A:C&T - Arrangements: Combinatorics and Topology

A:C&T

Arrangements: Combinatorics & Topology what is this?

UniFr SNSF

Topical:

Lecture: Combinatorial methods in Topology (Spring `19).

Lecture: Combinatorial polynomials (Spring `20).

Events

July 19, 2019, 14:00. Room: Lonza: Benjamin Schroeter (SUNY Binghamton)
Correlation constants of fields and matroids
Abstract
October 18, 2019, 14:00. Room: --: InterCity Seminar (Bern)
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November 22, 2019, 14:00. Room: --: InterCity Seminar (Neuchatel)
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December 13, 2019, 14:00. Room: --: InterCity Seminar (Fribourg)
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Past semesters

May 3, 2019, 14:00. Room: --: --
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March 29, 2019, 14:00. Room: --: --
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March 1, 2019, 14:00. Room: --: --
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December 19, 2018, 14:00. Room: --: --
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November 30, 2018, 14:00. Room: --: --
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October 26, 2018, 14:00. Room: --: --
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May 16, 2018, 14:00. Room: Lonza: Lisa Lamberti (ETHZ)
On tensor diagrams
Abstract
June 26, 2018, 14:00. Room: Lonza: Oscar Papini (U Pisa)
Computational aspects of line and toric arrangements
Abstract
May 25, 2018, 14:00. Room: --: --
Abstract
May 9, 2018, 14:00. Room: Lonza: Giovanni Paolini (SNS Pisa)
Algorithmic aspects of discrete Morse theory
Abstract Discrete Morse theory is a recent combinatorial version of Morse theory. It gives a way to simplify a simplicial complex (or, in general, a CW complex) and obtain a smaller complex with the same homotopy type. In this talk I will introduce discrete Morse theory and discuss some computational problems related to it.
April 13, 2018, 14:00. Room: --: --
Abstract
April 4, 2018, 14:00. Room: --: No seminar (Easter break)
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March 21, 2018, 14:00. Room: Lonza: Viola Siconolfi (Roma 2)
Wonderful models for generalized Dowling lattices
Abstract Given a subspace arrangement, De Concini and Procesi in the '90s described the construction of a variety associated to it, namely its wonderful model. An important feature of these model is that some of its geometric aspects are linked to some combinatorical properties of the subspace arrangement, in particular the description of its boundary and its Betti numbers. Some examples of computation of Betti numbers have been studied, among others, by Gaiffi, Henderson and Yuzwinsky. During the talk I will consider the subspace arrangement associated to a generalized Dowling lattice, a combinatorial object introduced by Hanlon. Our aim is to study the wonderful model associated to it and to give a description of its boundary. To deal with this I will use a bijection between the set of boundary components of the wonderful model and a family of graphs. The results presented have been obtained in collaboration with G. Gaiffi.
March 7, 2018, 14:00. Room: Lonza: Enrico Cecini (Genoa)
Matroid structures in Graph Signal Processing
Abstract I will start by reviewing a setting of rising relevance in some applications, namely the so called Graph Signal Processing, that is the study of spaces of functions defined on the vertexes of a graph, whose properties are induced by the eigensystem of a combinatorial laplacian operator. Within this setting I will show how certain natural notions of independent interest, such as admissible sampling subsets, Poisson-like summation formulae and translation operators, can be interpreted in terms of bases and circuits of a certain class of matroids. Next I will derive the defining properties of such matroids, striving to place them within the theory of Coxeter matroids. (This seminar is a report of an ongoing PhD project.)
October 6, 2017, 14:00. Room: Fribourg: Abstract Please click on the image for the program and abstracts.
June 2, 2017, 14:00. Room: Fribourg: Abstract Please click on the image for the program and abstracts.
April 28, 2017, 14:00. Room: Bern: Abstract Please click on the image for the program and abstracts.
