The Magic Number Problem for Subregular Language Families

We investigate the magic number problem, that is, the question whether there exists a minimal n-state nondeterministic finite automaton (NFA) whose equivalent minimal deterministic finite automaton (DFA) has alpha states, for all n and alpha satisfying n less or equal to alpha less or equal to exp(2,n). A number alpha not satisfying this condition is called a magic number (for n). It was shown in [11] that no magic numbers exist for general regular languages, while in [5] trivial and non-trivial magic numbers for unary regular languages were identified. We obtain similar results for automata accepting subregular languages like, for example, combinational languages, star-free, prefix-, suffix-, and infix-closed languages, and prefix-, suffix-, and infix-free languages, showing that there are only trivial magic numbers, when they exist. For finite languages we obtain some partial results showing that certain numbers are non-magic.Comment: In Proceedings DCFS 2010, arXiv:1008.127

Quotient Complexity of Regular Languages

The past research on the state complexity of operations on regular languages is examined, and a new approach based on an old method (derivatives of regular expressions) is presented. Since state complexity is a property of a language, it is appropriate to define it in formal-language terms as the number of distinct quotients of the language, and to call it "quotient complexity". The problem of finding the quotient complexity of a language f(K,L) is considered, where K and L are regular languages and f is a regular operation, for example, union or concatenation. Since quotients can be represented by derivatives, one can find a formula for the typical quotient of f(K,L) in terms of the quotients of K and L. To obtain an upper bound on the number of quotients of f(K,L) all one has to do is count how many such quotients are possible, and this makes automaton constructions unnecessary. The advantages of this point of view are illustrated by many examples. Moreover, new general observations are presented to help in the estimation of the upper bounds on quotient complexity of regular operations

A Hecke Algebra Approach to p-adic Functionals

University of Minnesota Ph.D. dissertation. August 2019. Major: Mathematics. Advisor: Benjamin Brubaker. 1 computer file (PDF); vi, 95 pages.Unique model spaces for representations of reductive groups over $p$-adic fields play an integral role in the theory of automorphic forms (where for our purposes, 'unique' means the decomposition of the model space is multiplicity-free). Uniqueness facilitates precise computation of special functions in the model as in the work of Casselman, Shalika, and Shintani \cite{casselman, casselman-shalika, shintani}, and is a common feature of local components of global integral representations of $L$-functions, as in Godement and Jacquet \cite{godement-jacquet}, and Ginzburg and Rallis \cite{ginzburg-rallis}. Here we study uniqueness of local model spaces with respect to 'universal unramified principal series' as outlined in Haines, Kottwitz, and Prasad \cite{HKP}. In this program, the convolution algebra of compactly supported, Iwahori-biinvariant functions on G(\Q_p) (the \emph{Iwahori-Hecke algebra}, henceforth denoted as \mc{H}), provides powerful algebraic structure to the theory of $p$-adic group representations and allows one to simultaneously study the full unramified principal series. A recurring theme is that unique models and unique functionals on unramified principal series representations are associated to Hecke algebra modules of the form \ind_{\mc{H}_0}^\mc{H} \varepsilon, where \mc{H}_0 is the finite Hecke algebra consisting of functions in \mc{H} supported on the integer points in G(\Q_p) and $\varepsilon$ is a linear character of \mc{H}_0. Brubaker, Bump, and Friedberg \cite{BBF} show that many standard unique functionals map to \emph{left} induced \mc{H}-modules of this form, and Chan and Savin \cite{chan-savin-bessel, chan-savin-iwahori} show that the Iwahori-fixed vectors in certain standard unique model spaces are associated to \emph{right} \mc{H}-modules of this form.\\ We explore and expand this program in several ways. We provide sufficient conditions for an \mc{H}-module to be of the form \ind_{\mc{H}_0}^\mc{H} \varepsilon, expanding the $GL_n$ case described in \cite{CS-BZ}. We show that \emph{left} \mc{H}-modules on the functional studied by Brubaker, Bump, and Friedberg \cite{BBF} are essentially the same as the \emph{right} \mc{H}-modules on the model side identified by Chan and Savin \cite{chan-savin-bessel, chan-savin-iwahori}. We then classify, under certain conditions, the \mc{H}-modules which are associated to either unique modules or unique functionals. Finally, we investigate possible generalizations of this theory to finite multiplicity (but not unique) model spaces, specifically the generalized Gelfand-Graev representations of Kawanaka \cite{KawI, KawS} (\emph{generalized Whittaker models} in the program of Moeglin and Waldspurger \cite{MW}) of both G(\F_q) and G(\Q_p)

Compound Du Val singularities, five-dimensional SCFTs and GV invariants

In this thesis work we introduce a new method to study the dynamics of M-theory on compound Du Val threefold singularities (cDV). Incidentally, this also furnishes a new way to systematically count the Gopakumar-Vafa invariants (GV) of these geometries and, reversely, to produce threefolds whose GV invariants display required properties. Our construction is inspired by the type IIA limit of M-theory on the considered singularities and rephrases the data of the threefolds in the language of seven-dimensional super Yang-Mills theory. This, more deeply, creates a connection between the algebraic properties of the ADE algebras and the geometric properties of the cDVs. We focused our analysis on two interesting classes of compound Du Val: the simple flops and the quasihomogeneous cDVs, obtaining in both cases complete information on the GV invariants (or, equivalently, on the Higgs Branch of M-theory reduced on these singularities). We also elucidate, during this procedure, the role of exotic type IIA branes bound states, called T-branes, that lack a clear interpretation in terms of the geometry of the threefold