Mathematics Dissertations

Free resolutions of orbit closures for representations with finitely many orbits

Federico Galetto — Wed, 22 May 2013 09:06:13 PDT

The irreducible representations of reductive groups with finitely many orbits are parametrized by graded simple Lie algebras. For the exceptional Lie algebras, Kraśkiewicz and Weyman exhibit the Hilbert polynomials and the expected minimal free resolutions of the normalization of the orbit closures. We present an interactive method to construct explicitly these and related resolutions in Macaulay2. The method is then used in the cases of the Lie algebras of type E₆, F₄, G₂, and select cases of type E₇ to confirm the shape of the expected resolutions as well as some geometric properties of the orbit closures.

...

Constructions of k-orbit abstract polytopes

Ilanit Shtull-Leber Helfand — Mon, 20 May 2013 13:31:14 PDT

The desire to derive new polytopes from old polytopes dates back to the classical study of polytopes, as many of the Archimedian solids can be obtained from Platonic solids through the act of truncation. In this dissertation, we apply these ideas to the setting of abstract polytopes. We present a number of constructions of abstract polytopes, which will generally share some properties with the polytopes from which they were derived. Most notably, we are interested in the circumstances under which the automorphism group of the derived polytope is isomorphic to the automorphism group of the original polytope. We construct polytopes...

Apolarity for the determinant and permanent

Masoumeh Shafiei — Fri, 10 May 2013 12:11:52 PDT

We show that the apolar ideals to the determinant and permanent of a generic matrix, the Pfaffian of a generic skew symmetric matrix and the determinant and the hafnian of a generic symmetric matrix are each generated in degree two. We also show that unlike the previous polynomials, the apolar ideal to the permanent of a generic symmetric matrix is generated in degrees two and three. In each case we specify the generators and give a Gröbner basis of the apolar ideal. As a consequence, using a result of K. Ranestad and F.-O. Schreyer we give lower bounds to the...

Discrete structures in finite type cluster algebras

Salvatore Stella — Thu, 09 May 2013 10:16:08 PDT

Due to their recursive definition, manipulating cluster algebras in an efficient way can be hard. Several combinatorial models have been developed in order to overcome this difficulty; here we investigate some of them in the finite type case.

In the first part of this thesis, using the parametrization of cluster variables by their g-vectors explicitly computed by S.-W. Yang and A. Zelevinsky, we extend the original construction of generalized associahedra by F. Chapoton, S. Fomin and A. Zelevinsky to any choice of acyclic initial cluster, and compare it to the one given by C. Hohlweg, C. Lange, and H. Thomas...

Internal and external invariance of abstract polytopes

Gabe Cunningham — Tue, 09 Apr 2013 08:38:27 PDT

In addition to the usual symmetries by reflections and rotations, abstract polytopes can have external symmetries such as self-duality and self-Petriality. In this dissertation, we describe and study a way of measuring the invariance of abstract polytopes under such external operations. We then present methods for constructing abstract polytopes with specified external symmetries. In particular, we describe how to construct polyhedra that are self-dual and self-Petrie, and how to construct polytopes that are self-dual and chiral.

...

Orbit closures of quiver representations

Kavita Sutar — Thu, 10 May 2012 05:51:07 PDT

Let $Q$ be a Dynkin quiver. We study orbit closures in $\mathrm{Rep}(Q, \underline{d})$, the affine space of quiver representations of a fixed dimension vector. The orbits arise from the action of $\mathrm{Gl}(\underline{d})$ on $\mathrm{Rep}(Q, \underline{d})$ and we consider their closure in the Zariski topology.

We investigate the properties of coordinate rings of orbit closures for quivers of type $A_3$ by considering the desingularization given by Reineke. We construct explicit minimal free resolutions of the defining ideals of the orbit closures thus giving us a minimal set of generators for the defining ideal. The resolution allows us to read off some...

Semi-invariants for gentle string algebras

Andrew Thomas Carroll — Tue, 24 Apr 2012 12:29:31 PDT

This thesis is devoted to the study of the geometry of representation spaces of string algebras. For each irreducible component C of a representation space of a gentle string algebra, we give an algorithm to determine the ring of semi-invariant functions on C. We show that these rings are semigroup rings (even coordinate rings of toric varieties) whose generators and relations can be described as walks on a particular graph. In addition, we determine the canonical decompositions of the modules in C. This decomposition allows a general discussion of the generating semi-invariants via Schofield's construction. This decomposition can be used...