April 10, 2017, 15:15. Room: Math I: Yash Lodha (EPFL)
A complex of clusters
Abstract We provide a certain generalisation to the notion of cube complexes, which we refer to as cluster complexes. A cluster is a CW subdivision of the regular Euclidean cube [0, 1]^n, obtained by considering the intersection pattern with a finite set of admissible hyperplanes. A cube complex is a CW complex obtained by gluing cubes along their faces, which are ``sub-cubes’’. A complex of clusters is a CW complex obtained by gluing clusters along sub-clusters. The key difference is that in a cluster complex, sub-clusters could be placed diagonally in a cluster. We build a cluster complex for a certain group of piecewise projective homeomorphisms of the real line, upon which it acts by cell permuting homeomorphisms. We conclude that this action establishes the property type F-infinity for the group, thereby making it the first example of a non amenable, type F-infinity group that does not contain nonabelian free subgroups.
March 24, 2017, 14:00. Room: Neuchatel: Abstract Please click on the image for the program and abstracts.
December 16, 2016, 14:00. Room: Neuchatel: Abstract Please click on the image for the program and abstracts.
November 25, 2016, 14:00. Room: Bern: Abstract Please click on the image for the program and abstracts.
September 30, 2016, 14:00. Room: Fribourg: Abstract Please click on the image for the program and abstracts.
May 17, 2016, 10:15. Room: Math 0.102: Nikita Kalinin (Geneva)
Tropical curves as scaling limits in sandpiles
Abstract A sandpile state S on a graph G is a collection of chips distributed at the vertices of G. If the number of chips at a vertex is at least its degree, then we can perform a toppling: this vertex sends one chip to each of its neighbours. A relaxation is doing topplings while it is possible. We denote by S’ the result of the relaxation of S. We will discuss a certain scaling properties of sandpiles. Let D be a domain in R^2, P be a finite collection of points of D, and G be the intersection of D with the integer lattice Z^2, dilated by a small h>0. Let S be the maximal stable stateon G, i.e. S constantly equal to 3. Let a state S_P be a perturbation of S obtained by adding chips to the vertices of G closest to the points p of P . Then S_P’ coincides with S almost everywhere and the set of all v in G with S(v) different from (S_P)’(v) constitutes a tropical curve as h tends to 0.
May 3, 2016, 10:15. Room: Math 0.102: Olivier Begassat (Nice)
TBA
Abstract
April 26, 2016, 10:15. Room: Math 0.102: TBD
TBA
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April 19, 2016, 10:15. Room:: No seminar

Abstract
April 12, 2016, 17:15. Room: Physics: Benjamin Sudakov
TBA (Kolloquium)
Abstract
April 7, 2016, 10:15. Room: Math 0.102: Davide Bolognini (Marburg/Genova)
Betti splitting of monomial ideals and simplicial complexes
Abstract Splitting a topological space in order to describe its invariants from the invariants of its pieces is a very old idea, due to Mayer and Vietoris. Nevertheless, the algebraic version of this approach for ideals and simplicial complexes, the so-called Betti splitting theory, is relatively new. We say that a homogeneous ideal I admits a Betti splitting I=J+K if the graded Betti numbers of I can be determined in terms of the graded Betti numbers of the ideals J, K and the intersection of K and J. We present some new results on this topic, generalizing several previous results in the literature and showing that they are suitable for recursive procedures in a combinatorial context. Moreover we get information on the Alexander dual ideals of simplicial complexes, in particular we relate some nice splitting properties of triangulated manifolds with orientability and Discrete Morse Theory.
March 8, 2016, 10:15. Room: Math 0.102: Alessandro Oneto (Stockholm University)
Power ideals - between Froeberg conjecture and fat points
Abstract Ideals generated by powers of linear forms in polynomial rings are called power ideals. In the last decades, power ideals have been largely studied especially due to their connections with different areas of commutative algebra, algebraic geometry and combinatorics. In this talk, we want to describe some of these algebraic and geometric connections: from an algebraic point of view, we relate the Hilbert series of power ideals to the Fröberg conjecture on Hilbert series of generic ideals; from a geometric point of view, we relate them, via apolarity theory, to ideals of fat points in projective spaces.
