Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 321-326). / In this thesis we generalize the construction of monopole Floer homology due to Kronheimer and Mrowka to the case of a gradient flow with Morse-Bott singularities. Focusing then on the special case of a three-manifold equipped equipped with a spinc structure which is isomorphic to its conjugate, we define the counterpart in this context of Manolescu's recent Pin(2)-equivariant Seiberg-Witten-Floer homology. In particular, we provide an alternative approach to his disproof of the celebrated Triangulation conjecture. Furthermore, we discuss the analogue in this setting of the surgery exact triangle, and perform some sample computations. / by Francesco Lin. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008. / Includes bibliographical references (p. 87-91). / We present a combined theoretical, experimental and numerical investigation of a sphere settling in a linearly stratified fluid at low Reynolds number (0.01 </= Re </= 2.1). We developed the microscale Synthetic Schlieren technique to study the wake of a microscale sphere settling through a density stratification. A video-microscope was used to magnify and image apparent displacements of a micron-sized random-dot pattern. Due to the nature of the wake, density gradient perturbations in the horizontal direction greatly exceed those in the vertical, requiring modification of a previously developed axisymmetric technique. We demonstrated that Schlieren could be extended to microscale (100 [mu]m) and obtained the first quantitative measurement of the density field in the wake of a sphere settling in a stratified fluid. As stratification breaks directional symmetry, the direction of motion strongly influences the dynamics, unlike in the homogeneous case. Previous work primarily focused on particles moving parallel to isopycnals. Here we investigate motion perpendicular to isopycnals. As the sphere settles, the particle draws lighter fluid downwards, generating buoyancy forces: this results in a long density wake, extending many particle diameters downstream. Using time-lapse photography, the drag on the sphere was measured and we have obtained the first experimental quantification of the added drag on a sphere due to stratification. We found that stratification increases the hydrodynamic drag, and that the added drag coefficient scales with the Richardson number Ri = a3N2/(vU) as Ri1/2, where a is the particle radius, U its speed, v the kinematic fluid viscosity and N the buoyancy frequency. These observations are confirmed by numerical simulations, and are in contrast with earlier results for higher Re. / (cont.) By analyzing the numerical velocity, pressure, density and vorticity fields around the sphere, we found that the pressure and viscous drags both increased with stratification. Combining these analyses with the investigations on isopycnal perturbations around the sphere and the buoyancy force in the wake, we conclude that the bulk of the wake does not contribute to the drag. Based on the experimental and numerical results, we derived a scaling argument which suggests that the added drag results from the buoyancy of the fluid in a small region of width (v/N)1/2 around the sphere. Here the physical mechanism responsible for the added drag in a stratified fluid at low Re is drastically different from mechanisms proposed at higher Re. The observed increase in drag could enhance retention time of particles at density interfaces as the parameter regime studied here applies to small particles in the ocean and affects the ecology of marine microorganisms by influencing particle-organism interactions. / by King Yeung Yick. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003. / Includes bibliographical references (p. 95-99). / In the field of genomics, this thesis presents algorithms for identifying non-coding RNA (ncRNA) genes. It describes a rapid and highly reliable comparative statistical method for identification of functionally significant base pairs in ncRNA genes in multiple sequence alignments of cross-species homologs, a divide-and-conquer approach to optimal assembly of exon predictions with O(n log n) time-complexity, (the standard algorithm for exon assembly has O(n²) time-complexity for ncRNA exon predictions,) and highly accurate statistical tests for exon boundaries based on recognition of non-contiguous patterns in known examples. It also describes a method for scanning cDNA for ncRNA genes. In the field of geometric measure theory, it proves that the set of cartesian currents given by integration over the graphs of smooth functions is dense in the set of all cartesian currents. / by Alex Coventry. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1972. / Vita. / Bibliography: leaves 123-124. