Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1980. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 48-49. / by David Jay Anick. / Ph.D.

The Poincaré series, Py(f) of a polynomial f was first introduced by Borevich and Shafarevich in [BS66], where they conjectured, that the series is always rational. Denef and Igusa independently proved this conjecture. However it is still of interest to explicitly compute the Poincaré series in special cases. In this direction several people looked at diagonal polynomials with restrictions on the coefficients or the exponents and computed its Poincaré series. However in this dissertation we consider a general diagonal polynomial without any restrictions and explicitly compute its Poincaré series, thus extending results of Goldman, Wang and Han. In a separate chapter some new results are also presented that give a criterion for an element to be an mth power in a complete discrete valuation ring.

This thesis is concerned with finding meromorphic extensions to a half-plane containing zero for certain generating functions. In particular, we generalise a result due to Morita and use it to show that the zeta function associated to the geodesic flow over a quotient of a Schottky group can be meromorphically extended to a half-plane containing zero. Moreover, we show that the special value at zero can be calculated. These results are then generalised to obtain meromorphic extensions past zero for L-functions defined on quotients of Schottky groups and to provide an expression for the special value at zero. Finally we show that Morita's method can be adapted to provide a meromorphic extension to a half-plane containing zero for Poincaré series defined for a Schottky group, and that in special circumstances the value at zero can be calculated.

Simplices are the "simplest" examples of polytopes, and yet they exhibit much of the rich and subtle combinatorics and commutative algebra of their more general cousins. In this way they are sufficiently complicated --- insights gained from their study can inform broader research in Ehrhart theory and associated fields. In this dissertation we consider two previously unstudied properties of lattice simplices; one algebraic and one combinatorial. The first is the Poincar\'e series of the associated semigroup algebra, which is substantially more complicated than the Hilbert series of that same algebra. The second is the partial ordering of the elements of the fundamental parallelepiped associated to the simplex. We conclude with a proof-of-concept for using machine learning techniques in algebraic combinatorics. Specifically, we attempt to model the integer decomposition property of a family of lattice simplices using a neural network.

On s'intéresse aux invariants pour l'action naturelle du groupe SL_2<br />sur l'algèbre B des coordonnées homogènes de la Grassmannienne des<br />pinceaux de formes quintiques binaires. La variété quotient<br />Proj(B^SL_2) est un candidat naturel pour la variété de modules des<br />quintiques gauches rationnelles.<br /><br />Un procédé connu établit une correspondance birationnelle et<br />équivariante entre la Grassmannienne des pinceaux de formes binaires<br />de degré d et l'espace projectif des formes binaires de degré 2d-2.<br />Lorsque le degré d est 5, cela suggère de comparer l'algèbre B^SL_2 et<br />l'algèbre des invariants d'une forme octique binaire. Cette algèbre a<br />été décrite en détail par T. Shioda en 1967.<br /><br />Nous établissons pour B^SL_2 un résultat analogue à celui de T.<br />Shioda : l'algèbre B^SL_2 est le quotient de l'algèbre de polynômes à<br />neuf indéterminées R=C[x_1,x_2,x_3,x'_3,x_4,x_5,x'_5,x_6,x_7] (les<br />indices donnent les degrés des indéterminées) par l'idéal des<br />4-Pfaffiens d'une matrice alternée 5x5 ; on identifie (numériquement)<br />la résolution libre minimale du R-module B^SL_2 ; enfin, on obtient<br />une famille génératrice minimale de l'algèbre B^SL_2.<br /><br />Pour y parvenir on commence par étendre la formule de T. Springer<br />(donnant la série de Poincaré de l'algèbre des invariants d'une forme<br />binaire) à l'algèbre des coordonnées homogènes d'une Grassmannienne.<br /><br /><br />Le point clé suivant consiste en l'identification d'un système de<br />paramètres homogènes. C'est possible grâce à une caractérisation, au<br />moyen du morphisme Wronskien, de la stabilité sur la Grassmannienne.<br />Il faut ensuite étudier les covariants d'ordre 4 et degré 2, ce qui<br />donne lieu à quelques énoncés de nature géométrique.<br /><br />Ces techniques permettent également de décrire les algèbres<br />d'invariants des pinceaux de cubiques et quartiques. Par ailleurs<br />l'étude du Wronskien conduit à de nouvelles formules de pléthysme.

We compute Ext groups between Soergel Bimodules associated to the infinite/finite dihedral group for a realization in characteristic 0 and show that they are free right 𝖱−modules with an explicit basis. We then give a diagrammatic presentation for the corresponding monoidal category of Ext-enhanced Soergel Bimodules. As applications, we compute reduced triply graded link homology 𝐇̅𝐇̅𝐇̅ of the connect sum of two Hopf links as an 𝖱−module and show that the Poincare series for the Hochschild homology of Soergel Bimodules of finite dihedral type categorifies Gomi's trace for finite dihedral groups.

The lattice point problem in dynamical systems investigates the distribution of certain objects with some length property in the space that the dynamics is defined. This problem in different contexts can be interpreted differently. In the context of symbolic dynamical systems, we are trying to investigate the growth of N(T), the number of finite words subject to a specific ergodic length T, as T tends to infinity. This problem has been investigated by Pollicott and Urbański to a great extent. We try to investigate it further, by relaxing a condition in the context of deterministic dynamical systems. Moreover, we investigate this problem in the context of random dynamical systems. The method for us is considering the Fourier-Stieltjes transform of N(T) and expressing it via a Poincaré series for which the spectral gap property of the transfer operator, enables us to apply some appropriate Tauberian theorems to understand asymptotic growth of N(T). For counting in the random dynamics, we use some results from probability theory.