# About the F1 World Seminar

This is an online seminar organized by Jaiung Jun, Oliver Lorscheid, Yuri Manin, Matt Szczesny, Koen Thas and Matt Young

The central theme of the seminar is the mathematics of F1, the “field with one element,” and its connections to other areas of mathematics, including arithmetic, geometry, representation theory and combinatorics. Topics covered by the seminar include, but are not limited to:

• Foundational theory (monoid schemes, relative schemes, F1-schemes, Lambda-schemes, generalized schemes, blue schemes);
• Arithmetic (relations to motives and Arakelov geometry, relations to modular forms, ideas around the Riemann Hypothesis);
• Tropical geometry and matroids;
• K-theory of F1-schemes;
• Representation theory (quivers, Hall algebras, degenerations of quantum groups);
• Combinatorics (finite field geometry).

WHEN and HOW?

We typically meet on alternating Wednesdays from 9:30 AM – 10:30 AM Eastern Standard Time. There will be room for mathematical discussion after each lecture.

A detailed schedule can be found here.

# Schedule Fall 2022

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 $\mathbb{F}_1$-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.

# Schedule Spring 2022

January 19, 2022: Alexander Smirnov, Steklov Institute

The 10th Discriminant and Tensor Powers of $\mathbb{Z}$

We plan to discuss very shortly certain achievements and disappointments of the $\mathbb{F}_1$-approach. In addition, we will consider a possibility to apply noncommutative tensor powers of $\mathbb{Z}$ to the Riemann Hypothesis. (Recording of Smirnov’s talk.)

February 2, 2022Alain Connes, IHES

$\mathbb{F}_1$, $q$ and $\zeta$

This work is joint work with C. Consani. I will start from the role of the limit $q \rightarrow 1$ in the classical number theory formulas of Hasse-Weil when dealing with Riemann’s zeta function, and will then explore the various geometric paradigms corresponding to the limit. First the paradigm of characteristic one, which is tropical and then the paradigm of the sphere spectrum which is based on Segal’s gamma rings and leads to a new algebraic geometry. (Recording of Connes’ talk.)

February 16, 2022: Alex Sistko, Manhattan College

$\mathbb{F}_1$-Representations of Quivers and Euler Characteristics of Quiver Grassmannians

To any quiver $Q$, we can associate its category of finite-dimensional representations over $\mathbb{F}_1$. This is a finitary proto-exact category, which admits a version of the Krull-Schmidt Theorem and a Hall algebra. For any field $k$, there is also a faithful functor which carries $\mathbb{F}_1$-representations to $k$-representations: the case where $k$ is the complex numbers is of particular interest, where Euler characteristics of quiver Grassmannians find relevance to cluster theory. Recently, it was shown that the category of $\mathbb{F}_1$-representations of $Q$ admits a description via coefficient quivers. In this talk, we show how this description helps us generalize existing techniques to compute Euler characteristics of quiver Grassmannians and place them in a new context. We introduce the nice length of an $\mathbb{F}_1$-representation, and show that when this quantity is finite, there is a simple combinatorial interpretation to the Euler characteristics of the quiver Grassmannians. We recover several results from the literature, and identify new classes of $\mathbb{F}_1$-representations towards which the classic techniques apply. Time permitting, we also discuss the category of $\mathbb{F}_1$-representations with finite nice length and recent efforts to describe it. This is joint work with Jaiung Jun. (Recording of Sistko’s talk.)

March 2, 2022: Chris Eur, Harvard University

Tautological classes of matroids

Algebraic geometry has furnished fruitful tools for studying matroids, which are combinatorial abstractions of hyperplane arrangements. We first survey some recent developments, pointing out how these developments remained partially disjoint. We then introduce certain vector bundles (K-classes) on permutohedral varieties, which we call “tautological bundles (classes)” of matroids, as a new framework that unifies, recovers, and extends these recent developments. Our framework leads to new questions that further probe the boundary between combinatorics and geometry. Joint work with Andrew Berget, Hunter Spink, and Dennis Tseng. (Recording of Eur’s talk.)

March 16, 2022: Oren Ben-Bassat, University of Haifa

Derived Analytic Geometry

I will start by reviewing some work of others: sketching a (higher) categorical approach to geometry. After that, I will explain how using this approach (derived) analytic geometry can be viewed in a precise way as a type of algebraic geometry. I will explain and use in a fundamental way Banach rings and categories of Banach modules over a Banach ring.  The theories of derived analytic geometry from this perspective include both archimedean and non-archimedean analytic theories.  I will give examples of homotopy epimorphisms between algebras of analytic nature. I will include examples and relevance in the rigid analytic context and an arithmetic context. I will discuss descent theorems and time permitting, other topics such as blow-ups. (Recording of Ben-Bassat’s talk.)

March 30, 2022: Kalina Mincheva, Tulane University

Tropical Geometry and the Commutative Algebra of Semirings

Tropical geometry provides a new set of purely combinatorial tools, which has been used to approach classical problems. In the recent years, there has been a lot of effort dedicated to developing the necessary tools for commutative algebra using different frameworks, among which prime congruences, tropical ideals, tropical schemes. These approaches allows for the exploration of the  properties of tropicalized spaces without tying them up to the original varieties and working with geometric structures inherently defined in characteristic one (that is, additively idempotent) semifields. In this talk we explore the relationship between tropical ideals and congruences and what they remember about the geometry of a tropical variety. The talk will give some overview of recent results and work in progress. (Recording of Mincheva’s talk.)

April 13, 2022: Manoel Jarra, Instituto de Matemática Pura e Aplicada

Flag matroids with coefficients

Matroids encode the combinatorics of independence in linear subspaces, as well flag matroids do for flags of subspaces. There is a way back: when representing a (flag) matroid over a field, we get a (flag of) linear subspace(s). In recent years, Baker and Bowler generalized this picture to a more general type of algebraic object that includes fields and hyperfields as particular cases. In this talk we explain how to extend this theory to flag matroids, including a geometric interpretation in terms of their moduli space. This is a joint work with Oliver Lorscheid. (Recording of Jarra’s talk.)

April 27, 2022: Jacob Matherne, University of Bonn

Equivariant log-concavity, matroids, and representation stability

Adiprasito, Huh, and Katz proved that the Betti numbers of the Orlik-Solomon algebra of any matroid form a log-concave sequence.  Now suppose that the matroid has symmetries. Then, the Orlik-Solomon algebra becomes a graded representation of that symmetry group.  In this situation, I will conjecture an equivariant version of the log-concavity result above.  Then, I will show how one can use the theory of representation stability to prove infinitely-many cases of this conjecture for the braid matroid, acted on by the symmetric group.  This is joint work with Dane Miyata, Nicholas Proudfoot, and Eric Ramos. (Recording of Matherne’s talk.)

May 11, 2022: No seminar

May 25, 2022: Hendrik Van Maldeghem, Ghent University

Classical and exceptional geometries of order one

In the spirit of Jacques Tits’ original observation about geometries over the field of order 1, we review properties of these geometries. In particular we demonstrate how such properties can tell us something new about their analogues over proper fields. Along the way we discuss Galois descent, a Magic Square, triality and split octonions, all over Fun. (Recording of Van Maldeghem’s talk.)