Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 223-229). / We present a collection of results on the relationship between intertwining operators for quantum groups and eigenfunctions for quantum integrable systems. First, we study the Etingof-Kirillov Jr. expression of Macdonald polynomials as traces of intertwiners of quantum groups in the Gelfand-Tsetlin basis. In the quasiclassical limit, we obtain a new Harish-Chandra type integral formula for Heckman- Opdam hypergeometric functions. This formula is related to an integral formula appearing in recent work of Borodin-Gorin by integration over Liouville tori of the Gelfand-Tsetlin integrable system. At the quantum level, we obtain a new proof of the branching rule for Macdonald polynomials which transparently relates branching of Macdonald polynomials to branching of quantum group representations. Second, we study traces of intertwiners for quantum affine algebras. In the sl2 case, we show that, when valued in the three-dimensional evaluation representation, such traces converge in a certain region of parameters and provide a representation-theoretic construction of Felder-Varchenko's hypergeometric solutions to the q-KZB heat equation. This gives the first proof that such a trace function converges and resolves the first case of a conjecture of Etingof-Varchenko. As an application, we prove Felder-Varchenko's conjecture that their elliptic Macdonald polynomials are related to Etingof-Kirillov Jr.'s affine Macdonald polynomials. In the general case, we modify the setting of the work of Etingof-Schiffmann-Varchenko to show that traces of such intertwiners satisfy four commuting systems of q-difference equations - the Macdonald-Ruijsenaars, dual Macdonald-Ruijsenaars, q-KZB, and dual q-KZB equations. / by Yi Sun. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997. / Includes bibliographical references (p. 157-159). / by Brian David Taylor. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998. / Includes bibliographical references (p. 67-68). / by Konstanze Christina Rietsch. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009. / Includes bibliographical references (p. 79-82). / Using the tools developed by Colding and Minicozzi in their study of the structure of embedded minimal surfaces in R3 [12, 19-22], we investigate the conformal and asymptotic properties of complete, embedded minimal surfaces of finite genus and one end. We first present a more geometric proof of the uniqueness of the helicoid than the original, due to Meeks and Rosenberg [45]. That is, the only properly embedded and complete minimal disks in R3 are the plane and the helicoid. We then extend these techniques to show that any complete, embedded minimal surface with one end and finite topology is conformal to a once -punctured compact Riemann surface. This completes the classification of the conformal type of embedded finite topology minimal surfaces in R3. Moreover, we show that such s surface has Weierstrass data asymptotic to that of the helicoid, and as a consequence is asymptotic to a helicoid (in a Hausdorff sense). As such, we call such surfaces genus-g helicoids. In addition, we sharpen results of Colding and Minicozzi on the shapes of embedded minimal disks in R3, giving a more precise scale on which minimal disks with large curvature are "helicoidal". Finally, we begin to study the finer properties of the structure of genus-g helicoids, in particular showing that the space of genus-one helicoids is compact (after a suitably normalization). / by Jacob Bernstein. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. / Includes bibliographical references (p. 127-130). / In this thesis, I introduce a new definition for orbispaces based a notion of stratified fibration and prove it's equivalence with other existing definitions. I study the notion of orbispace structures on a given stratified space. I then set up two parallel theories of stratified fibrations, one for topological spaces, and one for simplicial sets. Modulo a technical comparison between the two theories, I construct a classifying space for orbispace structures. Using a conjectural obstruction theory, I then prove that every compact orbispace is equivalent to the quotient of a compact space by the action of a compact Lie group. / by André Gil Henriques. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 105-110). / We study singularity models of mean curvature flow ("MCF") and their generalizations. In the first part, we focus on rigidity and curvature estimates for self-shrinkers. We give a rigidity theorem proving that any self-shrinker which is graphical in a large ball must be a hyperplane. This result gives a stronger version of the Bernstein type theorem for shrinkers proved by Ecker-Huisken. One key ingredient is a curvature estimate for almost stable shrinkers. By proving curvature estimates for mean convex shrinkers, we show that any shrinker which is mean convex in a large ball must be a round cylinder. This generalizes a result by Colding-Ilmanen-Minicozzi : no curvature bound assumption is needed. This part is joint work with Jonathan Zhu. In the second part, we consider [lambda]-hypersurfaces which can be thought of as a generalization of shrinkers. We first give various gap and rigidity theorems. We