March 1, 2016, 10:15. Room: Math 0.102: No seminar
(Group meeting)
Abstract
February 22, 2016, 13:15. Room: Math 0.102: Kolja Knauer (Aix-Marseille)
Complexes of oriented matroids
Abstract We introduce complexes of oriented matroids (COMs) and discuss their basic theory. COMs are a natural common generalization of oriented matroids and lopsided sets, where the latter were originally introduced by Lawrence as an asymmetric counterpart to oriented matroids. In fact, the generalization is so natural that it is surprising, that nobody did it before, e.g., realizable COMs, arise from intersecting a central hyperplane arrangement with a polyhedron. As a particular instance of a realizable COM we discuss the set of weak extensions of a poset. Generally, notions such as covectors, cocircuits, minors, and tope graphs carry over to COMs. Finally, we establish an amalgamation process allowing to construct any COM by glueing oriented matroids, leading to new classes of (locally realizable) COMs such as non-positively curved Coxeter zonotopal complexes. (joint with H.-J. Bandelt and V. Chepoi)
January 22, 2016, 00:00. Room: Lonza: Farhad Babaee (ENS Paris)
On approximability of extremal tropical currents
Abstract I will briefly explain complex tropical currents, and will address their extremality, intersection theory, and approximation problems. I will discuss how in joint work with June Huh, we constructed an example of a non-approximable tropical current, which, in turn, refutes a strong version of the Hodge conjecture for positive currents.
December 17, 2015, 10:15. Room: Math I 0.101: Mentor Stafa (ETHZ)
The topology of the commuting variety in a Lie group
Abstract We study the space of pairwise commuting elements in a compact and connected Lie group. In this talk we will focus on the rational and later integral cohomology of these spaces. The space of commuting pairs, or commuting variety, can be used to describe the integral cohomology of these spaces via the Mayer-Vietoris spectral sequence. I will give a brief overview on the current state of the problem and then describe some possible ways to approach it.
December 10, 2015, 10:15. Room: Math I 0.101: Olof Bergvall (Stockholm U)
Equivariant cohomology of toric arrangements associated to root systems
Abstract An arrangement is a finite set of closed subvarieties of a variety. Despite their simple definition, arrangements are of interest to a wide range of areas of mathematics such as algebraic geometry, topology, combinatorics, Lie theory and singularity theory. Classically, most attention has been given to arrangements of hyperplanes in an affine or projective space but during the past two decades toric arrangements, i.e. arrangements of subtori in a torus of higher dimension, have gained interest. This talk will give a brief introduction to the subject of arrangements and say something about what happens beyond the hyperplanes. In particular, we will see how to compute the Poincaré polynomial of (the complement of) an arrangement of hypertori and how this can be applied to compute cohomology of a moduli space of curves with level two structure. If time permits we will also see how to compute the total cohomology of (the complement of) the toric arrangement associated to the root system A_n as a representation of the corresponding Weyl group.
December 7, 2015, 17:30. Room: Perolles II G120: Bernd Sturmfels (UC Berkeley)
Plancherel lecture 2015 [website]
Beyond Linear Algebra
Abstract Linear algebra is the foundation of scientific computing and its numerous applications. Yet, the world is nonlinear. In this lecture we argue that it pays off to work with models that are described by nonlinear polynomials, while still taking advantage of the power of numerical linear algebra. We offer a glimpse of applied algebraic geometry, by discussing current trends in tensor decomposition, polynomial optimization, and algebraic statistics. Students will especially enjoy the illustrations of these concepts by many colorful pictures.
December 3, 2015, 10:15. Room: Math I 0.101: Giovanni Gaiffi (Pisa)
Models of reflection arrangements and related combinatorial structures
Abstract Let us consider, in a real or complex vector space V, an hyperplane arrangement A whose hyperplanes generate a (real or complex) finite reflection group W. We will focus on the topological and combinatorial properties of the De Concini-Procesi models associated with A. Some of these properties are well known: for instance, if A is the braid arrangement (and therefore W = Sn), the minimal complex model associated to it is the moduli space of stable genus 0 curves with n + 1 points. We will point out the existence of a combinatorial action of a ‘big’ symmetric group on the boundary strata of these models and we will show how this action leads to find non recursive formulas for the computation of Betti numbers of the models and of the faces of some polytopes (nestohedra) associated to this construction.
