311.265 (24W) Selected Topics in Discrete Mathematics: Combinatorial Reciprocity via Geometry
Überblick
- Lehrende/r
- LV-Titel englisch Selected Topics in Discrete Mathematics: Combinatorial Reciprocity via Geometry
- LV-Art Vorlesung
- LV-Modell Präsenzlehrveranstaltung
- Semesterstunde/n 2.0
- ECTS-Anrechnungspunkte 3.0
- Anmeldungen 13
- Organisationseinheit
- Unterrichtssprache Englisch
- LV-Beginn 03.10.2024
- eLearning zum Moodle-Kurs
-
Anmerkungen
Die Lehrveranstaltung wird sowohl für das Bachelorstudium als auch als fürs Masterstudium angerechnet. Über die Unterrichtssprache wird am Beginn des Semesters entschieden.
Zeit und Ort
LV-Beschreibung
Intendierte Lernergebnisse
After successfully completing this course, students will know the main results concerning combinatorial reciprocity: They can explain and prove interesting connections between counting sequences of different combinatorial objects. Moreover, they can apply combinatorial and geometric methods to derive combinatorial reciprocity results, and they will also be able to verify special cases of reciprocity results using a computer algebra system.
Lehrmethodik
Lecture with interactive elements
Inhalt/e
The content of this course lies in the interplay between enumerative and geometric combinatorics. The main theme will be the study of a fascinating phenomenon known as combinatorial reciprocity, which relates the enumeration of two families of combinatorial objects through the evaluation of a polynomial at positive and negative integers. The main objective is to develop combinatorial and geometric tools to derive and explain combinatorial reciprocities. In the process, we will learn about
- partially ordered sets and order polynomials,
- colorings of graphs,
- acyclic orientations and flows, and
- counting regions of hyperplane arrangements.
We will also see how the geometry of convex polytopes and polyhedra can beautifully explain many of the combinatorial results presented in this course.
Erwartete Vorkenntnisse
Basic knowledge in combinatorics and graph theory as taught in the course „Kombinatorische Strukturen“, programming skills as taught in the course „Computermathematik“, and profound knowledge of proof techniques are assumed.
Literatur
- Matthias Beck and Raman Sanyal, Combinatorial Reciprocity Theorems: An Invitation To Enumerative Geometric Combinatorics, Graduate Studies in Mathematics, 195. American Mathematical Society, 2018.
- Richard Stanley, Combinatorial reciprocity theorems, Advances in Mathematics 14 (1974), 194–253.
- Günter Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, 152. Springer-Verlag, New York, 1995.
Prüfungsinformationen
Prüfungsmethode/n
Oral exam (approx. 45 minutes)
Prüfungsinhalt/e
Contents of the lecture
Beurteilungskriterien/-maßstäbe
Emphasis is laid on reasonable knowledge of the definitions, methods and facts and thorough understanding of the material of the course including the proofs. In case of partial fulfillment, the grade will be awarded freely depending on the knowledge shown.
Beurteilungsschema
Note BenotungsschemaPosition im Curriculum
- Doktoratsprogramm Modeling-Analysis-Optimization of discrete, continuous and stochastic systems
(SKZ: ---, Version: 16W.1)
-
Fach: Modeling-Analysis-Optimization of discrete, continuous and stochastic systems
(Pflichtfach)
-
Modeling-Analysis - Optimization of discrete, continuous and stochastic systems (
0.0h XX / 0.0 ECTS)
- 311.265 Selected Topics in Discrete Mathematics: Combinatorial Reciprocity via Geometry (2.0h VO / 3.0 ECTS)
-
Modeling-Analysis - Optimization of discrete, continuous and stochastic systems (
0.0h XX / 0.0 ECTS)
-
Fach: Modeling-Analysis-Optimization of discrete, continuous and stochastic systems
(Pflichtfach)
- Bachelorstudium Technische Mathematik
(SKZ: 201, Version: 17W.1)
-
Fach: Diskrete Mathematik
(Wahlfach)
-
10.8 Ausgewählte Kapitel der Diskreten Mathematik (
2.0h VO / 3.0 ECTS)
- 311.265 Selected Topics in Discrete Mathematics: Combinatorial Reciprocity via Geometry (2.0h VO / 3.0 ECTS) Absolvierung im 5., 6. Semester empfohlen
-
10.8 Ausgewählte Kapitel der Diskreten Mathematik (
2.0h VO / 3.0 ECTS)
-
Fach: Diskrete Mathematik
(Wahlfach)
- Bachelorstudium Technische Mathematik
(SKZ: 201, Version: 22W.1)
-
Fach: Diskrete Mathematik
(Wahlfach)
-
10.8 Ausgewählte Kapitel der Diskreten Mathematik (
2.0h VO / 3.0 ECTS)
- 311.265 Selected Topics in Discrete Mathematics: Combinatorial Reciprocity via Geometry (2.0h VO / 3.0 ECTS)
-
10.8 Ausgewählte Kapitel der Diskreten Mathematik (
2.0h VO / 3.0 ECTS)
-
Fach: Diskrete Mathematik
(Wahlfach)
- Masterstudium Mathematics
(SKZ: 401, Version: 18W.1)
-
Fach: Discrete Mathematics
(Wahlfach)
-
6.7 Selected Topics in Discrete Mathematics (
2.0h VO / 3.0 ECTS)
- 311.265 Selected Topics in Discrete Mathematics: Combinatorial Reciprocity via Geometry (2.0h VO / 3.0 ECTS)
-
6.7 Selected Topics in Discrete Mathematics (
2.0h VO / 3.0 ECTS)
-
Fach: Discrete Mathematics
(Wahlfach)
- Masterstudium Mathematics
(SKZ: 401, Version: 18W.1)
-
Fach: Applied Mathematics
(Wahlfach)
-
Lehrveranstaltungen aus den Vertiefungsfächern (
0.0h XX / 12.0 ECTS)
- 311.265 Selected Topics in Discrete Mathematics: Combinatorial Reciprocity via Geometry (2.0h VO / 3.0 ECTS)
-
Lehrveranstaltungen aus den Vertiefungsfächern (
0.0h XX / 12.0 ECTS)
-
Fach: Applied Mathematics
(Wahlfach)
- Masterstudium Mathematics
(SKZ: 401, Version: 22W.1)
-
Fach: Discrete Mathematics
(Wahlfach)
-
5.7 Selected Topics in Discrete Mathematics (
2.0h VO / 3.0 ECTS)
- 311.265 Selected Topics in Discrete Mathematics: Combinatorial Reciprocity via Geometry (2.0h VO / 3.0 ECTS) Absolvierung im 1., 2., 3. Semester empfohlen
-
5.7 Selected Topics in Discrete Mathematics (
2.0h VO / 3.0 ECTS)
-
Fach: Discrete Mathematics
(Wahlfach)
- Masterstudium Mathematics
(SKZ: 401, Version: 22W.1)
-
Fach: Applied Mathematics
(Wahlfach)
-
7.1 Wahl von weiteren Lehrveranstaltungen aus den Vertiefungsfächern (
0.0h XX / 12.0 ECTS)
- 311.265 Selected Topics in Discrete Mathematics: Combinatorial Reciprocity via Geometry (2.0h VO / 3.0 ECTS) Absolvierung im 2., 3. Semester empfohlen
-
7.1 Wahl von weiteren Lehrveranstaltungen aus den Vertiefungsfächern (
0.0h XX / 12.0 ECTS)
-
Fach: Applied Mathematics
(Wahlfach)