Ph.D. Recipients and their Thesis Abstracts

Algebra

Algebra, Analysis, Combinatorics, Differential Equations / Dynamical Systems, Differential Geometry, Geometry, Interdisciplinary, Lie Groups, Logic, Mathematical Physics, PDE / Numerical Analysis, Probability, Statistics, Topology

Quasisymmetric Functions and Flag Enumeration in Eulerian Posets

Abstract: We study the algebraic and enumerative combinatorial aspects of Eulerian posets and their quasisymmetric generating functions. These generating functions span the so-called peak algebra Pi, which originated with Stembridge's theory of enriched P-partitions. Remarkably, many constructs in the peak algebra that are natural in the context of enriched P-partitions are also important from the viewpoint of flag enumeration. For example, we show that the fundamental basis of peak functions arising from enriched P-partitions of chains is precisely the basis that is needed to properly encode the cd-index, a common invariant in the study of convex polytopes and Eulerian posets. As another example, the descents-to-peaks map, which relates the ordinary and enriched theories of P-partitions, turns out to be important for computing the flag-enumerative information of an oriented matroid based on that of the underlying matroid.

We introduce a family of Eulerian posets, called posets of signed order ideals. Each of these posets is defined by appropriately "signing" a distributive lattice. We establish that the flag-enumerative relationship between a poset of signed order ideals and its underlying distributive lattice is completely analogous to that between an oriented matroid and its underlying matroid. Furthermore, we show that these posets are EL-shellable, they have non-negative cd-indices, and their quasisymmetric generating functions enumerate the enriched P-partitions of certain labeled posets.

We analyze the (external) Hopf algebra structure on Pi, showing it to be dual to the concatenation Hopf algebra on letters of odd degree. Consequently, Pi is freely generated as a commutative algebra. Upon closer inspection of the generators, we find that Pi is free as a module over the ring of Schur Q-functions. Much of our analysis builds on basic properties of a monomial-like basis for the peak algebra, as well as a recursively defined basis of eigenvectors for the descents-to-peaks map. We give a simple criterion, in terms of this monomial basis, for determining when an element of Pi is symmetric.

Homological Results in Commutative Algebra

Abstract: We discuss a bound on the multiplicity of ideals in the first chapter. Herzog and Srinivasan have conjectured that for any homogeneous k-algebra of codimension d, the multiplicity is bounded by
$$
\prod_{i=1}^d\frac{M_i}{d!},
$$
where $M_i$ is the maximal degree of an ith syzygy. Using the minimal free resolutions as constructed by Peeva and Sturmfels, we show that this bound holds for codimension 2 lattice ideals.

The topic of chapter two is ideals having three generators. It is known due to work by Burch and by Kohn that any projective dimension may be realized by a 3-generated ideal. Buchsbaum and Eisenbud conjectured that the tail of any resolution may be realized by a 3-generated ideal. This result and a generalization was shown by Bruns. We introduce a family of ideals and prove that for each n the resolution of the ideal in n variables from the family has the same tail as the Koszul resolution on those n variables.

In chapter three we attempt to extend the result of chapter two to ideals generated by binomials. We display binomial examples in 4, 5 and 6 variables. A binomial example having the same tail as the Koszul complex for n = 7 is not found, but we do display 3-generated binomial examples having projective dimensions seven and eight.

Shuffles and Shellings via Projection Maps

Abstract: Projection maps which appear in the theory of buildings and oriented matroids are closely related to the notion of shellability. This was first observed by Björner. In the first chapter, we give an axiomatic treatment of either concept and show their equivalence. We also axiomatize duality in this setting. As applications of these ideas, we prove a duality theorem on buildings and give a geometric interpretation of the flag h vector. The former may be regarded as a q-analogue of the Dehn-Sommerville equations. We also briefly discuss the connection with the random walks introduced by Bidigare, Hanlon and Rockmore.

The random-to-top and the riffle shuffle are two well-studied methods for shuffling a deck of cards. These correspond to the symmetric group S_n, i.e., the Coxeter group of type A_{n–1}. In the second chapter, we give analogous shuffles for the Coxeter groups of type B_n and D_n. These can be interpreted as shuffles on a "signed'' deck of cards. With these examples as motivation, we abstract the notion of a shuffle algebra which captures the connection between the algebraic structure of the shuffles and the geometry of the Coxeter groups. We also give new joker shuffles of type A_{n–1} and briefly discuss the generalization to buildings which leads to q-analogues.