November 25, 2015, 15:15. Room: Lonza: Fatemeh Mohammadi (TU Berlin)
The geometry and combinatorics of syzygies: divisor theory of graphs and system reliability theory
Abstract I will talk about the application of syzygy tool from commutative algebra in two concrete examples arisen in divisor theory of graphs and system reliability theory. Attached to any finite graph G, there is a natural lattice (the ``lattice of integral cuts'') whose Delaunay cell decomposition is related to graphic hyperplane arrangements. There is also a canonical ideal I_G which encodes the linear equivalences of divisors on G. The ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices can be used to read the Betti numbers of I_G in terms of the number of faces of various dimensions in the graphic hyperplane arrangement, or equivalently, the number of orbits of the Delaunay cells of various dimensions in the cut lattice. The ideal I_G is also appearing in the theory of system reliability, and its Hilbert series encodes the reliability of the system which is the probability that G is connected (assuming that the vertices are reliable but each edge may fail with the probability p_e). This develops new connections between the theory of oriented matroid, the theory of divisors on graphs, and the theory of system reliability.
November 19, 2015, 11:15. Room: Math I 0.101: Hugo Parlier (Fribourg)
Triangulations and combinatorial moduli spaces
Abstract A convenient way to define a distance between surface triangulations is to measure how many flip transformations are needed to go from triangulation to another. For a fixed surface type, you get a graph (a flip graph) with triangulations as vertices and edges between triangulations related by a single flip. Flip graphs come up in a variety of contexts and, depending on the surface type, can either be finite or infinite graphs. A well studied example of a finite flip graph is when the surface is a polygon. However, once the surface has enough topology, the graphs are infinite and as metric spaces turn out to be quasi-isometric to the underlying mapping class groups (groups of self homeomorphisms of the surface up to isotopy). Surface homeomorphisms act naturally on these graphs and the quotient graphs are a type of combinatorial moduli space.
November 13, 2015, 11:15. Room: Lonza: Alberto Ravagnani (Neuchatel)
Coding theory and combinatorics
Abstract In coding theory, a MacWilliams identity expresses linear relations between the weight distribution of a code and the weight distribution of the dual code. The transformation is described by certain numbers called Krawtchouk coefficients. When studying additive codes in finite abelian groups, the code and the dual code are subsets of different ambient spaces, and their weight enumerators refer in general to different weight functions. Invertible MacWilliams identities hold when the weights are mutually compatible. A major problem in this area is the construction of mutually compatible weights, and the computation of the associated Krawtchouk coefficients. Using a combinatorial method, we construct a family of mutually compatible weight functions on finite abelian groups that yield invertible MacWilliams identities. The weights are obtained composing a support map with the rank function of a graded lattice with certain regularity properties. We express the corresponding Krawtchouk coefficients in terms of the combinatorial invariants of the underlying lattice, giving a method to compute them. The most important weight functions studied in coding theory belong, up to equivalence, to the class of weight functions that we introduce. This allows in particular to compute some classical Krawtchouk coefficients with combinatorial techniques. In the last part of the talk we show applications of our approach to some enumerative problems over finite fields.
October 27, 2015, 10:15. Room: Math I 0.101: Spencer Backman (Roma 1)
Graph Fourientations and the Tutte Polynomial
Abstract A fourientation of a graph is a choice for each edge whether to orient that edge in either direction, leave it unoriented, or biorient it. Fourientations can naturally be viewed as a mixture of graph orientations and subgraphs where unoriented and bioriented edges play the roles of absent and present edges, respectively. The Tutte polynomial is the most general bivariate polynomial associated to a graph which can be defined using the deletion and contraction of edges. I will describe joint work with Sam Hopkins where we investigate properties of cuts and cycles in fourientations which determine classes of fourientations which are enumerated by Tutte polynomial evaluations. Time permitting, I will illustrate how several of these classes relate to algebraic, combinatorial, and geometric topics such as Riemann-Roch theory for graphs, bigraphical arrangements, Lawrence ideals, zonotopal algebras, and the reliability polynomial.