Three contributions in representation theory: (1) cluster algebras and Grassmannians of type $G_2$ (2) Yangians and quantum loop algebras (3) monodromy of the trigonometric Casimir connection of $\mathfrak{sl}_2$

Sachin Gautam — Wed, 26 Oct 2011 10:00:21 PDT

The aim of the current dissertation is to address certain problems in the representation theory of simple Lie algebras and associated quantum algebras.

In Part I, we study the simple Lie group of type $G_2$ from the cluster point of view. We prove a conjecture of Geiss, Leclerc and Schröer, relating the geometry of the partial flag varieties to cluster algebras in the case of $G_2$.

In Part II, we establish a concrete relationship between certain infinite dimensional quantum groups, namely the quantum loop algebra $U_{ \hbar}(L \mathfrak{g})$ and the Yangian $Y_{ \hbar}(\mathfrak{g})$, associated with a simple Lie algebra $\mathfrak{g}$....

Unbounded solutions of the modified Korteweg-De Vries equation

John Gonzalez — Thu, 07 Apr 2011 12:17:38 PDT

In this dissertation we prove local existence and uniqueness of solutions of the focusing modified Korteweg - de Vries equation u_t + u2u_x + u_xxx = 0 in classes of unbounded functions that admit an asymptotic expansion at infinity in decreasing powers of x. We show that an asymptotic solution differs from a genuine solution by a smooth function that is of Schwartz class with respect to x and that solves a generalized version of the focusing mKdV equation. The latter equation is solved by discretization methods.

...

Quivers with potentials associated with triangulations of Riemann surfaces

Daniel Labardini-Fragoso — Tue, 08 Mar 2011 07:34:02 PST

We study the behavior of quivers with potentials and their mutations introduced by Derksen-Weyman-Zelevinsky in the combinatorial framework developed by Fomin-Shapiro-Thurston for cluster algebras that arise from bordered Riemann surfaces with marked points.

In Part I we associate to each ideal triangulation of a bordered surface with marked points a quiver with potential, in such a way that whenever two ideal triangulations are related by a flip of an arc, the respective quivers with potentials are related by a mutation with respect to the flipped arc. We prove that if the surface has non-empty boundary, then the quivers with potentials...

Quantum F-polynomials in the theory of cluster algebras

Thao Tran — Fri, 24 Sep 2010 06:43:54 PDT

F-polynomials and g-vectors were defined by Fomin and Zelevinsky to give a formula which expresses cluster variables in a cluster algebra in terms of the initial cluster data. A quantum cluster algebra is a certain noncommutative deformation of a cluster algebra. In this thesis, we define and prove the existence of analogous quantum F -polynomials for quantum cluster algebras. We compute quantum F-polynomials and g-vectors for a certain class of cluster variables, which includes all cluster variables in type A quantum cluster algebras. Finally, we give formulas for F-polynomials and quantum F-polynomials in classical types when the initial exchange matrix...

Standard bases for coordinate rings of cotangent varieties

Marcus Lloyd Fries — Thu, 23 Sep 2010 10:01:57 PDT

We extend the results of the Standard Monomial Theory to the cotangent variety of the Grassmannian, $T*Grass(k,n)$. We exhibit a standard basis in terms of triples of tableaux. We are also able to show that $T*Grass(k,n)$ is arithmetically Cohen-Macaulay and normal in all characteristics. Results on Cohen-Macaulayness and normality are shown for other algebraic groups and cominuscule parabolic subgroups, but only in characteristic zero.

Cluster algebras of finite type via semisimple groups and generalized minors

Shih-Wei Yang — Thu, 23 Sep 2010 06:46:20 PDT

In this thesis, we give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in the simply connected semisimple algebraic group of the same Cartan-Killing type. In this realization, the cluster variables appear as certain (generalized) principal minors.

Based on this realization, we give combinatorial formulas for $F$-polynomials in cluster algebras of classical types in terms of the weighted paths in certain directed graphs. As a consequence we prove the positivity of $F$-polynomials...

Transitivity of graphs associated with highly symmetric polytopes

Mark Mixer — Fri, 10 Sep 2010 12:23:18 PDT

This dissertation deals with highly symmetric abstract polytopes. Abstract polytopes are combinatorial structures which generalize the classical notion of convex polytopes. There are many different natural graphs associated with an abstract polytope. Particular attention is given to two of these graphs, namely the comparability graph of a polytope and the Hasse diagram of a polytope. We study various types of transitivity in these graphs.