Subspace Subcodes and Generalized Coset Codes

Abstract: This dissertation considers codes that are formed using certain subsets of algebraic error-control codes and signal space codes. We first consider subspace subcodes of Reed-Solomon (SSRS) codes. We prove a conjecture of Hattori concerning how to identify subspaces that give an SSRS code whose dimension exceeds a certain lower bound.

We next consider generalized coset codes, which are built using partitions of signal space codes. We extend the concepts of generalized coset codes in Euclidean space by defining generalized coset codes in Lee space. We prove that all linear codes over Z_m are realizable as these new Lee-generalized coset codes. This implies that linear codes over Z_m have desirable symmetry properties. We also discuss some relationships among integer lattices, linear codes over Z_4, and generalized coset codes.

Enumeration in Geometric Lattices and the
Symmetric Group

Abstract: This work primarily deals with questions relating to the enumeration of chains in geometric lattices. The study of geometric lattices arose in the process of characterizing the lattices of subspaces formed by a finite set of points in projective or affine space, and ordered by inclusion. More generally geometric lattices encode the structure of matroids (or combinatorial geometries), which take an axiomatic approach to the concept of dependence.

The flag Whitney numbers of a geometric lattice count the number of chains of the lattice with elements having specified ranks. We are interested in the linear inequalities satisfied by the flag Whitney numbers.

We present a lower bond on the flag Whitney numbers of a lattice in terms of its rank and number of atoms. We also show that all flag Whitney numbers are simultaneously minimized by the lattice corresponding to the near pencil arrangement of points in R^n. Next we focus on rank 3 geometric lattices and give a collection of inequalities which imply all the linear inequalities satisfied by the flag Whitney numbers of rank 3 geometric lattices. We further describe the smallest closed convex set containing the flag Whitney numbers of rank 3 lattices corresponding to oriented matroids.

The ab-index is a polynomial associated to a lattice which contains all of the information regarding the flag Whitney numbers. We give a recurrence relation for the ab-index of families of lattices satisfying a particular set of conditions. We also give a collection of inequalities for the ab-index of geometric lattices.

Finally, we turn our attention to the group algebra of the symmetric group. The peak set of a permutation $\sigma$ is the set $\{i:\sigma(i–1)<\sigma(i)>\sigma(i+1)\}$. We prove the existence of a subalgebra of Solomon's descent algebra in which elements are sums of permutations that share a common peak set.

Linear Inequalities for Flag f-Vectors of Polytopes

Abstract: Here we study the combinatorics of polytopes. A polytope P is the convex hull of a finite set of points in R^d, and its boundary is a collection of lower-dimensional polytopes known as the faces of P. The flag f-vector of P counts the faces of each dimension and their incidences with one another. We would like to know what linear inequalities the entries of the flag f-vector satisfy.

First we present some of the history of this problem, along with the necessary mathematical background. We discuss several special classes of polytopes, including simplicials, simples, cubicals and zonotopes, whose flag f-vectors satisfy inequalities not satisfied by all polytopes.

Then we define Stanley's toric g-vector, which can be used to generate many linear inequalities for flag f-vectors. We prove Meisinger's conjecture that some of these inequalities are implied by others. In addition, we consider the cd-index, another source of many inequalities. We show that not all of these are consequences of the non-negativity of the toric g-vector.

We then use linear inequalities satisfied by lower-dimensional polytopes to generate linear relations satisfied by simplicial, simple, k-simplicial and k-simple polytopes, and cubical zonotopes. We also examine a g-vector for cubical polytopes proposed by Adin and give evidence that supports the conjecture g_2 <= 0. In particular, we show this to hold for the class of almost simple cubical polytopes, where one might expect it is most likely to fail. Next we improve upon previously known linear inequalities satisfied by zonotopes. Finally, we construct examples of another special class of polytopes, the self-dual polytopes.

Derived Categories of Twisted Sheaves on Calabi-Yau Manifolds

Abstract: This dissertation is primarily concerned with the study of derived categories of twisted sheaves on Calabi-Yau manifolds. Twisted sheaves occur naturally in a variety of problems, but the most important situation where they are relevant is in the study of moduli problems of semistable sheaves on varieties. Although universal sheaves may not exist as such, in many cases one can construct them as twisted universal sheaves. In fact, the twisting is an intrinsic property of the moduli problem under consideration.

