Kalina Mincheva

Algebra and Combinatorics Seminar

We meet weekly on Wednesday at 3:00 PM (Central time), TBA

Organizers: Daniel Bernstein, Mahir Can, Tài Huy Hà, Kalina Mincheva.

Spring 2023

• Kalina Mincheva, Tulane University.
  Tropical power series I - algebraic aspects. abstract.
  Working towards endowing tropical varieties with extra structure, we study the algebra of convergent tropical power series and the topological spaces (of prime congruences) it corresponds to. We characterize the (nice) prime congruences of this algebra and we show that the dimension behaves as expected.

• Nati Friedenberg, Tulane University.
  Tropical power series II - geometric aspects. abstract.
  Continuing last week's talk, we start with explaining some of the nuance of prime congruences. We then use this to show that the points of our topological space are in bijection with equivalence classes of flags of polyhedra and give some pretty pictures.

• Gene Kopp, LSU.
  Class field theory for orders of number fields. abstract.
  Class field theory abstractly specifies the abelian Galois extensions of a number field in terms of data intrinsic to the base field. The classical formulation involves ray class groups and associated ray class fields. Not every abelian extension is a ray class field, but every abelian extension is contained in some ray class field. There are also ring class groups associated to arbitrary orders in the base field, with associated ring class fields, this time not containing or generating arbitrary abelian extensions, but arising naturally, for example, in the theory of complex multiplication. We define a "ray class group of an order" and a "ray class field of an order," common generalizations of the ray and ring class concepts. We explain how these objects fit in to class field theory and its applications. Along the way, we encounter some of the pitfalls of working with non-maximal orders. This is joint work with Jeffrey Lagarias.

• Larry Rolen, Vanderbilt University.
  L-functions for Harmonic Maass Forms. abstract.
  The theory of harmonic Maass forms and mock modular forms has seen an explosion of activity in the past 20 years, with applications to physics, partitions, enumerative geometry, and many other topics. Along the way, much has been developed in the theory of harmonic Maass forms. However, until recently, harmonic Maass form theory lacked analogues of key structures that exist for classical holomorphic modular forms and Maass waveforms, such as the theory of L-functions. Recent work with Diamantis, Lee and Raji will be described which gives the first general such theory. In particular, I will describe how we obtain new Weil-type Converse Theorems and a Voronoi-type summation formula in these settings. I will also describe connections with the construction of differential operators on these spaces and a more thorough explanation of a previous formula for a central L-value of the j-invariant, which had been discovered heuristically by Zagier and proven in that case by Bruinier, Funke, and Imamoglu.

• Hernán Iriarte, Uinversity of Texas, Austin.
  Higher rank tropical and polyhedral geometry. abstract.
  We start by giving an overview of what is currently known about tropicalization of algebraic varieties with respect to valuations of rank different from one. In this context, a tropical variety is given by a fibration in which the base and each fiber are tropical varieties as usual. These fibrations admit the structure of a polyhedral complex with coefficients in the ordered ring R[x]/(xk). Moreover, we will show how in the case of hypersurfaces, we can completely understand the combinatorics of this fibration from a layered regular subdivision of its Newton polytope.

• Peter Marcus, Tulane University.
  Transfer Systems. abstract.
  Transfer systems are combinatorial objects that arise in equivariant homotopy theory. They are defined as a certain type of partial order on the set of subgroups of a fixed finite group. The central question is enumerating all possible transfer systems for a given group. I will discuss this and other related results.

• May Cai, Georgia Tech.
  Symmetric (tropical) rank 2 matrix completion. abstract.
  The matrix completion problem asks what partially-filled-in matrices can be "completed" to a matrix of certain desired properties. For this talk, we are concerned with completion to a symmetric rank 2 matrix. Equivalently, we may ask for the independent sets of the algebraic matroid arising from the variety of symmetric $n \times n$ rank 2 matrices. In this talk, we describe a combinatorial characterisation of these independent sets. We solve the problem by tropicalizing the variety, which is to say solving the analogous problem for symmetric tropical rank 2 matrices. Based on joint work with Cvetelina Hill, Kisun Lee, and Josephine Yu.

• Sergey Grigorian, University of Texas Rio Grande Valley.
  Lie-like structures on parallelizable manifolds. abstract.
  In this talk we will explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold $\mathbb{L}$, there exists a global trivialization of the tangent bundle, which defines a map $\rho_p:\mathfrak{l} \longrightarrow T_p \mathbb{L}$ for each point $p \in \mathbb{L}$, where $\mathfrak{l}$ is some vector space. This allows us to define a particular class of vector fields, known as fundamental vector fields, that correspond to each element of $\mathfrak{l}$. Furthermore, flows of these vector fields give rise to a product between elements of $\mathfrak{l}$ and $\mathbb{L}$, which in turn induces a local loop structure (i.e. a non-associative analog of a group). Furthermore, we also define a generalization of a Lie algebra structure on $\mathfrak{l}$. We will describe the properties and applications of these constructions.

• Jaiung Jun, SUNY New Paltz.
  Hall algebras in a non-additive setting and combinatorial Hopf algebras. abstract.
  The Hall algebra of an abelian category (satisfying some finiteness conditions) naturally encodes the structures of the space of extensions between objects in an abelian category. Recently, Dyckerhoff and Kapranov introduced a notion of proto-exact categories, as a non-additive generalization of an exact category, allowing one to associate Hall algebras. This framework is well suited to study categories whose objects are ``combinatorial''. In this talk, I will discuss several examples of proto-exact categories and its connections to combinatorial Hopf algebras. The talk is based on collaborations with C. Eppolito, A. Sistko, and M. Szczesny.

• TBA.
  TBA. abstract.
  TBA.