Both the comparability graph and the Hasse diagram for a polytope inherit nicely the rank function associated with a polytope. Using this rank function, we can study specific subgraphs of each graph, by restricting to vertices...

Tensor structure on smooth motives

Anandam Banerjee — Wed, 30 Jun 2010 10:35:48 PDT

Grothendieck first defined the notion of a "motif" as a way of finding a universal cohomology theory for algebraic varieties. Although this program has not been realized, Voevodsky has constructed a triangulated category of geometric motives over a perfect field, which has many of the properties expected of the derived category of the conjectural abelian category of motives. The construction of the triangulated category of motives has been extended by Cisinski-Deglise to a triangulated category of motives over a base-scheme S. Recently, Bondarko constructed a DG category of motives, whose homotopy category is equivalent to Voevodsky's category of effective geometric...

Exhaustive weakly wandering sequences for alpha type transformations

John Lindhe — Tue, 15 Jun 2010 10:55:14 PDT

In this thesis, new conditions are given which allow for the control of the exhaustive weakly wandering sequence of an ergodic, infinite measure preserving transformation. It is also shown how to control the α-type of a transformation. These are then extended to permit the simultaneous control of both the exhaustive weakly wandering sequence and the α-type. Applying these results, new transformations are presented. The first known examples are given of 0-type and 1/3-type with known exhaustive weakly wandering sequences. In addition, we present an explicit sequence of integers which is exhaustive weakly wandering for α-type transformations of every α ∈...

Various parameters of subgraphs and supergraphs of the hypercube

Elizabeth A. Donovan — Fri, 18 Dec 2009 08:21:45 PST

The hypercube, in its various forms, has been greatly studied over the last sixty years. Our research involving this graph primarily uses a concept of ""levels"" in this graph, where the i-th level of the n-cube is defined as the set of all vertices of weight i. When examining more than one level, we are, in fact, just viewing the induced subgraph whose vertex set consists of those vertices of desired weights. We begin by considering the vertices of any two consecutive levels on the lower half of the cube sorted in colexicographic order and obtain a perfect matching of...

Some geometric properties of certain toric varieties and Schubert varieties

Justin Allen Brown — Fri, 18 Dec 2009 08:21:44 PST

This thesis has three distinct chapters: Bruhat-Hibi toric varieties, Gorenstein Schubert varieties in a minuscule G/P, and Wahl's conjecture for a minuscule G/P. We begin with a study of toric varieties associated to Bruhat lattices for a minuscule G/P. Our main result is a combinatorial characterization of the singular loci of these toric varieties. In the next chapter, we are concerned with Schubert varieties in a minuscule G/P, for which we give a combinatorial characterization for them to be arithmetically Gorenstein. In the last chapter, we prove Wahl's conjecture for a minuscule G/P.

...

Validation of a probabilistic model of language acquisition in children

Kenneth Jerold Straus — Fri, 18 Dec 2009 08:21:44 PST

Charles D. Yang has developed a mathematical model of how a child learns his first language. This model, based on Noam Chomsky's Principles and Parameters theory of Universal Grammar, represents the actual process of a child learning the parameter settings for the grammar of his native language. The process is one of ""natural selection"" wherein the correct settings (for his native language) gradually ""prevail"" and the incorrect ones ""die out"". Mathematically, this is expressed as probabilities associated with each of the possible settings, probabilities which give the likelihood that the child will use a particular parameter setting when he parses...

Singularities of a certain class of toric varieties

Himadri Mukherjee — Fri, 18 Dec 2009 08:21:43 PST

Hibi considered the class of algebras $k[\mathcal{L}]=k[x_{\alpha} \res \alpha \in \mathcal{L}]$ with straightening laws associated to a finite distributive lattice $\mathcal{L}$ in his paper \cite{hibi}. In that paper he proves that these algebras are normal and integral domains. This result along with the work of Sturmfels and Eisenbud \cite{ES} on binomial prime ideals implies that the affine varieties associated to the algebra $k[\mathcal{L}]$ are normal toric varieties. In the present work we will consider the toric variety $X(\mathcal{L})=\mathrm{spec}( k[\mathcal{L}])$, we will give the combinatorial description of the cone $\sigma$ associated to it. The final result will be to give a...