A fundamental construction due to Mukai associates to a universal sheaf is a transform between the derived category of the original space and the derived category of the moduli space, which often turns out to be an equivalence. In the present work we study what happens when the universal sheaf is replaced by a twisted one. Under these circumstances we obtain a transform between the derived category of sheaves on the original space and the derived category of twisted sheaves on the moduli space.

The dissertation is divided into two parts. The first part presents the main technical tools: the Brauer group, twisted sheaves and their derived category, as well as a criterion for checking whether an integral transform is an equivalence (a so-called Fourier-Mukai transform). When this is the case we also obtain results regarding the cohomological transforms associated to the ones on the level of derived categories.

In the second part we apply the theoretical results of the first part to a large set of relevant examples. We study smooth elliptic fibrations and the relationship between the theory of twisted sheaves and Ogg-Shafarevich theory, K_3 surfaces, and elliptic Calabi-Yau threefolds. In particular, the study of elliptic Calabi-Yau threefolds leads us to an example which is likely to provide a counterexample to the generalization of the Torelli theorem from K_3 surfaces to threefolds. A similarity between the examples we study and certain examples considered by Vafa-Witten and Aspinwall-Morrison shows up, although we can only guess the relationship between these two situations at the moment.

Representations and Characters of an Extension of SL(3,R) by an Outer Automorphism

Abstract: Not available.

Centralizers of Finite Subgroups of Automorphisms and Outer Automorphisms of Free Groups

Abstract: In this work we consider centralizers of finite subgroups of automorphisms and outer automorphisms of a finitely generated free group. The work of S. Krstic reveals that the centralizer of a finite subgroup of outer automorphisms (resp. automorphisms) of a free group is generated by isomorphisms and Nielsen transformations of reduced G-graphs (resp. pointed G-graphs). Here this characterization is exploited to find necessary and sufficient conditions for the centralizer of a finite subgroup of outer automorphisms (resp. automorphisms) to be finite.

Cohomology of Aut(F_n)

Abstract: For odd primes p, we examine the Farrell cohomology {\hat H}^*(Aut(F_{2(p–1)}); Z_{(p)}) of the group of automorphisms of a free group F_{2(p–1)} on 2(p–1) generators, with coefficients in the integers localized at the prime (p)\subset Z. This extends results in [12] by Glover and Mislin, whose calculations yield {\hat H}^*(Aut(F_n); Z_{(p)}) for n\in{p–1, p} and is concurrent with work by Chen in [9] where he calculates {\hat H}^*(Aut(F_n); Z_{(p)}) for n\in{p+1, p+2}. The main tools used are Ken Brown's "Normalizer spectral sequence" from [7], a modification of Krstic and Vogtmann's proof of the contractibility of fixed point sets for outer space in [19], and a modification of the Degree Theorem of Hatcher and Vogtmann in [15].

Other cohomological calculations in the paper yield that H^5(Q_m; Z) never stabilizes as m\to\infty, where Q_m is the quotient of the spine X_m of "auter space" introduced in [15] by Hatcher and Vogtmann. This contrasts with the theorems in [15] where various stability results are shown for H^n(Aut(F_m); Z), H^n(Aut(F_m); Q), and H^n(Q_m; Q).

Actions of Artin Groups and Automorphism Groups on R-Trees

Abstract: This dissertation determines the ways in which two classes of groups act by isometries on R-trees. The first groups studied are the two-generator Artin groups A_{2n+1}, where 2n+1 is half the length of the relator. In the case where 2n+1 is prime, all abelian and non-abelian A_{2n+1}-actions are found, thus determining by Bass-Serre theory all of the graph of groups decompositions of A_{2n+1}. The same techniques are applied to a three-generator Artin group, the braid group on four strands, enabling a description of all of its actions on R-trees. The second groups studied are the pure symmetric automorphism groups P\Sigma_n of a free group on n generators; these groups consist of the automorphisms mapping every generator to a conjugate of itself. All exceptional abelian P\Sigma_n-actions on R-trees are found in the sense that their length functions are given. This information determines the Bieri-Neumann-Strebel invariant of P\Sigma_n using Brown's characterization. The invariant tells which normal subgroups of P\Sigma_n with abelian quotient are finitely generated.

