Polyhedral geometry for lecture hall partitions

Lecture hall partitions are a fundamental combinatorial structure which have been studied extensively over the past two decades. These objects have produced new results, as well as reinterpretations and generalizations of classicial results, which are of interest in combinatorial number theory, enumerative combinatorics, and convex geometry. In a recent survey of Savage \cite{Savage-LHP-Survey}, a wide variety of these results are nicely presented. However, since the publication of this survey, there have been many new developments related to the polyhedral geometry and Ehrhart theory arising from lecture hall partitions. Subsequently, in this survey article, we focus exclusively on the polyhedral geometric results in the theory of lecture hall partitions in an effort to showcase these new developments. In particular, we highlight results on lecture hall cones, lecture hall simplices, and lecture hall order polytopes. We conclude with an extensive list of open problems and conjectures in this area.Comment: 20 pages; To appear in to proceedings of the 2018 Summer Workshop on Lattice Polytopes at Osaka Universit

HILBERT BASES, DESCENT STATISTICS, AND COMBINATORIAL SEMIGROUP ALGEBRAS

The broad topic of this dissertation is the study of algebraic structure arising from polyhedral geometric objects. There are three distinct topics covered over three main chapters. However, each of these topics are further linked by a connection to the Eulerian polynomials. Chapter 2 studies Euler-Mahonian identities arising from both the symmetric group and generalized permutation groups. Specifically, we study the algebraic structure of unit cube semigroup algebra using Gröbner basis methods to acquire these identities. Moreover, this serves as a bridge between previous methods involving polyhedral geometry and triangulations with descent bases methods arising in representation theory. In Chapter 3, the aim is to characterize Hilbert basis elements of certain -lecture hall cones. In particular, the main focus is the classification of the Hilbert bases for the 1 mod cones and the -sequence cones, both of which generalize a previous known result. Additionally, there is much broader characterization of Hilbert bases in dimension ≤ 4 for -generated Gorenstein lecture hall cones. Finally, Chapter 4 focuses on certain algebraic and geometric properties of -lecture hall polytopes. This consists of partial classification results for the Gorenstein property, the integer-decomposition property, and the existence of regular, unimodular triangulations

Lattice polytopes from Schur and symmetric Grothendieck polynomials

Given a family of lattice polytopes, two common questions in Ehrhart Theory are determining when a polytope has the integer decomposition property and determining when a polytope is reflexive. While these properties are of independent interest, the confluence of these properties is a source of active investigation due to conjectures regarding the unimodality of the $h^\ast$-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomials. In both cases, we prove that these polytopes have the integer decomposition property by using the fact that both families of polynomials have saturated Newton polytope. Furthermore, in both cases, we provide a complete characterization of when these polytopes are reflexive. We conclude with some explicit formulas and unimodality implications of the $h^\ast$-vector in the case of Schur polynomials.Comment: 37 pages, 3 tables, 4 figures; Comments Welcome; Version 2: updated references to acknowledge one result was previously known, corrected values in Table 1 and reference correct OEIS sequence; Version 3: Final Version. To appear in Electronic Journal of Combinatoric

Benzodiazepine receptor binding in cerebellar degenerations studied with positron emission tomography

We used positron emission tomography with [ 11 C]flumazenil to study gamma‐aminobutyric acid type A/benzodiazepine receptor binding quantitatively in the cerebral hemispheres, basal ganglia, thalamus, cerebellum, and brainstem of 72 subjects, including 14 with multiple system atrophy of the ataxic (olivopontocerebellar atrophy) type, 5 with multiple system atrophy of the extrapyramidal/autonomic (Shy‐Drager syndrome) type, 18 with sporadic olivoponto‐cerebellar atrophy, 15 with dominantly inherited olivopontocerebellar atrophy, and 20 normal control subjects with similar age and sex distributions. In comparison with data obtained from the normal control subjects, we found significantly Decemberreased ligand influx in the cerebellum and brainstem of multiple system atrophy patients of the olivopontocerebellar atrophy type and in patients with sporadic olivopontocerebellar atrophy, but not in patients with multiple system atrophy of the Shy‐Drager syndrome type. Despite these differences in ligand influx, benzodiazepine binding was largely preserved in the cerebral hemispheres, basal ganglia, thalamus, cerebellum, and brainstem in patients with multiple system atrophy of both types as well as those with sporadic or dominantly inhierited olivoponto‐cerebellar atrophy as compared with normal control subjects. The finding of relative preservation of benzodiazepine receptors indicates that these sites are available for pharmacological therapy in these disorders.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/92122/1/410380209_ftp.pd