October 16, 2015, 14:15. Room:: No seminar (Hyperbolic geometry days)
Abstract TBA
October 9, 2015, 14:15. Room: Lonza: Ioan Manolescu (Fribourg)
Minicourse on random maps
Abstract TBA
September 18, 2015, 14:15. Room: Lonza: Katharina Jochemko (TU Wien)
Combinatorial positivity of translation-invariant valuations and a discrete Hadwiger theorem
Abstract For a lattice polytope P and a natural number n the function counting lattice points in the n-th dilate nP is given by a polynomial E(n) in n -- the Ehrhart polynomial of P. Motivated by applications in algebra and combinatorics, the problem of characterizing Ehrhart polynomials has received considerable attention. A fundamental result in this direction was obtained by Stanley who showed with methods from commutative algebra that the entries of the Ehrhart h^*-vector are always nonnegative integers and satisfy a monotonicity property. We introduce the notion of combinatorial positivity and combinatorial monotonicity of general translation-invariant valuations on convex polytopes that extends the nonnegativity and monotonicity of Ehrhart h^*-vectors. We show that combinatorial positivity and combinatorial monotonicity are, in fact, equivalent and we give a simple characterization of all valuations with these properties. This in turn implies Stanley's nonnegativity and monotonicity of Ehrhart h^*-vectors and generalizes work of Beck et al. (2010) from solid-angle polynomials to all translation-invariant simple valuations. For general polytopes, this yields a new characterization of the volume as the unique combinatorially positive valuation up to scaling. For lattice polytopes our results extend work of Betke--Kneser (1985) and give a discrete Hadwiger theorem: There is essentially a unique combinatorially positive basis for the space of lattice-invariant valuations. This is joint work with Raman Sanyal.
May 26, 2015, 11:00. Room: Lonza: Relinde Jurrius (Neuchatel)
Application of hyperplane arrangements to weight enumeration
Abstract 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.
May 8, 2015, 11:15. Room: Math I: Nicholas Proudfoot (U Oregon)
Toric and Hypertoric combinatorics
Abstract 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.
April 24, 2015, 13:00. Room: Lonza: Tim Lindemann and Daniel Ris
`Seminaire Libre'/'Freies Seminar' on polytope theory
Abstract For a lattice polytope P and a natural number n the function counting lattice points in the n-th dilate nP is given by a polynomial E(n) in n -- the Ehrhart polynomial of P. Motivated by applications in algebra and combinatorics, the problem of characterizing Ehrhart polynomials has received considerable attention. A fundamental result in this direction was obtained by Stanley who showed with methods from commutative algebra that the entries of the Ehrhart h^*-vector are always nonnegative integers and satisfy a monotonicity property. We introduce the notion of combinatorial positivity and combinatorial monotonicity of general translation-invariant valuations on convex polytopes that extends the nonnegativity and monotonicity of Ehrhart h^*-vectors. We show that combinatorial positivity and combinatorial monotonicity are, in fact, equivalent and we give a simple characterization of all valuations with these properties. This in turn implies Stanley's nonnegativity and monotonicity of Ehrhart h^*-vectors and generalizes work of Beck et al. (2010) from solid-angle polynomials to all translation-invariant simple valuations. For general polytopes, this yields a new characterization of the volume as the unique combinatorially positive valuation up to scaling. For lattice polytopes our results extend work of Betke--Kneser (1985) and give a discrete Hadwiger theorem: There is essentially a unique combinatorially positive basis for the space of lattice-invariant valuations. This is joint work with Raman Sanyal.