Internal Categories and Quantum Groups

Abstract: Let {\Cal S} be a monoidal category with equalizes that are preserved by the tensor product. The notion of categories internal to {\Cal S}, and extending the usual notion of internal categories, which is obtained when {\Cal S} is a category with products and equalizers.

The basic theory of internal categories is developed and several applications to quantum groups are found. Delta categories are defined; these are algebraic objects that generalize groups or bialgebras, in the sense that attached to them there is a monoidal category of representations. Quantum groups are constructed from delta categories. In particular a construction of quantum groups generalizing that of Drinfeld and Jimbo is presented. An invariant of finite dimensional quasitriangular Hopf algebras is constructed.

Homological Methods in the Theory of Splines

Abstract: This thesis is devoted to the study of splines, which are piecewise polynomial functions of a prescribed order of smoothness on a triangulated (or otherwise polyhedrally subdivided) region \Delta of R^d. The set of splines of degree at most k (i.e. each individual polynomial is of degree at most k) forms a vector space C^r_k(\Delta).

It turns out that a good way to study C^r_k(\Delta) is to embed \Delta in R^{d+1}, and form the cone \hat\Delta of \Delta with the origin. Then the set of splines (of all degrees) on \hat\Delta is a graded module C^r(\hat\Delta) over the polynomial ring R in d+1 variables, and the dimension of C^r_k(\Delta) is the dimension of C^r(\hat\Delta) in degree exactly k.

The thesis follows the work of Billera and Rose, who defined a chain complex such that C^r_k(\Delta) appears as the top homology module, and approaches the problem from the viewpoint of homological algebra (but using a different chain complex (\Cal R/\Cal J)).

First, we analyze the lower homology modules in the planar case, and prove that the module C^r(\hat\Delta) is free if and only if the module H_1(\Cal R/\Cal J) vanishes. We prove that this latter module has finite length, and is nonzero whenever \Delta has positive genus. For a fixed configuration \Delta, we prove a criterion which gives sufficient conditions for C^r(\hat\Delta) to be nonfree, i.e. for H_1(\Cal R/\Cal J) to be nonzero.

Next, we relate the local geometry of \Delta (the arrangement of lines at each vertex) to ideals generated by powers of homogeneous linear forms in R[x, y], and completely describe the resolutions possible for such ideals. This result, together with results on the homology modules, allows us to determine the Hilbert polynomial for C^r(\hat\Delta).

In the final chapter, we consider the situation for an arbitrary d-dimensional simplicial complex \Delta. Generalizing the planar results, we obtain bounds on the dimension of the homology modules H_i(\Cal R/\Cal J) for all i< d. Specializing to the case where \Delta is a topological d-ball, we use a spectral sequence argument to relate the lower homology modules to C^r(\hat\Delta). For example, we prove that C^r(\hat\Delta) is free iff H_i(\Cal R/\Cal J) is zero for all i< d. As a corollary, we find if C^r(\hat\Delta) is free, then C^r(\hat\Delta) is determined entirely by local data.

Solving Sparse Polynomial Systems

Abstract: The computation of all solutions of a system of polynomial equations is a basic and in general hard problem. The main techniques for accomplishing this task in an algorithmic manner, Gröbner bases, multivariate resultants and continuation methods have received a great deal of attention recently. This thesis describes continuation methods for numerically determining all complex solutions of a system of polynomial equations. The emphasis is on the set up of continuation problems which require as few paths as possible, rather than on the numerical process of path following.

The key to efficient continuation methods is to realize the system to be solved as a generic element in a one dimensional parameterized family. Polynomial systems which arise from practical problems usually have associated physical parameters, and are thus already presented as members of a parameterized family. In the first chapter we discuss the total space of such a parameterized family and its implications for continuation methods.

