Mathematics  Ph.D. / Sc.D.
http://hdl.handle.net/1721.1/7843
20160930T18:38:31Z

Studies on quasisymmetric functions
http://hdl.handle.net/1721.1/104468
Studies on quasisymmetric functions
Grinberg, Darij
In 1983, Ira Gessel introduced the ring of quasisymmetric functions (QSym), an extension of the ring of symmetric functions and nowadays one of the standard examples of a combinatorial Hopf algebra. In this thesis, I elucidate three aspects of its theory: 1) Gessel's Ppartition enumerators are quasisymmetric functions that generalize, and share many properties of, the Schur functions; their Hopfalgebraic antipode satisfies a simple and explicit formula. Malvenuto and Reutenauer have generalized this formula to quasisymmetric functions "associated to a set of equalities and inequalities". I reformulate their generalization in the handier terminology of double posets, and present a new proof and an even further generalization in which a group acts on the double poset. 2) There is a second bialgebra structure on QSym, with its own "internal" comultiplication. I show how this bialgebra can be constructed using the AguiarBergeron Sottile universal property of QSym by extending the base ring; the same approach also constructs the socalled "Bernstein homomorphism", which makes any connected graded commutative Hopf algebra into a comodule over this second bialgebra QSym. 3) I prove a recursive formula for the "dual immaculate quasisymmetric functions" (a certain special case of Ppartition enumerators) conjectured by Mike Zabrocki. The proof introduces a dendriform algebra structure on QSym. Two further results appearing in this thesis, but not directly connected to QSym, are: 4) generalizations of Whitney's formula for the chromatic polynomial of a graph in terms of broken circuits. One of these generalizations involves weights assigned to the broken circuits. A formula for the chromatic symmetric function is also obtained. 5) a proof of a conjecture by Bergeron, Ceballos and LabbĂ© on reducedword graphs in Coxeter groups (joint work with Alexander Postnikov). Given an element of a Coxeter group, we can form a graph whose vertices are the reduced expressions of this element, and whose edges connect two reduced expressions which are "a single braid move apart". The simplest part of the conjecture says that this graph is bipartite; we show finer claims about its cycles.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.; This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.; Cataloged from studentsubmitted PDF version of thesis.; Includes bibliographical references (pages 297302).
20160101T00:00:00Z

Faster algorithms for convex and combinatorial optimization
http://hdl.handle.net/1721.1/104467
Faster algorithms for convex and combinatorial optimization
Lee, Yin Tat
In this thesis, we revisit three algorithmic techniques: sparsification, cutting and collapsing. We use them to obtain the following results on convex and combinatorial optimization: Linear Programming: We obtain the first improvement to the running time for linear programming in 25 years. The convergence rate of this randomized algorithm nearly matches the universal barrier for interior point methods. As a corollary, we obtain the first ... time randomized algorithm for solving the maximum flow problem on directed graphs with m edges and n vertices. This improves upon the previous fastest running time of achieved over 15 years ago by Goldberg and Rao. Maximum Flow Problem: We obtain one of the first almostlinear time randomized algorithms for approximating the maximum flow in undirected graphs. As a corollary, we improve the running time of a wide range of algorithms that use the computation of maximum flows as a subroutine. NonSmooth Convex Optimization: We obtain the first nearlycubic time randomized algorithm for convex problems under the black box model. As a corollary, this implies a polynomially faster algorithm for three fundamental problems in computer science: submodular function minimization, matroid intersection, and semidefinite programming. Graph Sparsification: We obtain the first almostlinear time randomized algorithm for spectrally approximating any graph by one with just a linear number of edges. This sparse graph approximately preserves all cut values of the original graph and is useful for solving a wide range of combinatorial problems. This algorithm improves all previous linearsized constructions, which required at least quadratic time. Numerical Linear Algebra: Multigrid is an efficient method for solving largescale linear systems which arise from graphs in low dimensions. It has been used extensively for 30 years in scientific computing. Unlike the previous approaches that make assumptions on the graphs, we give the first generalization of the multigrid that provably solves Laplacian systems of any graphs in nearlylinear expected time.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.; This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.; Cataloged from studentsubmitted PDF version of thesis.; Includes bibliographical references (pages 443458).
20160101T00:00:00Z

An investigation of oscillations in flows over curved surfaces
http://hdl.handle.net/1721.1/104162
An investigation of oscillations in flows over curved surfaces
Battin, Richard H
Thesis (Ph.D.) Massachusetts Institute of Technology. Dept. of Mathematics, 1951.; Vita.; Bibliography: leaves 170174.
19510101T00:00:00Z

Some irreducible complex representations of a finite group with BN pair
http://hdl.handle.net/1721.1/104159
Some irreducible complex representations of a finite group with BN pair
Kilmoyer, Robert William, 1939
Thesis (Ph. D.)Massachusetts Institute of Technology, Dept. of Mathematics, 1969.; Vita.; Bibliography: leaves 104105.
19690101T00:00:00Z