Factorizations of tropical and sign polynomials

In this text, we study factorizations of polynomials over the tropical hyperfield and the sign hyperfield, which we call `tropical polynomials' and `sign polynomials', respectively. We classify all irreducible polynomials in either case. We show that tropical polynomials factor uniquely into irreducible factors, but that unique factorization fails for sign polyomials. We describe division algorithms for tropical and sign polynomials by linear terms that correspond to roots of the polynomials

Algebraic KK-theory and Grothendieck-Witt theory of monoid schemes

We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic $K$-theory space of an integral monoid scheme $X$ in terms of its Picard group $\operatorname{Pic}(X)$ and pointed monoid of regular functions $\Gamma(X, \mathcal{O}_X)$ and a description of the Grothendieck-Witt space of $X$ in terms of an additional involution on $\operatorname{Pic}(X)$. We also prove space-level projective bundle formulae in both settings

Group completion in the K-theory and Grothendieck-Witt theory of proto-exact categories

We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories, with a particular focus on classes of examples of $\mathbb{F}_1$-linear nature. Our main results are analogues of theorems of Quillen and Schlichting, relating the $K$-theory or Grothendieck-Witt theories of proto-exact categories defined using the (hermitian) $Q$-construction and group completion

Toroidal Automorphic Forms for Function Fields

The definition of a toroidal automorphic form is due to Don Zagier, who showed in a paper in 1979 that the vanishing of certain integrals of Eisenstein series over tori in GL(2) is related to the vanishing of the Riemann zeta function at the weight of the Eisenstein series; and thus a relation between the unitarizability of the space of unramified toroidal automorphic forms and the Riemann hypothesis. In the same paper, Zagier asked the question of what happens in the context of a function field of a curve over a finite field. In this thesis, we interpret automorphic forms over global function fields as functions on the vertices of certain graphs that can be described in terms of rank-2 bundles on this curve. By providing a reinterpretation of Hecke operators in terms of these graphs and the reinterpretation of cuspidality and toroidality, we are able to compute spaces of cusp forms and toroidal forms by looking at systems of linear equations. In particular, the question of the validity of the Riemann hypothesis can be reformulated as an eigenvalue problem