Thesis (Ph.D.) Massachusetts Institute of Technology. Dept. of Mathematics, 1951. / Vita. / Bibliography: leaves 170-174. / by Richard Horace Battin. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 74-78). / Stanley's Cauchy identity for Jack symmetric functions defines a Jack measure, a model random partitions for every analytic real function v(w) on the unit circle and parameters E2 < 0 < E1. Jacks are eigenfunctions of the Hamiltonian ... of the quantum Benjamin-Ono equation with periodic boundary conditions, dispersion and quantization corresponding E1 + E2 and -E1E2 respectively. From this point of view, Jack measures are the random energy distribution of a coherent state around a classical configuration v(w). Taking E2 --> 0 <-- E1 at a comparable rate [beta]/2, we prove that the slopes of the profiles of the random partition concentrate on a limit shape independent of [beta], the push-forward of the uniform measure on the circle along v. This is the conserved density of the classical inviscid Hopf hierarchy on the circle, following Dubrovin (2014). At the quantum Hopf hierarchy ([beta] = 2), we recover Okounkov's limit shape for Schur measures (2003) as a verification of the correspondence principle. Our main result is the computation of macroscopic fluctuations of random profiles around the limit shape: they converge to the push-forward along v of the restriction to the circle of a Gaussian free field on the upper half-plane whose covariance is independent of [beta]. At [beta] = 2, our result matches Breuer-Duits' central limit theorem (2013) for Borodin's biorthogonal ensembles. Our limit theorems follow from a diagrammatic all-order convergent expansion of joint cumulants of linear statistics over "ribbon paths". This expansion has the same form as the 1/N refined topological asymptotic expansion over ribbon graphs on surfaces for [beta]-ensembles on the line in one-cut potentials V due to Chekhov-Eynard (2006) and Borot-Guionnet (2012). Our analysis relies on the Lax operator L for the quantum Benjamin-Ono hierarchy introduced by Nazarov-Sklyanin (2013). L is expressed through Toeplitz operators whose symbols are affine Kac-Moody currents for gl1. We use the spectral shift function of L to construct a generating function y(u) of local Hamiltonians commuting with y3. This explicit y(u) is a special case of the y-operator defined implicitly by functional calculus in Nekrasov (2016). / by Alexander Moll. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995. / "May 1995." / Includes bibliographical references (p. 33-34). / by Dmitry Kaledin. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. / 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. 131-134) and index. / The main result is a computation of the Nahm transform of a SU(2)-instanton over R x T³, called spatially-periodic instanton. It is a singular monopole over T³, a solution to the Bogomolny equation, whose rank is computed and behavior at the singular points is understood under certain conditions. A full description of the Riemannian ADHMN construction of instantons on R⁴ is given, preceding a description of the heuristic behind the theory of instantons on quotients of R⁴. The Fredholm theory of twisted Dirac operators on cylindrical manifolds is derived, the spectra of spin Dirac operators on spheres and on product manifolds are computed. A brief discussion on the decay of spatially-periodic and doubly-periodic instantons is included. / by Benoit Charbonneau. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006. / Includes bibliographical references (leaf 55). / In this thesis we study the processes by which university students solve problems in multivariable calculus. Our data consists of a series of questionnaires and interviews with students enrolled in a vector calculus class at MIT. We interpret our observations in the light of previous research into the acquisition of mathematical knowledge and understanding. / by Francesca Deserti. / Ph.D.

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1931. / by Hsin P. Soh. / M.S.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013. / Cataloged from PDF version of thesis. / Includes bibliographical references. / In this dissertation we apply computer-assisted proof techniques to two problems, one in discrete geometry and one in celestial mechanics. Our main tool is an effective inverse function theorem which shows that, in favorable conditions, the existence of an approximate solution to a system of equations implies the existence of an exact solution nearby. This allows us to leverage approximate computational techniques for finding solutions into rigorous computational techniques for proving the existence of solutions. Our first application is to tight codes in compact spaces, i.e., optimal codes whose optimality follows from linear programming bounds. In particular, we show the existence of many hitherto unknown tight regular simplices in quaternionic projective spaces and in the octonionic projective plane. We also consider regular simplices in real Grassmannians. The second application is to gravitational choreographies, i.e., periodic trajectories of point particles under Newtonian gravity such that all of the particles follow the same curve. Many numerical examples of choreographies, but few existence proofs, were previously known. We present a method for computer-assisted proof of existence and demonstrate its effectiveness by applying it to a wide-ranging set of choreographies. / by Gregory T. Minton. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1967. / Vita. / Bibliography: leaves 148-149. / by David G. Ebin. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1990. / Includes bibliographical references (p. 113-114). / by Julian West. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995. / Includes bibliographical references (p. 209-214) and index. / by Alain Isaac Saias. / Ph.D.