The main body of the thesis deals with the application of toric deformations, and polyhedral subdivisions to produce continuation methods which are optimal for systems with a given collection of support sets {\Cal A} = {{\Cal A}_1,...,{\Cal A}_n}. We identify monomials x^q=x_1^{q_1}... x_n^{q_n} with lattice points q=(q_1,...,q_n)\in Z^n and allow polynomials of the form

f_i(x)=\sum_{q\in{\Cal A}_i} c_{i,q}x^q

where i=1,...,n and the coefficients c_{i,q} are complex. A famous result of Bernstein sates that, for generic choices of coefficients, the mixed volume of the Newton polytopes {\Cal Q}_i= conv({\Cal A}_i) is equal to the number of isolated solutions in (C*)^n. In Chapter 2, the concepts of Minkowski addition, mixed volume, and fine mixed subdivisions are presented. The use of fine mixed subdivisions of the Minkowski sum {\Cal Q}_i+...+{\Cal Q}_n in computing mixed volumes generalizes the use of triangulations in volume computation. Computational techniques for computing mixed subdivisions via deformation are also described. In Chapter 3, these techniques are adapted to provide a constructive proof of Bernstein's theorem, as well as a generalization of Bernstein's theorem to counting roots in C^n. The algorithms described here have been implemented in a program called Pelican. Chapter 4 describes Pelican and provides a tutorial and reference manual.

Algebraic K-Theory and Assembly for Complexes of Groups

Abstract: I define the algebraic K-theory K_i(RG(X)) for a complex of groups G(X) and a ring R, incorporating the combinatorial structure of X as well as group-theoretic data. If G is the fundamental group of G(X), I construct a comparison map K_i(RG(X))\to K_i(RG) and show that it is an isomorphism under certain geometric conditions on G(X).

In addition I define an assembly map H_i(G(X);K(R))\to K_i(RG(X)) relating the homology of G(X) to its K-theory. There is a natural transformation from the assembly map for G(X) to the ordinary assembly map for the group G, which is an equivalence when the above geometric hypothesis holds.

Normsets and Rings of Algebraic Integers

Abstract: Let F be a finite field extension of K which is, in turn, a finite field extension of Q, the rational numbers with rings of integers R and T respectively. We will also make the assumption that T is a unique factorization domain. It is well-known that the norm function N_K^F from F to K maps R into a multiplicative submonoid S of T which we will call the normset of R (with respect to T).

Intuitively, the richer and more intricate multiplicative structure of the ring R could be lost in the translation to the normset. Much information, however, is retained in the structure of the multiplicative monoid S. It is the aim of this dissertation to investigate certain properties of the monoid S and deduce what those properties tell us about R. We will also ask what certain properties of R tell us about S.

In the first chapter, we investigate unique factorization properties. There is a natural way to define the concept of unique factorization in the normset S. The most important result of the first chapter is the fact that if K is Galois over F, then R has unique factorization if and only if S has unique factorization. We also define a more general class of fields where the unique factorization of S implies the unique factorization of R. In the second chapter, we look at the relationship between the class group of R and the structure of the normset will be investigated. It will be shown that a property of a normset called saturation will measure how the Galois group acts on the class group, and from this some consequences about the structure of the class group of R can be deduced.

In the third chapter it is shown that in the Galois case, the normset determines the field extension. We also give an example of two non-Galois extensions such that the prime decomposition laws are identical, but the normsets are not.

In the final chapter, we use the techniques of this thesis to prove some factorization results about orders in quadratic number fields.

Geometry and Combinatorics of Chow Forms

Abstract: The Chow form of a variety is similar to a resultant. Given d+1 linear forms on a d-dimensional projective variety, the Chow form is a polynomial in the coefficients of the linear forms (unique up to multiplication by a nonzero constant) which is zero if and only if the linear forms have a common zero on the variety. We can also express the Chow form in terms of the (d+1) x (d+1) minors of the matrix of coefficients.

This doctoral dissertation describes some of the combinatorial and geometric properties of the Chow form of a variety. In chapter 1, we give an introduction to Chow forms and describe how to perform some basic geometric operations on varieties by manipulating their Chow forms. In chapter 2, we study the combinatorics of Chow forms of point sets in projective spaces and compare and contrast this theory with the theory of binary forms (which are Chow forms of point sets in P^1). We also consider the inhomogeneous version of this theory, the theory of multisymmetric functions. We give proofs for many of the assertions made in this paper, and also a counterexample to one of them. In chapter 3, we consider the Chow polytope as a combinatorial invariant of the Chow form. We show that the Chow polytope of the ruled join of two varieties (a construction used in intersection theory) is the Cartesian product of the Chow polytopes after a change of scale. We also investigate the stratification of the Chow variety of plane conics (whose points are the coefficient vectors of their Chow forms) induced by the different Chow polytopes. In chapter 4, we present a geometric algorithm for computing the Chow form of a ruled join in stages, and give examples of its use. Chapter 5 is somewhat disconnected from the remainder of the thesis; it proves that all connected complexes of coordinate lines in P^3 are initial complexes of space curves. This is a partial converse to the cited result of Kalkbrener and Sturmfels that all initial complexes of irreducible varieties are connected. Appendix A describes some MAPLE code written by the author to implement the algorithms of Section 2.1.3 and to make it easier to execute the algorithm of Section 4.2.

