Lecture - Minicourses - Seminar - Guests
Combinatorial Methods
Thematic semester - Spring 2015 - University of Fribourg

Full calendar:
February 26, 10-12 & 13-15 - Lonza annex
Lecture. Introduction
March 3, 17:00. - Physik II (Kolloquium)
Seminar. Michael Cuntz (Hannover)
Simplicial Arrangements
Arrangements of lines which triangulate the plane, the so-called simplicial arrangements, appear to be rare. Many of them have been found during the last 70 years and it is conjectured that they are all known. In my talk I will present some old and some very recent observations, results, and conjectures on simplicial arrangements as well as several applications to these results.
March 5, 10-12 & 13-15 - Lonza annex
Lecture. Basic poset topology
March 6, 13:00. - Lonza
Seminar. Max Wakefield (USNA Annapolis)
A deformed Moebius algebra and a matroid Kazhdan-Lusztig basis
The spectrum of the Orlik-Terao algebra, also called the reciprocal plane, is useful in studying various properties of the associated hyperplane arrangement. We begin this discussion by trying to compute the related Poincar\'e polynomial of the intersection cohomology. This story closely parallels the classical Kazhdan-Lusztig polynomials in the study of Hecke algebras for Coxeter groups. We will discuss an algebraic analog in the setting of M\"obius algebras. One can deform the M\"obius algebra and then study the related Kazhdan-Lusztig basis. This deformed algebra has many nice properties, but there are many related open problems.
March 12, 10-12 & 13-15 - Lonza annex
Lecture. Poset topology II: carrier lemma and Quillen lemma
March 13, 13:00. - Lonza
Seminar. Ornella Greco (KTH Stockholm)
The Betti table of Veronese Modules
In this talk, I will concentrate on the minimal free resolution of the Veronese modules, and I will give a formula for their Betti numbers in terms of the reduced homology of some skeleton of a simplicial complex. As applications, I will characterize their Cohen-Macaulayness and the linearity of their resolution. Moreover, I will give a complete description of the Betti table of all the Veronese modules in 2 variables.
March 19, 10-12 & 13-15 - Lonza annex
Lecture. Nerve Lemma, Salvetti's complex
March 20, 10-12 & 13-15 - Lonza
Minicourse (Ivan Martino).
Introduction to toric varieties
March 25-27
CUSO doctoral school ``Let's matroid''
in Neuchatel [go to webpage]
April 2, 10-12 & 13-15 - Lonza annex
Lecture. CW complexes and CW posets
April 14, 17:00. - Physik II (Kolloquium)
Seminar. Mario Salvetti (Pisa)
TBA
April 16, 10-12 & 13-15 - Lonza annex
Lecture. Discrete Morse Theory I
April 22, 10:00. - Lonza (joint w./ Oberseminar Geometrie)
Seminar. Viktoriya Ozornova (Bremen / MPI Bonn)
Factorability structures and rewriting systems
Let X be a projective subvariety of the projective space over an algebraically closed field. The dual graph of X, denoted by G(X), has the irreducible components of X as nodes, and two nodes are connected by an edge if and only if the corresponding irreducible components intersect in codimension 1. A classical result of Hartshorne states that, if the (quotient by) an ideal defining X is Cohen-Macaulay, then G(X) is connected. In particular G(X) is connected whenever X is the zero locus of c polynomials, where c is the codimension of X. If the (only) radical ideal defining X is defined by c polynomials (i.e. if X is a complete intersection) and X is the union of linear subspaces (i.e. X is a subspace arrangement) we recently proved with Bruno Benedetti that G(X) is r-connected, where r = d_1+…+d_c-c and the d_i’s are the degrees of the polynomials. In the talk I will discuss this result, do some related examples and, time permitting, explain a more general version of the theorem holding true for any arithmetically Gorenstein projective scheme.
April 23, 10-12 & 13-15 - Lonza annex
Lecture. Discrete Morse Theory II
April 24, 11:00. - DOKPE 0.26
Seminar. Matteo Varbaro (Genova)
