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, , Room: Math I 0. 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, , 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. This talk will give a brief introduction to the subject of arrangements and say something about what happens beyond the hyperplanes. December 7, , 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, , We will focus on the topological and combinatorial properties of the De Concini-Procesi models associated with A. 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.
This develops new connections between the theory of oriented matroid, the theory of divisors on graphs, and the theory of system reliability. November 19, , 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, , 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, , 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, , 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. We show that combinatorial positivity and combinatorial monotonicity are, in fact, equivalent and we give a simple characterization of all valuations with these properties. 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 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, , 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. Main article: Structural rigidity.
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Main article: Incidence structure. Main article: Oriented matroid.
Main article: Geometric graph theory. Main article: Simplicial complex. Main article: Topological combinatorics. Main articles: Lattice group and discrete group. Main article: Digital geometry. Main article: Discrete differential geometry. Bokowski, Chapter 1.
Alexander Postnikov: Combinatorics and Geometry
Ziegler, Chapter 7. Bokowski, Chapters For learning about oriented matroids, a good preparation is to study the textbook on linear optimization by Nering and Tucker, which is infused with oriented-matroid ideas, and then to proceed to Ziegler's lectures on polytopes.
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History of mathematics Recreational mathematics Mathematics and art Mathematics education. Category Portal Commons WikiProject. However, simplicial complexes are much more powerful for the discrete modeling of physical objects for the following reason: The topological space described by a combinatorial simplicial complex comes with a canonical piecewise affine structure.
We now elaborate what this means. The main property that makes simplicial complexes useful for physical modeling is the following easy-to-prove. Then there is a unique map. You must be logged in to post a comment.
Related Geometrical Combinatorial Topology: v. 2
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