Algebraic and Combinatorial Methods for Face Enumeration in Polytopes

Abstract: Let P be a d-dimensional polytope and S a subset of {0, 1, ... , d–1}. We denote by f_S(P) the number of chains of faces whose dimensions make up S. The flag f-vector is the vector made up of all such flag numbers. This thesis studies linear and non-linear conditions on the flag f-vectors of several classes of polytopes.

We define a noncommutative algebra whose elements can be viewed as linear forms of flag numbers of polytopes. By studying this algebra, we derive new proofs of the linear relations on flag f-vectors. We also obtain results about the CD-index and symmetric linear forms having same value on any polytope and its dual. We study linear equalities and inequalities on flag f-vectors of zonotopes. The method is to construct a class of zonotopes to establish a lower bound for the span of flag f-vectors. We also find a class of inequalities on flag numbers of zonotopes; some of the inequalities from this class do not hold for general polytopes. In the case of the cube, we prove the g-conjecture in a stronger form, i.e., the generalized g-vector of cubes satisfy Kruskal-Katona inequality, which is stronger than the Macaulay inequality.

On Semi-Linear Representations Over Local Fields

Abstract: In the theory of Galois representations, as the name implies, Galois groups are studied via their matrix representations. In case the ground field is a p-adic field, the set of (continuous) representations of its absolute Galois group by integral p-adic matrices has the natural structure of a metric space. One of the main results of this thesis is that this metric space is bounded. Some related computations are given, describing the metric in detail for some special representations. Also discussed are some natural invariants of the representations, and the extent to which they determine the representations.

The Semistability at Infinity for Multiple Extension Groups

Abstract: Semistability at infinity is a geometric invariant used to study the ends of a locally finite, connected CW complex. One can use this definition to define the notion of the semistability at infinity for a finitely presented group. Much work has been done (especially by M. Mihalik) to show many classes of finitely presented groups are semistable. He has also given a definition of semistability for finitely generated groups that generalizes the original definition. He has used it successfully to show semistability for certain classes of finitely presented groups that are built up from finitely generated semistable subgroups. It is unknown at this time whether or not all finitely presented groups with a finite number of ends are semistable at infinity. The purpose of this dissertation is to use this alternative notion of semistability for finitely generated groups and prove the class of finitely generated groups containing an infinite, finitely generated subnormal subgroup of finite index are semistable at infinity. This would generalize two earlier results of Mihalik. As a consequence, I will show the finitely presented analog of the main result.

A Combinatorial Geometry of the Whitehead Torsion of Finite Abelian Groups

Abstract: Given an odd prime p and an arbitrary finite abelian p-group G, except for the exponent, not much is known about the invariants of the Whitehead torsion SK_1(Z[G]). Roger C. Alperin, R. Keith Dennis, Robert Oliver and Michael R. Stein have succeeded in using class field theory to give a simplified combinatorial description using norm residue symbols and the integral logarithm of L. Taylor and R. Oliver.

In this thesis, inspired by the delicate explicit calculations of specific groups by Oliver, we attempt to develop a combinatorial geometry that may facilitate a better understanding of the complications involved in these computations. Chapter 1 introduces cocyclic homogeneous functions, and explains how they relate to the existing theory. Chapter 2 builds the combinatorial machinery that is relevant to the calculations in this essay. The resulting geometry seems to borrow the notion of a tree from graph theory, and amalgamate it with the idea of a characteristic function from analysis! Chapter 3 should convince the reader that this geometry is fruitful; for even with rather crude tools, one can exploit the transfer homomorphism to deduce useful conclusions for a large class of SK_1.

