Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008. / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / Includes bibliographical references (p. 135-137). / Symplectic reflection algebras are attached to any finite group G of automorphisms of a symplectic vector space V , and are a multi-parameter deformation of the smash product TV ?G, where TV is the tensor algebra. Their representations have been studied in connection with different subjects, such as symplectic quotient singularities, Hilbert scheme of points in the plane and combinatorics. Let ... SL(2,C) be a finite subgroup, and let Sn be the symmetric group on n letters. We study finite dimensional representations of the wreath product symplectic reflection algebra ... of rank n, attached to the wreath product group ... and to the parameters (k, c), where k is a complex number (occurring only for n > 1), and c a class function on the set of nontrivial elements of ... In particular, we construct, for the first time, families of irreducible finite dimensional modules when ... is not cyclic, n > 1, and (k, c) vary in some linear subspace of the space of parameters. The method is deformation theoretic and uses properties of the Hochschild cohomology of H1,k,c(...), and a Morita equivalence, established by Crawley-Boevey and Holland, between the rank one algebra H1, ... and the deformed preprojective algebra ?Q), where Q is the extended Dynkin quiver attached to ?? via the McKay correspondence. We carry out a similar construction for continuous wreath product symplectic reflection algebras, a generalization to the case when ... SL(2,C) is infinite reductive. This time the main tool is the definition of a continuous analog of the deformed preprojective algebras for the infinite affine Dynkin quivers corresponding to the infinite reductive subgroups of SL(2,C). / by Silvia Montarani. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1991. / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / Includes bibliographical references (p. 130-132). / by Gísli Másson. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1974. / Vita. / Bibliography: leaves 111-114. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 44-46). / Given a prime number p, Bloch and Kato showed how the p Selmer group of an abelian variety A over a number field K is determined by the p-adic Tate module. In general, the pm1-Selmer group Selpmn A need not be determined by the mod pm Galois representation A[pm]; we show, however, that this is the case if p is large enough. More precisely, we exhibit a finite explicit set of rational primes E depending on K and A, such that Selpm A is determined by A[pm] for all ... In the course of the argument we describe the flat cohomology group ... of the ring of integers of K with coefficients in the pm- torsion A[pm] of the Neron model of A by local conditions for p V E, compare them with the local conditions defining Selm 2A, and prove that A[p't ] itself is determined by A[pm] for such p. Our method sharpens the relationship between Selpm A and ... which was observed by Mazur and continues to work for other isogenies 0 between abelian varieties over global fields provided that deg o is constrained appropriately. To illustrate it, we exhibit resulting explicit rank predictions for the elliptic curve 11A1 over certain families of number fields. Standard glueing techniques developed in the course of the proofs have applications to finite flat group schemes over global bases, permitting us to transfer many of the known local results to the global setting. / by Kęstutis Česnavičius. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 153-157). / Allocation or scheduling of resources among various entities contending for their access is one of the fundamental problem in engineering and operations research. To design a large, scalable networked systems, scheduling algorithms are required to be computationally very simple and distributed (or message passing). In this thesis, we present a novel method to design performance optimal, simple and distributed algorithms for a variety of scheduling problems. The algorithmic method is explained in detail in the context of wireless medium access. However, it naturally extends to essentially any instance of stochastic processing network (cf. [23]). In a wireless network, multiple transmitters are attempting to utilize common wireless medium for the purpose of communication. Due to nature of wireless communication, two simultaneous transmissions may interfere with each other. To avoid such destructive interference, a scheduling algorithm, known as medium access control (MAC), is required. The question of (design efficient MAC has been extensively studied starting with the ALOHA network [1]. Although certain types of MAC algorithms are used in practice (e.g. those confirming to IEEE 802.11)., a provably performance efficient algorithm has remained mystery for more than four decades. As an important contribution of this thesis, we resolve this challenge by presenting a novel, randomized medium access control (MAC) that is provably performance optimal. Like the solutions utilized in practice, it is a "randomized" or "back-off-like" algorithm and uses "carrier sense" information. This is the first instance of MAC that is proven to be performance optimal for general interference topology. Our solution blends the classical Metropolis-Hastings sampling mechanism with insights obtained from analysis of time-varying queueing dynamics. Methodically, our theoretical framework is applicable to design of efficient distributed scheduling algorithms for a wide class of combinatorial resource allocation problem in stochastic processing networks, including scheduling in input-queued switches and optical core network. / by Jinwoo Shin. