton, J. Harris Representation Theory: A First Course (Springer 1991)

Riemann Surfaces and Algebraic Curves

  • Lecturer: Daniele Agostini, Rainer Sinn
  • Date: Wednesday 15:15 - 16:45 (lectures), Wednesday 11:00 - 12:30 (exercise sessions)
  • Room: The course will take place on Zoom. Please send an email to Daniele Agostini for the link.
  • Keywords: Riemann surfaces, algebraic curves
  • Prerequisites: abstract algebra and familiarity with differential or algebraic geometry
  • Remarks: A lecture log, some notes and the exercise sheets will appear on the group page.

Abstract

The course will be a first introduction to Riemann surfaces and algebraic curves. These are beautiful objects which sit at the intersection of algebra, geometry and analysis. Indeed, on one side these are complex manifolds of dimension one, and on the other they are varieties defined as a zero locus of polynomial equations. Furthermore, they are ubiquitous throughout mathematics, from diophantine equations in number theory to water waves in mathematical physics and Teichmüller theory in dynamical systems.

We will aim to cover the theorems of Riemann-Hurwitz and Riemann-Roch, meromorphic functions and their zeroes and poles, plane curves and elliptic curves, abelian integrals, the theorem of Abel-Jacobi and the construction of Jacobian varieties. Time permitting, we might touch upon further topics such as canonical curves, moduli spaces, the Schottky problem and tropical curves.

Prerequisites: abstract algebra and familiarity with differential or algebraic geometry.


References : Notes for some of the lectures will appear below. We will not follow exactly any particular book, but the main inspirations for the course will be

There are many other beautiful references for this topic. Some of them are:


Other Lectures at MPI MIS

Please follow this link for other regular lectures at the Max-Planck-Institute.

22.01.2021, 18:34