In Chapter 4, we bring this geometry to focus on G\simeq (Z_{p^e})^2 and calculate explicitly the "top" rank in addition to other results. The main difference between the approach of Alperin-Dennis-Stein and ours is that instead of using the full flexibility in the choice of representatives of the orbits in the character group of G, we specialize to one particular choice. We close the essay by debunking a hitherto hope that monomorphisms of finite abelian p-groups should induce monomorphisms on the SK_1's.

Explicit Weight Two Motivic Cohomology Complexes and Algebraic K-Theory

Abstract: Beilinson et al.'s Motivic Cohomology Complex is related to a motivic complex (considered earlier by Lichtenbaum) formed using groups related to the scissors congruence groups. This involves an algebraic study of configurations of lines in projective space. One thereby obtains a more precise relationship between the cohomology of Beilinson's complex and lower dimensional K-groups. Most of the proofs omitted in Beilinson et al. are supplied here. There is also a list of certain torsional elements of the third K-group for fields between the rationales and the real cyclotomics. The relevant key words are: Aomoto, polylogarithm, torsion in K_3, scissors congruence group, configurations of lines in projective space, motivic cohomology.

Combinatorial Structure on the Hilbert Scheme

Abstract: In this thesis, the component structure of the Hilbert scheme is investigated. The techniques used to carry out this investigation include Gröbner bases, state polytopes, Hartshorne's fans, normal bundles, Borel-fixed ideals, and methods from deformation theory.

Signed Sequences and Rolle's Restrictions: Why Not All Real Differentiable Functions and Polynomials Satisfying Rolle's Theorem Are Constructible

Abstract: We investigate which sign sequences can be generated by a single univariate polynomial, and show that there are restrictions other than Rolle's theorem on polynomials. We then prove a fifth order Rolle's theorem, showing that an arrangement of roots satisfying the classical Rolle's theorem is not constructable by a real differentiable function.

Rewriting Systems for Coxeter Groups

Abstract: A finite complete rewriting system for a group is a finite presentation which gives a solution to the word problem and a regular language of normal forms for the group. In this thesis it is shown that the fundamental group of an orientable closed surface of genus g has a finite complete rewriting system, using the usual generators a_1,..., a_g, b_1,..., b_g along with generators representing their inverses. Constructions of finite complete rewriting systems are also given for any Coxeter group G satisfying one of the following hypotheses.

1) G has three or fewer generators.

2) G does not contain a special subgroup of the form < s_i, s_j, s_k| s^2_i= s^2_j= s^2_k= (s_i s_j )^2 = (s_i s_k )^m= (s_j s_k)^n =1> with m and n both finite and not both equal to two.

Endomorphisms of Negatively Curved Polygonal Groups

Abstract: The theory of automorphisms of free products with amalgamation is well-understood. We examine the dimension 2 analogue of a free product with amalgamation, a polygonal amalgam, and show that in general the automorphisms of polygonal amalgams are similarly well-behaved.

Special Values of L-Functions of Curves Over Finite Fields

Abstract: The central theme is to establish an equality "L= \chi," relating the value of L-function to an appropriate Euler characteristic. The first chapter studies various cohomologies, giving the definitions of the Euler characteristics and proving necessary properties. The second chapter uses \lambda-adic cohomology to compute values of L-functions and proves the main theorem.

Groups of Piecewise Linear Homeomorphisms

Abstract: Given a compact subinterval of the real line, the order preserving piecewise linear homeomorphisms of the interval form a group under composition. In this thesis we study subgroups of such groups obtained by restricting the slopes and singularities occurring in the homeomorphisms. We also consider analogous groups of homeomorphisms and bijections of the circle.

This class of groups contains examples which have furnished the first examples of finitely presented infinite simple groups. We extend this by exhibiting simple subgroups in all of these groups, and showing them to be finitely presented in some cases.

We then consider subgroups obtained by restricting slopes to a finitely generated subgroup of the rationals generated by integers, say n_1, n_2,...,n_k, and insisting that the singularities lie in Z[1 / (n_1 n_2...n_k)].

We construct contractible CW complexes on which these groups act with finite stabilizers, yielding classifying spaces as the quotients. We obtain combinatorial information about the groups by studying these classifying spaces. We compute homology and obtain finite presentations.

Last modified: February 10, 2004