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 67-69). / Adinkras are graphical tools created to study representations of supersymmetry algebras. Besides having inherent interest for physicists, the study of adinkras has already shown nontrivial connections with coding theory and Clifford algebras. Furthermore, adinkras offer many easy-to-state and accessible mathematical problems of algebraic, combinatorial, and computational nature. In this work, we make a self-contained treatment of the mathematical foundations of adinkras that slightly generalizes the existing literature. Then, we make new connections to other areas including homological algebra, theory of polytopes, Pfaffian orientations, graph coloring, and poset theory. Selected results include the enumeration of odd dashings for all adinkraizable chromotopologies, the notion of Stiefel-Whitney classes for codes and their vanishing conditions, and the enumeration of all Hamming cube adinkras up through dimension 5. / by Yan Zhang. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 57-58). / In this thesis, I study unstable homotopy theory with chromatic methods. Using the v, self maps provided by the Hopkins-Smith periodicity theorem, we can decompose the unstable homotopy groups of a space into its periodic parts, except some lower stems. For fixed n, using the Bousfield-Kuhn functor [Phi]n, we can associate to any space a spectrum, which captures the vo-periodic part of its homotopy groups. I study the homotopy type of the spectra LK(n)[Phi]nfSk, which would tell us much about the vn-periodic part of the homotopy groups of spheres provided we have a good understanding of the telescope conjecture. I make use the Goodwillie tower of the identity functor, which resolves the unstable spheres into spectra which are the Steinberg summands of classifying spaces of the additive groups of vector spaces over F,. By understanding the attaching maps of the Goodwillie tower after applying the Bousfield-Kuhn functor, we would be able to determine the homotopy type of LK(n)[Phi]nSk. As an example of how this works in concrete computations, I will compute the homotopy groups of LK(2)[Phi]nS3 at primes p >/= 5. The computations show that the unstable homotopy groups not only have finite p-torsion, their K(2)-local parts also have finite vo-torsion, which indicates there might be a more general finite v-torsion phenomena in the unstable world. / by Guozhen Wang. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1964. / Vita. / Includes bibliographical references (leaf 68). / by Peter Allan Trojan. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. / Includes bibliographical references (p. 57-59). / We define a moduli functor parametrizing finite maps from a projective (locally) Cohen-Macaulay curve to a fixed projective space. The definition of the functor includes a number of technical conditions, but the most important is that the map is almost everywhere an isomorphism onto its image. The motivation for this definition comes from trying to interpolate between the Hilbert scheme and the Kontsevich mapping space. The main result of this thesis is that our functor is represented by a proper algebraic space. As an application we obtain interesting compactifications of the spaces of smooth curves in projective space. We illustrate this in the case of rational quartics, where the resulting space appears easier than the Hilbert scheme. / by Morten Oskar Hønsen. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014. / 47 / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 61-64). / We construct counterexamples to a number of questions related to positivity properties of line bundles on algebraic varieties. The examples are based on studying the geometry of varieties that admit pseudo automorphisms of positive entropy, and in particular on the action of standard Cremona transformations on blow-ups of projective space at configurations of points. The main examples include the following: nefness is not an open condition in families; the diminished base locus of a divisor is not always a closed set; Zariski decompositions do not necessarily exist in dimension three; asymptotic multiplicity invariants are not always finite in the relative setting; and the number of Fourier- Mukai partners of a variety can be infinite. / by John Lesieutre. / Ph. D.