then establish the Bernstein type theorem for [lambda]-hypersurfaces and classify [lambda]-curves. In the last part, we study translating solitons of MCF from four aspects: volume growth, entropy, stability, and curvature estimates. First, we show that every properly immersed translator has at least linear volume growth. Second, using Huisken's monotonicity formula, we compute the entropy of the grim reaper and the bowl solitons. Third, we estimate the spectrum of the stability operator L for translators and give a rigidity result of L-stable translators. Finally, we provide curvature estimates for L-stable translators, graphical translators and translators with small entropy. / by Qiang Guang. / Ph. D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 107-111). / Parallel repetition is a fundamental operation for amplifying the hardness inherent in multiplayer games. Through the efforts of many researchers in the past two decades (e.g. Feige, Kilian, Raz, Holentstein, Rao, Braverman, etc.), parallel repetition of two-player classical games has become relatively well-understood. On the other hand, games with entangled players (quantum games), crucial to the study of quantum non-locality and quantum cryptography, and multi-player games were poorly understood until recently. In this thesis, we resolve some of the major problems regarding the parallel repetition of quantum and multi-player games by establishing the first exponential-rate hardness amplification results for these games and hence extend the classes of games where exponential decay rates is known considerably. We consider two different methods for obtaining these hardness amplification results. For our first method, we draw from the recent work of Moshkovitz on parallel repetition of fortified games. We introduce an analytic reformulation of Moshkovitz's fortification framework. This reformulation allows us to expand the scope of the fortification method to new settings. In particular, we prove parallel repetition and fortification theorems for games with players sharing quantum entanglement, and games with more than two players in this new framework. An important component of our work is a variant of the fortification transformation, called ordered fortification, that preserves the entangled value of a game. For our second method, we introduce a class of games we call anchored. Anchoring is a simple transformation on games inspired in part by the transformation proposed in the pioneering work of Feige-Kilian. Unlike the Feige-Kilian transformation, our anchoring transformation is completeness preserving. We prove an exponential-decay parallel repetition theorem for anchored games that involve any number of entangled players. We also prove a threshold version of our parallel repetition theorem for anchored games. / by Mohammad Bavarian. / Ph. D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 75-76). / Faltings' theorem states that curves of genus g > 2 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the upper bound on the number of rational points, XI, [paragraph]2, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is smaller than g, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. We draw ideas from nonarchimedean geometry to show that we can also give an effective bound on the number of rational points outside of the special set of Symd X, where X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at most g > d. / by Jennifer Mun Young Park. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1993. / Includes bibliographical references (leaves 69-70). / by Farshid Hajir. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. / Includes bibliographical references (p. 153-155). / In the Traveling Salesman Path Problem, we are given a set of cities, traveling costs between city pairs and fixed source and destination cities. The objective is to find a minimum cost path from the source to destination visiting all cities exactly once. The problem is a generalization of the Traveling Salesman Problem with many important applications. In this thesis, we study polyhedral and combinatorial properties of a variant we call the Traveling Salesman Walk Problem, in which the minimum cost walk from the source to destination visits all cities at least once. Using the approach of linear programming, we study properties of the polyhedron corresponding to a linear programming relaxation of the traveling salesman walk problem. Our results relate the structure of the underlying graph of the problem instance with polyhedral properties of the corresponding fractional walk polyhedron. We first characterize traveling salesman walk perfect graphs, graphs for which the convex hull of incidence vectors of traveling salesman walks can be described by linear inequalities. We show these graphs have a description by way of forbidden minors and also characterize them constructively. / (cont.) We extend these results to relate the underlying graph structure to the integrality gap of the corresponding fractional walk polyhedron. We present several graph operations which preserve integrality gap; these operations allow us to find the integrality gap of graphs built from smaller bricks, whose integrality gaps can be found by computational or other methods. / by Fumei Lam. / Ph.D.