Abstract: We present an algorithm which computes a minimal degree rational family of curves on any rational complex projective surface. The algorithm computes a chain of surfaces connected by morphishms to the minimal surface. A minimal surface is a representative in a birational equivalence class, which is simplest in some sence. We use the classification of minimal surfaces, and give minimal degree rational families on them. Then we pull back such a minimal family through the chain in a controlled way. This gives the required family on the input surface.

Abstract: We describe a linear upper bound on the degrees of the essential generators of the ring of projective invariants of $n$ labeled points in the projective plane under the action of $SL_3(C)$. More generally, we use results of H. Derksen and V. L. Popov to show that the projective coordinate ring of a type-A weight variety is bounded by the dimension of that variety. In the case of $n$ points in the projective plane, we show that this linear bound contrasts strikingly with the toric degeneration to a Gelfand--Tsetlin semigroup ring, where we show that the bounds on the degrees of essential generators can grow exponentially in $n$.

Abstract: The Birkhoff polytope, also known as assignement polytope, is one of the most studied polytopes. It is the convex hull of all permutation matrices.
In this talk I will propose the systematic study of Permutation polytopes. They are the convex hull of a subgroup of the group of n x n permutation matrices. We will see examples, discuss different notations of equivalence and present some structural results. As an application we will classify the ≤ 4 -dimensional faces of permutation polytopes.
This is joint work with C. Haase, B. Nill and A. Paffenholz.

Abstract: In this talk, we take a significant step towards confirming a long-standing and far-reaching conjecture of Peter J. Cameron and Cheryl E. Praeger. They conjectured in 1993 that there are no non-trivial block-transitive 6-designs. We essentially prove that the Cameron-Praeger conjecture is true for the important case of Steiner 6-designs, i.e. for 6-(v,k,λ) designs with λ=1. Consequently, the result also contributes to the more general and fundamental problem in combinatorial design theory whether there exists any non-trivial Steiner 6-design.

Abstract: A theorem of Howe states that every 3-dimensional lattice polytope P whose only lattice points are its vertices, is a Cayley polytope, i.e. P is the convex hull of two lattice polygons with distance one. We want to generalize this result by classifying 3-dimensional lattice polytopes without interior lattice points. The main result will be, that they are up to finite many exceptions either Cayley polytopes or there is a projection, which maps the polytope to the double unimodular 2-simplex.

Abstract: The f-vectors of centrally-symmetric convex polytopes are the subject of three conjectures A, B, C of increasing strength by Kalai from 1989. We disprove Conjecture C for d>4, and Conjecture B for d>4, and speculate a bit about Conjecture A. (Joint work with Raman Sanyal and Axel Werner)

Abstract: For fixed t, -1 < t < 1, consider the graph whose vertices are the points of the (n-1)-dimensional unit sphere and in which two vertices are adjacent if their inner product is exactly t; this is an example of a distance graph on the sphere. For this and similar graphs, notions of stability number and chromatic number can be defined in analogy with the case of finite graphs. The measurable chromatic number, for instance, is the minimum number of colors needed to color the points of the sphere in such a way that no two points whose inner product is t get the same color; moreover we require the sets of points that receive the same color to be Lebesgue measurable. In this talk, we will show how to generalize the Lovasz theta number for finite graphs to distance graphs on the sphere. This will provide an upper bound for the stability number and a lower bound for the measurable chromatic number of these graphs. By exploiting the symmetry of such graphs, it is possible to actually compute these bounds, even analytically. This also provides, for dimensions n = 3, ..., 24, better lower bounds for the measurable chromatic number of the n-dimensional Euclidean space, a notion introduced by Falconer (1981) as a restriction of the notion of chromatic number of the Euclidean space, which was first considered by Nelson and Hadwiger.
(Joint work with C. Bachoc, G. Nebe, and F. Vallentin)
References:
* C. Bachoc, G. Nebe, F.M. de Oliveira Filho, and F. Vallentin, "Lower bounds for measurable chromatic numbers", arXiv:0801.1059v2 [math.CO], to appear in GAFA, 18 pp, 2008.
* F.M. de Oliveira Filho and F. Vallentin, "Fourier analysis, linear programming, and densities of distance avoiding sets in R^n", arXiv:0808.1822v1 [math.CO], 10pp, 2008.

Abstract: We show that an 'almost strong Lefschetz' property holds for the barycentric subdivision of a shellable complex. From this we conclude that for the barycentric subdivision of a Cohen-Macaulay complex, the h-vector is unimodal, peaks in its middle degree (one of them if the dimension of the complex is even), and that its g-vector is an M-sequence. In particular, the (combinatorial) g-conjecture is verified for barycentric subdivisions of homology spheres. In addition, using the above algebraic result, we derive new inequalities on a refinement of the Eulerian statistics on permutations, where permutations are grouped by the number of descents and the image of 1.