On dual graphs of complete intersections
Let X be a projective subvariety of the projective space over an algebraically closed field. The dual graph of X, denoted by G(X), has the irreducible components of X as nodes, and two nodes are connected by an edge if and only if the corresponding irreducible components intersect in codimension 1. A classical result of Hartshorne states that, if the (quotient by) an ideal defining X is Cohen-Macaulay, then G(X) is connected. In particular G(X) is connected whenever X is the zero locus of c polynomials, where c is the codimension of X. If the (only) radical ideal defining X is defined by c polynomials (i.e. if X is a complete intersection) and X is the union of linear subspaces (i.e. X is a subspace arrangement) we recently proved with Bruno Benedetti that G(X) is r-connected, where r = d_1+…+d_c-c and the d_i’s are the degrees of the polynomials. In the talk I will discuss this result, do some related examples and, time permitting, explain a more general version of the theorem holding true for any arithmetically Gorenstein projective scheme.
April 24, 13:00. - Lonza
Seminar. Tim Lindemann and Daniel Ris
`Seminaire Libre'/'Freies Seminar' on polytope theory
April 30, 10-12 & 13-15 - Lonza annex
Lecture. Topological representation of oriented matroids
May 7, 10-12 & 13-15 - Lonza annex
Lecture. Hyperplane arrangements
May 8, 11:15. - Math I
Seminar. Nicholas Proudfoot (Oregon U)
Toric and hypertoric combinatorics
To any rational polytope one can associate a projective algebraic variety called a toric variety, and there is a rich interaction between the combinatorics of the polytope and the geometry of the variety. For example, the Betti numbers of the variety are certain combinatorial invariants of the polytope, and this fact was used to solve an important combinatorial problem about these numbers. Hypertoric varieties are quaternionic analogues of toric varieties that relate in a similar way to the combinatorics of hyperplane arrangements. In this talk I will define toric and hypertoric varieties and survey some of the aforementioned results.
May 21, 10-12 & 13-15 - Lonza annex
Lecture. Homotopy colimits
May 26, 11:00. - (!) unusual slot - Lonza
Seminar. Relinde Jurrius (Neuchatel)
Application of hyperplane arrangements to weight enumeration
Many research in coding theory is focussed on linear error-correcting codes. Since these codes are subspaces, linear algebra plays a prominent role in studying them.
An important polynomial invariant of linear error-correcting codes is the (extended) weight enumerator. The weight enumerator gives information about the probability of undetected errors in error-detection, and about the probability of decoding errors in bounded distance decoding. Furthermore, the extended weight enumerator is equivalent to the Tutte polynomial of the matroid associated to the code.
Linear codes are closely connected to hyperplane arrangements: the columns of the generator matrix of a code can be viewed as the coordinates of a hyperplane arrangement over a finite field. Using this correspondence, the problem of determining the extended weight enumerator can be transformed into a counting problem on a hyperplane arrangement. In fact, the extended weight enumerator is equivalent to the coboundary polynomial (or two-variable characteristic polynomial) of the associated hyperplane arrangement.
In this talk, we will examine this application of hyperplane arrangements to weight enumeration in more detail. The practical use of the theory will be motivated by several examples.


Host:
A:T&C group
[to the group page]
Guests:

Filippo Callegaro (Pisa)
Michael Cuntz (Hannover)
Ornella Greco (KTH Stockholm)
Relinde Jurrius (Neuchâtel)
Luca Moci
(Paris VII)
Viktoriya Ozornova (Bremen / MPI Bonn)
Mario Salvetti (Pisa)
Matteo Varbaro (Genova)
Max Wakefield
(USNA Annapolis)
Rade Živaljević (SANU Belgrade)


Visiting students:

Michela Di Marca (Genova)
Tim Lindemann (Bremen)