Venue: University of Fribourg, "Pérolles II" (Building PER 21), room G 230.
Maps: [map of the Pérolles-campus] [location of Pérolles inside the town]

Contact: Emanuele Delucchi, emanuele.delucchi "at'' unifr.ch

Schedule:

		Monday, August 28th	Tuesday, August 29th
	9:00	Laura Anderson (SUNY Binghamton)	Jaiung Jun (SUNY Binghamton)
	10:00	Coffee break	Coffee break
	10:20	Alex Fink (Queen Mary, London)	Rudi Pendavingh (TU Eindhoven)
	11:20	Masahiko Yoshinaga (Hokkaido)	Elia Saini (Fribourg)
		Lunch	Lunch
	13:45	Jan Draisma (Bern)	Problem session / discussions
	14:40	Jeffrey Giansiracusa (Swansea)	(open ended)

Titles and abstracts of talks (Chronological)

• Laura Anderson (SUNY Binghamton)
  Vectors of matroids over hyperfields
  
• Alex Fink (Queen Mary, London)
  Why "matroids over rings" aren't so far from matroids over hyperfields
  
• Masahiko Yoshinaga (Hokkaido)
  G-Tutte polynomials
  Given a list of integer vectors, one can associate several mathematical objects, e.g., matroids, hyperplane arrangements, toric arrangements, zonotopes etc. Each has certain (quasi)polynomial invariants which possesses rich topological and enumerative information. Among others the Tutte polynomial detects Betti numbers of the complement of a complex hyperplane arrangement and the arithmetic Tutte polynomial acts similarly for toric arrangements. In this talk, we introduce G-Tutte polynomial for an abelian group G (with a weak assumption on the finiteness of torsions). Main examples are abelian Lie groups with finitely many connected components. It is a generalization of "Tutte polynomials" in the sense that G=C and C^* recovers Tutte and arithmetic Tutte polynomial, respectively. We see that many well known properties are shared also by G-Tutte polynomials. We also discuss the topology of the complement of corresponding "arrangements" for non-compact group G. This is a joint work with Ye Liu and Tan Nhat Tran (Hokkaido).
• Jan Draisma (Bern)
  Algebraic matroids and Frobenius Flocks
  In characteristic zero, every algebraic matroid admits a linear representation. In positive characteristic, it turns out that every algebraic matroid admits a representation by a Frobenius flock: a lattice worth of vector spaces that are connected by two simple axioms. These two axioms lead to a surprisingly rich theory: flocks always define matroids, they have contractions, deletions, and dual flocks, and they give rise to a partition of the lattice into cells that are both max-plus and min-plus closed. Flocks arise from other sources than algebraic matroids, as well, e.g. from linear spaces over valued fields---and even the Vamos matroid is flock-representable! Certain matroids are so rigid, that the cell structure of any flock representing them is necessarily a fan, and in this case the matroid is algebraic if and only if it admits a linear representation. This streamlines existing non-algebraicity results (notably by Lindstrom), and points towards generalisations of these results.
• Jeffrey Giansiracusa (Swansea)
  Idempotent exterior algebra, Matroids, and Grassmannians
  Grassmannians are an important example of moduli spaces in algebraic geometry (over rings). In this talk I will explore and contrast several distinct notions of Grassmannians that appear when passing to the setting of idempotent semirings. There is: (1) a moduli space of locally free quotients, which has an interesting topology but might not be representable; (2) the Dressian, which is described elegantly in terms of an idempotent version of exterior algebra; and (3) various tropicalizations of Grassmannians. I will discuss these objects from a scheme-theoretic and moduli-theoretic point of view.
• Jaiung Jun (SUNY Binghamton)
  On the relation between hyperrings and fuzzy rings.
  Abstract: We construct a full embedding of the category of hyperfields into Dress's category of fuzzy rings and explicitly characterize the essential image. This embedding provides an identification of Baker's theory of matroids over hyperfields with Dress's theory of matroids over fuzzy rings
• Rudi Pendavingh (TU Eindhoven)
  An introduction to parafields.
  
• Elia Saini (Fribourg)
  Rescaling classes of matroids over hyperfields
  We review axiom systems for matroids over hyperfields in terms of orthogonal circuit and cocircuit signatures, we introduce the notion of rescaling classes and we present some basic properties of the ``moduli space'' of rescaling classes. Next, we give a combinatorial characterization of our spaces. To do this, we define hyperfield projective classes of a matroid in terms of its cir- cuits and cocircuits and we algebraically describe those hyperfields for which there exists a one-to-one correspondence between the spaces of rescaling classes and the spaces of projective classes. Exploiting the structure of the inner Tutte group of a matroid, we give an algebraic description of the space of rescaling classes and we provide a geometric characterizations of the space of weak hyperfield projective classes of a matroid. We then focus on the phase hyperfield and we prove that in the phased case the bijections established in the general context of matroids over hy- perfields induce actual homeomorphic equivalences. Using some topolog- ical properties of these spaces we deduce the existence of non-realizable uniform weak phased matroids.