Organizers: Günter M. Ziegler
Phan Thị Hà Dương
I. Lecturers
- Günter M. Ziegler (Professor of FU Berlin), http://page.mi.fu-berlin.de/gmziegler/ Raman Sanyal (Professor of FU Berlin),http://page.mi.fu-berlin.de/sanyal/
- Matthias Beck (Associate Professor of San Francisco State University), http://math.sfsu.edu/beck/
- Phan Thị Hà Dương (Associate Professor of Institute of Mathematics – Vietnam Academy of Science and Technology), http://vie.math.ac.vn/~phduong/
- Lê Anh Vinh (Lecturer of Hanoi National University), http://www.math.harvard.edu/~vinh
II. Lectures
0. “Introduction to Enumerative Combinatorics and Geometric Combinatorics.”
(in Vietnamese)
Lecturers: Phan Thi Ha Duong and Le Anh Vinh
February 19-25.
Abstract: We first give an introduction to Enumerative Combinatorics: notions of set and multiset, permutations, integer partitions, Euler number, Stirling numbers, etc, some methods to count the cardinality of a given set, generating functions. In the second part, we present some preliminary definitions and ingredients of Geometric Combinatorics, in particular, some notions of polytopes.
1. “Interesting Polytopes”
Lecturer: Günter M. Ziegler (FU Berlin)
March 4-7
Abstract: Regular convex polytopes have been studied since antiquity, General convex polytopes since Descartes' work on the Euler-Schläfli-Poincaré formula. Interesting examples arise from many different mathematical contexts, including geometry, combinatorics, optimization, algebra, and representation theory. In this workshop we will construct, compute and explore classes of examples, such as the regular polytopes, hypersimplices, stacked polytopes, cut polytopes, 2s2s 4-polytopes, neighborly cubical polytopes, and many others.
2. "Combinatorics and Valuations"
Lecturer: Raman Sanyal (FU Berlin)
March 7,8 and 11,12
Abstract: Inclusion-exclusion and, more generally, Moebius inversion is the combinatorial art of relating the value of "the whole" to the values of "its pieces". In geometry, the incarnation of this principle is the theory of valuations with fine examples being the volume and the discrete volume (i.e., the number of integer points). In these lectures we will develop the theory of valuations on polytopes from a discrete-geometric perspective. To this end, we will study subdivisions and dissections of polyhedra, their geometry and their combinatorics and, as a reward, we will see classical combinatorial objects such as spanning trees, permutations, Stirling numbers, and Euler numbers in a geometric light.
3. "Combinatorial Reciprocity Theorems"
Lecturer: Matthias Beck (San Francisco State University)
March 12-15
Abstract: A common theme of enumerative combinatorics is formed by counting functions that are polynomials. For example, one proves in any introductory graph theory course that the number of proper k-colorings of a given graph G is a polynomial in k, the chromatic polynomial of G. Combinatorics is abundant with polynomials that count something when evaluated at positive integers, and many of these polynomials have a (completely different) interpretation when evaluated at negative integers: these instances go by the name of combinatorial reciprocity theorems. For example, when we evaluate the chromatic polynomial of G at -1, we obtain (up to a sign) the number of acyclic orientations of G, that is, those orientations of G that do not contain a coherently oriented cycle. Combinatorial reciprocity theorems appear all over combinatorics. We will attempt to show some of the charm (and usefulness!) these theorems exhibit. Our goal is to weave a unifying thread through various combinatorial reciprocity theorems, by looking at them through the lens of geometry.
III. Tentative Schedule
Morning: Lectures 8:30-10:00 and Exercises 10:30-12:00
Afternoon: Lectures 14:00-15:30 and Exercises 16:00-17:30
Week 1 (Feb. 18-22): (morning| afternoon)
February 19: | Phan
February 20: Phan |
February 21: | Phan
February 22: Phan |
Week 2 (Feb. 25-Mar. 1) (afternoon). Speakers: Dr. Tran Nam Trung, Dr. Ha Minh Lam
- Tuesday, Feb. 25, 2013: Representation of polytopes:+ Main theorem for polytopes
+ Schlegel Diagrams
+ Gale Diagrams
+ Classes of examples
- Wednesday, Feb. 26, 2013: Triangulations of point configurations
- Thursday, Feb. 27, 2013: Ehrhart polynomial
Week 3 (Mar. 4-8): (morning| afternoon)
March 4: Ziegler | Ziegler
March 5: Ziegler | Ziegler
March 6: Ziegler |
March 7: Sanyal | Ziegler
March 8: Sanyal |
Week 4 (Mar. 11-15) (morning| afternoon)
March 11: Sanyal | Sanyal
March 12: Beck | Beck
March 13: Sanyal |
March 14: Beck | Beck
March 15: Beck |
IV. Inscription
Please write a message with subject “Spring School 2013” to This email address is being protected from spambots. You need JavaScript enabled to view it. , not later than February 05, 2013.
