Ehrhart-Theorie/Ehrhart Theory

Sommersemester 2022

Introduction

Linear algebra deals with solutions of linear equations. When the equalities are replaced by inequalities, the set of solutions is called a polyhedron. Bounded polyhedra are termed polytopes. The study of polytopes dates back at least to the ancient Babylonian and Egypt in the construction of pyramids.
Ehrhart theory deals with the problem of counting the number of integer points in a polytope. When the polytope has integer vertices, the number of its integer points is the value at 1 of a polynomial function, called the Ehrhart polynomial of the polytope. The leading coefficient of the Ehrhart polynomial is the volume of the polytope. In recent years this topic received a considerable amount of attentions from geometric, algebraic and combinatorial points of view.
The goal of this seminar is to get acquaintance to the Ehrhart theory, and to discuss connections/applications to combinatorics, number theory and geometry.

References

Our main reference is the following book of Beck and Robins. The first edition of the book of Beck and Robins is translated into German, and should be downloadable from Springer using Uni-Köln VPN. The second edition is available only in English.
  • M. Beck, S. Robins, Computing the continuous discretely, 2nd ed. Springer Verlag, 2015. https://matthbeck.github.io/ccd.html
  • M. Beck, S. Robins, Das Kontinuum diskret berechnen, Springer Lehrbuch, 2008.
    • The following book serves as a complement to some topics in the main reference.
  • M. Beck, R. Sanyal, Combinatorial Reciprocity Theorems: An Invitation To Enumerative Geometric Combinatorics, Graduate studies in mathematics 195. Manuscript is available here: https://matthbeck.github.io/crt.html.

    • Organization

  • Termine: The seminar is a Blockseminar, and will be held on 7, 8 and 14, April, 2022, 14h-17h30.
  • Vorbesprechung: The Vorbesprechung of the seminar will take place on 25.01.2022, 11.30 Uhr via zoom:
    https://uni-koeln.zoom.us/j/94806711957?pwd=d1VEOGtFSmVnaU1sZmJUbmxxVWZYZz09
    Meeting-ID: 948 0671 1957
    Passwort: 719614
  • Sprache/Language: Für Bachelorstudenten können Sie zwischen Deutsch und Englisch wählen (für den Beamer und das Skript). For Master students the talk and the script should be in English.
  • It is not known whether this seminar will be online, but we prepare it as an online seminar.
  • Every participant should prepare an online (beamer) talk for 45 minutes, and a detailed script of the material covered in the talk (no more than 10 pages). The talk contributes 60/100 to the grade, and the script contributes 40/100.

    • Registration

    For registration please send an e-mail to Dr. Xin Fang (xfang@mi.uni-koeln.de) with the following information:
    Name, Vorname, Matrikelnummer, Studium, Semester, and at least 3 different choices of topics between Topic 1 and 9 (see below).
    According to the regulation, registrations e-mail must be sent between 28.01.2022 00:00 and 02.02.2022 15:00.

    Plan of talks

  • Talk 1: Polytopes, examples of Ehrhart polynomials (Chapter 2, Section 2.1-2.6).
  • Talk 2: Generating functions, triangulations, cones (Chapter 2, 3, Section 2.7, 2.8, 3.1-3.3)
  • Talk 3: Ehrhart theory (Chapter 3, Section 3.4-3.8).
  • Talk 4: Ehrhart-MacDonald reciprocity (Chapter 4).
  • Talk 5: Dehn-Sommerville relations and Ehrhart coefficients (Chapter 5).
  • Talk 6: Dedekind sums, Mordell-Pommersheim tetrahedron (Chapter 8).
  • Talk 7: Zonotopes and their Ehrhart polynomial (Chapter 9).
  • Talk 8: Tangent cone and Brion's theorem (Chapter 11).
  • Talk 9: Euler-Maclaurin summation (Chapter 12).

    • Arbeitsgruppe Algebra und Zahlentheorie, Mathematisches Institut, Universität zu Köln