April 24, 2015, 11:00. Room: DOKPE 0.26: Matteo Varbaro (Genova)
On dual graphs of complete intersections
Abstract 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 22, 2015, 10:00. Room: Lonza (joint w./ Oberseminar Geometrie): Viktoriya Ozornova (Bremen / MPI Bonn)
Factorability structures and rewriting systems
Abstract
April 14, 2015, 17:00. Room: Physik II (Kolloquium): Mario Salvetti (Pisa)
TBA
Abstract
March 25-27, 2015: CUSO doctoral school ``Let's matroid''
in Neuchatel [go to webpage]
March 20, 2015, 10:00. Room: Lonza: Ivan Martino
Minicourse: Toric varieties
Abstract I plan to give the definition of a toric variety and to play with several toy examples. I plan to construct a basis for the (co)homology ring of a smooth, complete and simplicial toric variety only using combinatorial tools as the notion of shellability (of a simplicial complex). The lectures will be at an introductory level and will only require a minimum of knowledge of Commutative Algebra (notion of affine algebraic variety and its ideal) and Combinatorics (notion of abstract simplicial complex).
March 13, 2015, 13:00. Room: Lonza: Ornella Greco (KTH Stockholm)
The Betti table of Veronese Modules
Abstract 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 6, 2015, 13:00. Room: Lonza: Max Wakefield (USNA Annapolis)
A deformed Moebius algebra and a matroid Kazhdan-Lusztig basis
Abstract 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 3, 2015, 17:00. Room: Physik II: Michael Cuntz (Hannover)
Simplicial arrangements
Abstract 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.
December 5, 2014, 16:00. Room: Lonza: Ann Kiefer (Fribourg)
Units of integral group rings and discontinuous group actions
Abstract
December 1-3, 2014: We are at the workshop on 'Arrangements of hyperplanes' at institute ALTA in Bremen.
November 27, 2014, 15:00. Room: Lonza: Michele Torielli (Hokkaido)
Supersolvable arrangements and their minimal CW-complex
Abstract In this talk I will recall the notion of supersolvable arrangements and introduce a combinatorial way to describe the minimal CW-complex structure of the complement. This is a joint work with S. Settepanella.
November 27, 2014, 14:00. Room: Lonza: Matthias Lenz (Oxford)
On the volume and the number of integer points of a convex polytope
Abstract Let \(X\) be a \( (d\times N) \)-matrix. We consider the variable polytope \(\Pi_X(u) = \{ w \ge 0 : X w = u \}\). It is known that the function \( T_X \) that assigns to a parameter \( u \in R^d \) the volume of the polytope \(\Pi_X(u) \) is piecewise polynomial. Formulas of Khovanskii-Pukhlikov, Brion-Vergne, De Concini-Procesi-Vergne, and the speaker imply that the number of integer points in \(\Pi_X(u)\) can be obtained by applying certain differential operators to the function \( T_X \). In this talk I will explain the facts mentioned above and talk about the underlying algebraic and combinatorial structures such as Dahmen-Micchelli spaces and their duals, matroids and arithmetic matroids, as well as hyperplane arrangements and toric arrangements.
November 6, 2014, 14:00. Room: Lonza: Elia Saini (Fribourg)
Betti number of Tutte group of a matroid.
Abstract
October 31, 2014, 14:00. Room: Lonza: Martin Herrmann (Fribourg)
Curvature and cohomology rings
Abstract
October 16, 2014, 14:00. Room: Lonza: Ivan Martino (Fribourg)
The cohomology of complete smooth toric varieties using shellability
Abstract
October 9, 2014, 15:00. Room: Lonza: Bram Petri (Fribourg)
Random surfaces
Abstract
October 9, 2014, 14:00. Room: Lonza: Federica Fanoni (Fribourg)
Hyperbolic surfaces, systoles and kissing numbers.
Abstract
October 3, 2014, 14:30. Room: 0.102: Emanuele Delucchi (Fribourg)
Minicourse part III: Spectral sequences
Abstract
October 3, 2014, 11:00. Room: 0.102: Emanuele Delucchi (Fribourg)
Minicourse part II: Homotopy colimits
Abstract
October 2, 2014, 14:30. Room: 0.102: Emanuele Delucchi (Fribourg)
Minicourse part I: Poset topology
Abstract
October 1-3, 2014: Sonja is visiting, we'll have a minicourse on combinatorial topology - check back for time and place.