Oberwolfach Workshop: Geometric Algebraic and Topological Combinatorics 2023, part I

This week I participate (remotely) in an exciting Oberwolfach meeting on Geometric Algebraic and Topological Combinatorics. See this post about a whole sequence of related meetings. Let me tell you first about the first two lectures.

Federico Ardila,  Recent developments in the Intersection theory of matroids.

The Chow ring records the intersection of subvarieties of an algebraic variety and Federico talked about four different way to define the Chow ring of a toric variety: One of the ways due to Lou Billera is based on the study of splines. Other ways are related to early works of Sturmfels and Ardila and  Klivans, and quite a few others. (Federico referred to a future survey paper where this notions are explained, and here is an older related survey by him The geometry of geometries: matroid theory, old and new.) Federico explained how the different representations of the Chow ring enable different proofs of combinatorial results such as unimodality of coefficients of chromatic polynomials.

Vic Reiner, Stirling numbers and Koszul algebras with symmetry

We start with combinatorial objects like the classic Stirling number of the first and second kind and move on to graded algebraic objects whose Hilbert functions (that record the dimensions of the graded part)  correspond to these combinatorial objects. (This is the theme of combinatorial commutative algebra.) Next we refine the Hilbert functions in terms of irreducible representations of  a natural group action. (This is the theme of combinatorial representation theory.) Vic discussed all sort of miracles including the Koszul property and the phenomenon of representation stability. Just when Vic talked about the definition of Koszul algebras there were sirens and I had to leave the lecture to a shelter for about five minutes.

In the third lecture, “A Regular Unimodular Triangulation of the Matroid Base Polytope” Gaku Liu presented (almost with full details) an ingenious inductive proof that every matroid polytope has a regular unimodular triangulation.

Of course, remote participation in a conference is not as good as real participation. There are many friends among the participants that I would like to talk with about their mathematical and other endeavors, and there are usually quite a few informal academic and other activities. Also the Oberwolfach atmosphere, scenery and library are quite inspiring. (I heard from colleagues also good things about the food.) On the other hand, zoom participating is not as bad as not participating at all, and possibly in remote lectures I find myself less day-dreaming and not paying attention to the talks, but I am not sure about it.

Let me give you a bird-eye view of the talks in the first three sessions. Our meetings in this series are eclectic and I once described them as meetings where each participant is a member of a very-appreciated minority. Naturally, I can be more precise regarding things I am more familiar with and, for the purpose of making this post timely, I will not do justice to speaker’s coauthors and collaborators. There are many things I would love to come back to later in the blog. Any corrections and additions, privately or in the comment section are welcome.

Monday morning

I already mentioned Gaku’s talk about unimodular triangulations and an open problem that I remember is to understand cases where there are such triangulations that are “flag” (no missing faces of dimension 2 or more.)

Monday afternoon

Eran Nevo gave a talk about rigidity expander graphs. Rigidity extends connectivity and expansion is a quantitative form of connectivity. Eran and his partners study the spectral notion of rigidity to -rigidity. Hailun Zheng, talked about Stress spaces, reconstruction problems and lower bound problems. Certain high dimensional stress spaces describe the – numbers of polytopes and Hailun and her collaborators answer questions that I raised long ago whether the spece of stresses and the corresponding skeleton, determine the combinatorial structure, and at times, even the affine structure of a simplicial polytope. Patricia Hersh, described a relaxation of the notion of recursive atom ordering that still implies CL-shellability. These terms refer to famous notions of shellability of posets that were especially useful for the study of Coxeter groups. The work provides new avenues for proving shellability.

Tuesday morning

Florian Frick talked about topological methods in zero-sum Ramsey theory. The starting point was a topological proof for the Erdos-Ginzburg-Ziv theorem. Erdos, Ginzburg and Ziv proved that every sequence of elements of ℤ_n with length at least 2n – 1 contains a subsequence of length n with a zero sum, and this may be the starting point to many results in additive combinatorics and Florian’s lecture may promise topological methods in this area. Pablo Soberon, talked about High-dimensional envy-free partitions. Suppose that you have several measures in say . It is a classical question to find partitions (of various kind) to equal parts with respect to all these measures. Envy-free partitions deal with the case that the th measure represent the utility of an individual, the -th individual. Envy free partition is a partition where every individial recieves at least as all others according her measures. This is a classic notion in game theory and now envy-free ham sandwiches could be a very nice direction. Kevin Piterman, talked about fixed points on contractible spaces. He emphasized group acting on 2-dimensional contractible spaces.   Here there are classical results and questions by Smith, Oliver, Quillen, Ashbacher, Segev, and others (I know Segev personally) and Kevin presented a solution to a central problem about the existence of fixed points for every finite group acting on compact complex. Roy Meshulam, lecture on random balanced Cayley complexes was a very rich blend of combinatorial, topological, Fourier-theoretical, and algebraic methods.

Let me stop here (publish the post in a drafty mode to be improved later) and continue for the other days in a separate post. In a few minutes our Problem Session starts!

Update: Let me tell you the fascinating last problem of the problem session by Pablo Soberon:

Consider lines in the plane with no pairs of parallel lines. Define the “measure” of a set as the maximum number of lines all of their pairwise intersections are in (or 1 otherwise). Consider a (Voronoi) partition of the plane to three cones . The conjecture is that the product of the measures is at least .