November 16, 2022: Matt Baker, Georgia Institute of Technology
Foundations of Matroids
Matroid theorists are interested in questions concerning representability of matroids over fields. More generally, one can ask about representability over partial fields in the sense of Semple and Whittle. Pendavingh and van Zwam introduced the universal partial field of a matroid, which governs the representations of over all partial fields. Unfortunately, most matroids are not representable over any partial field, and in this case, the universal partial field is not defined. Oliver Lorscheid and I have introduced a generalization of the universal partial field which we call the foundation of a matroid; it is always well-defined. The foundation is a type of algebraic object which we call a pasture; pastures include both hyperfields and partial fields. As a particular application of this point of view, I will explain the classification of all possible foundations for matroids having no minor isomorphic to U(2,5) or U(3,5). Among other things, this provides a short and conceptual proof of the 1997 theorem of Lee and Scobee which says that a matroid is both ternary and orientable if and only if it is dyadic. (Recording of Baker’s talk.)
November 30, 2022: Samarpita Ray, Indiana University
The topological shadow of -geometry: congruence spaces
In this talk, we introduce a topological shadow for monoid schemes, which we call the congruence space. It is constructed from monoid congruences (instead of ideals) and carries the “right” topological information about closed subsets. This allows us to extend topological characterizations (and definitions) of several notions from usual scheme theory to monoid schemes, such as closed immersions and subschemes, as well as separated and proper morphisms. This is a joint work with Oliver Lorscheid, and soon to appear.
January 25, 2023: Jaeho Shin, Seoul National University
Birational geometry of matroids and abstract hyperplane arrangements
A matroid is a machine that captures linearity of mathematical objects and produces combinatorial structures. Matroids arise everywhere, as linearity does. One of the most natural appearances of matroids is from hyperplane arrangements. Although much research on the three things: matroids, matroid polytopes, and hyperplane arrangements in a trilateral relation has been done, the gaps in our knowledge base remain substantial. For example, we do not have matroidal counterparts corresponding to matroid subdivisions, and only some matroid subdivisions are associated with stable hyperplane arrangements, which indicates that the trilateral relation needs improvement. Further, we need to learn more about the face structure of a matroid polytope and how to glue or subdivide base polytopes, which are inarguably essential questions. It is also an interesting question if the geometry of hyperplane arrangements can be implemented over matroids, and then how much of it. This talk will discuss how to develop a theory combining the three things into a trinity relation to answer as many questions as possible, such as the previously mentioned ones.
February 22, 2023: Netanel Friedenberg, Tulane University
Towards tropical adic spaces
After a brief introduction to tropical geometry, I will introduce a new approach to endowing tropical varieties with additional structure. The approach will be analytic in nature, and so we will focus on the algebra of tropical power series. We introduce the continuous spectrum of a topological semiring, and show that, in the case of tropical power series, it can be understood as a set of prime congruences on the algebra of tropical polynomials. If time permits, we will see that the dimension behaves as expected and that the points the continuous spectrum can be interpreted in terms of more classical geometry. (Recording of Fridenberg’s talk.)
March 22, 2023: Olivia Caramello, University of Insubria
Grothendieck toposes as unifying `bridges’ in mathematics
I will explain the sense in which Grothendieck toposes can act as unifying ‘bridges’ for relating different mathematical theories to each other and studying them from a multiplicity of points of view. I shall first present the general techniques underpinning this theory and then discuss a number of selected applications in different mathematical fields. (Recording of Caramello’ talk.)
April 5, 2023: Joachim Kock, Universitat Autonoma de Barcelona
Some basic steps in objective linear algebra
The ‘objective method’, advocated by Lawvere, seeks to calculate directly with combinatorial objects, rather than with their numbers. In algebraic combinatorics, this is largely a question of doing linear algebra over the ‘ground field’ of finite sets (or groupoids or infinity-groupoids). The role of vector spaces is played by slice categories, and the role of linear maps is played by linear functors (which here means colimit-preserving functors), which in turn can be represented by spans, so that ‘matrix multiplication’ is given by pullbacks. The ordinary vector-space level is recovered from the objective level by taking (homotopy) cardinality. After explaining the basic theory, the talk will focus on finiteness conditions and the objective version of the duality between vector spaces and pro-finite-dimensional vector spaces, including a neat combinatorial interpretation of continuity. To finish I will briefly outline how the basic set-up supports more elaborate algebraic structures, such as incidence bialgebras, Möbius inversion, and antipodes.
This is based on joint work with Imma Gálvez and Andy Tonks. (Recording of Kock’s talk.)
April 20, 2023: Inna Zakharevich, Cornell University
Coinvariants, assembler K-theory, and scissors congruence
For a geometry (such as Euclidean, spherical, or hyperbolic) with isometry group the scissors congruence group is defined to be the free abelian group generated by polytopes in , modulo the relation that for polytopes and that intersect only on the boundary, , and for , . This group classifies polytopes up to “scissors congruence,” i.e. cutting up into pieces, rearranging the pieces, and gluing them back together. With some basic group homology one can see that . Using combinatorial -theory can be expressed as the of a spectrum . In this talk, we will generalize this formula to show that, in fact, , and in fact more generally that this is true for any assembler with a -action. This is joint work with Anna Marie Bohmann, Teena Gerhardt, Cary Malkiewich, and Mona Merling. (Recording of Zakharevich‘s talk.)
May 3, 2023: Omid Amini, l’Ecole Polytechnique
May 17, 2023: Hernan Iriarte, University of Texas at Austin
Weak continuity on the variation of Newton Okounkov bodies
We start by presenting new tools and results suitable for the study of valuations of higher rank on function fields of algebraic
varieties. This will be based on a study of higher rank quasi-monomial valuations taking values in the lexicographically ordered group . This gives us a space of higher rank valuations that we endow with a weak “tropical” topology. In this setting, we show that the Newton Okounkov bodies of a given line bundle vary continuously with respect to the valuation. We explain how this result fits in the literature and how it gives us a restriction in the existence of mutations of Newton Okounkov bodies. Joint work with Omid Amini.