Abstract: One of the main open questions in liaison theory is whether every homogeneous Cohen-Macaulay ideal in a polynomial ring is glicci, i.e. if it is in the G-liaison class of a complete intersection. We give an affirmative answer to this question for Stanley-Reisner ideals defined by simplicial complexes that are weakly vertex-decomposable. This class of complexes includes matroid, shifted and Gorenstein complexes respectively. Moreover, we present a simplicial complex which shows that the property of being glicci depends on the characteristic of the base field. The talk is based on joint work with Uwe Nagel.

Abstract: Let $(a_n)_{n \geq 0}$ be a sequence of complex numbers such that its generating series satisfies $\sum_{n \geq 0} a_nt^n = \frac{h(t)}{(1-t)^d}$ for some polynomial $h(t)$. For any $r \geq 1$ we study the transformation of the coefficient series of $h(t)$ to that of $h^{\langle r \rangle}(t)$ where $\sum_{n \geq 0} a_{nr} t^n = \frac{h^{\langle r \rangle}(t)}{(1-t)^d}$. We give a precise description of this transformation and show that under some natural mild hypotheses the roots of $h^{\langle r \rangle}(t)$ converge when $r$ goes to infinity. In particular, this holds if $\sum_{n \geq 0} a_n t^n$ is the Hilbert series of a standard graded $k$-algebra $A$. If in addition $A$ is Cohen-Macaulay then the coefficients of $h^{\langle r \rangle}(t)$ are monotonely increasing with $r$. If $A$ is the Stanley-Reisner ring of a simplicial complex $\Delta$ then this relates to the $r$th edgewise subdivision of $\Delta$ a subdivision operation relevant in computational geometry and graphics which in turn allows some corollaries on the behavior of the respective $f$-vectors.

Abstract: We bound the location of roots of polynomials that have nonnegative coeffcients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most d. For this, we interpret the basis polynomials as vector fields in the real plane, and at each point in the plane analyze the combinatorics of the Gale dual vector configuration. We apply our technique to bound the location of roots of Ehrhart and chromatic polynomials. Finally, we give an explanation for the clustering seen in plots of roots of random polynomials.

Abstract: The image of an algebraic variety under the coordinatewise valuation map from the field of Puiseux series to the rational numbers is a polyhedral complex called a tropical variety. While most information is lost when going from an algebraic variety to this tropical shadow some information such as the dimension is preserved. In tropical algebraic geometry two kind of computations appear. One is purely polyhedral concerned with Minkowski sums of Newton polytopes while the other is in commutative algebra taking cancellation of terms into account. In this talk we give examples of algorithms in tropical geometry while also mentioning how tropical computations have been applied outside their field.

Abstract: We describe the first implementation of the Barvinok-Woods (2003) algorithm, which computes a short rational generating function for an integer projection of the set of integer points in a polytope in polynomial time, when the dimension is fixed. The projection algorithm is based on Kannan's partitioning lemma and the application of set operations to generating functions that correspond to these sets. We use a variant of the recent strengthening of the partitioning lemma due to Eisenbrand and Shmonin (2007) and provide several algorithmic refinements to avoid performing redundant set operations. (Joint work with Sven Verdoolaege and Kevin Woods)

Abstract: Nestohedra are simple polytopes, dual to the nested set complexes associated to the "building sets" of DeConcini and Procesi. They include many favorite examples, such as simplices, cubes, permutohedra, associahedra, cyclohedra, and Stanley-Pitman polytopes.
We interpret their (f- and) h-vectors in terms of permutation statistics. We also characterize when they are _flag_ simple polytopes, and interpret their \gamma-vectors (arising in Gal's generalization of the Charney-Davis conjecture for flag spheres) via permutation statistics in the interesting special case of _chordal_ building sets, which generalize chordal graphs.
(joint with A. Postnikov and L. Williams)

Abstract: It has been known for some time that the associahedron can be obtained from the permutahedron by omitting certain defining inequalities. A natural question is to describe different sets of inequalities such that their omission yields an associahedron. Other questions are whether this construction works for other realizations of the permutahedron and whether this construction generalizes to other finite reflection groups.
We prove the following theorem: Let W be a finite reflection group acting on R^n with associated hyperplane arrangement H of reflection hyperplanes. Let v be a point of the complement of H. The W-permutahedron P_v(W) is the convex hull of the W-orbit of v. Every Coxeter element c of W determines a set of defining inequalities of P_v(W) such that the omission of these inequalities yields a generalized associahedron A_{v,c}(W) as defined by S. Fomin and A. Zelevinsky in the context of cluster algebras. Moreover, the normal fan of this polytope is the cambrian fan F_c as described by N. Reading.
This work is joint work with C. Hohlweg and H. Thomas.