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008. / Includes bibliographical references (p. 75-76). / We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation [sigma] = [sigma] 1 [sigma] 2 an defined as the set of indices i such that either i is odd and ai > ui+l, or i is even and au < au+l. We show that this statistic is equidistributed with the 3-descent set statistic on permutations [sigma] = [sigma] 1 [sigma] 2 ... [sigma] n+1 with al = 1, defined to be the set of indices i such that the triple [sigma] i [sigma] i + [sigma] i +2 forms an odd permutation of size 3. We then introduce Mahonian inversion statistics corresponding to the two new variations of descents and show that the joint distributions of the resulting descent-inversion pairs are the same. We examine the generating functions involving alternating Eulerian polynomials, defined by analogy with the classical Eulerian polynomials ... using alternating descents. By looking at the number of alternating inversions in alternating (down-up) permutations, we obtain a new qanalog of the Euler number En and show how it emerges in a q-analog of an identity expressing E, as a weighted sum of Dyck paths. Other parts of this thesis are devoted to polytopes relevant to the descent statistic. One such polytope is a "signed" version of the Pitman-Stanley parking function polytope, which can be viewed as a generalization of the chain polytope of the zigzag poset. We also discuss the family of descent polytopes, also known as order polytopes of ribbon posets, giving ways to compute their f-vectors and looking further into their combinatorial structure. / by Denis Chebikin. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 153-155). / The theme of this thesis is the novel application of techniques of algebraic topology (specifically, Steenrod's operations and Smith's localization theory) to representation theory (especially in the context of the geometric Satake equivalence). In Chapter 2, we use Steenrod's construction to prove that the quantum Coulomb branch is a Frobenius-constant quantization. We also demonstrate the corresponding result for the K-theoretic version of the quantum Coulomb branch. In Chapter 3, we develop the theory of parity sheaves with coefficients in the Tate spectrum, and use it to give a geometric construction of the Frobenius-contraction functor. In Chapter 4, we discuss some related results, including a geometric construction of the Frobenius twist functor, and also discuss future directions of research. The content of Chapter 3 is joint work with S. Leslie. / by Gus Lonergan. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 97-99). / We study global monotone solutions of the free boundary problem that arises from minimizing the energy functional I(u) = f lVul2 + V(U), where V(u) is the characteristic function of the interval (-1, 1). This functional is a close relative of the scalar Ginzburg-Landau functional J(u) = f lVul2 + W(u), where W(u) = (1 - u2 )2/2 is a standard double-well potential. According to a famous conjecture of De Giorgi, global critical points of J that are bounded and monotone in one direction have levell sets that are hyperplanes, at least up to dimension 8. Recently, Del Pino, Kowalczyk and Wei gave an intricate fixed-point-argument construction of a counterexample in dimension 9, whose level sets "follow" the entire minimal non-planar graph, built by Bombieri, De Giorgi and Giusti (BdGG). In this thesis, we turn to the free boundary variant of the problem and we construct the analogous example; the advantage here is that of geometric transparency as the interphase {lul < 1} will be contained within a unit-width band around the BdGG graph. Furthermore, we avoid the technicalities of Del Pino, Kowalczyk and Wei's fixed-point argument by using barriers only. / by Nikola Kamburov. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. / Includes bibliographical references (p. 111-116). / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / This thesis concerns the enumeration of pattern-avoiding permutations with respect to certain statistics. Our first result is that the joint distribution of the pair of statistics 'number of fixed points' and 'number of excedances' is the same in 321-avoiding as in 132-avoiding permutations. This generalizes a recent result of Robertson, Saracino and Zeilberger, for which we also give another, more direct proof. The key ideas are to introduce a new class of statistics on Dyck paths, based on what we call a tunnel, and to use a new technique involving diagonals of non-rational generating functions. Next we present a new statistic-preserving family of bijections from the set of Dyck paths to itself. They map statistics that appear in the study of pattern-avoiding permutations into classical statistics on Dyck paths, whose distribution is easy to obtain. In particular, this gives a simple bijective proof of the equidistribution of fixed points in the above two sets of restricted permutations. / (cont.) Then we introduce a bijection between 321- and 132-avoiding permutations that preserves the number of fixed points and the number of excedances. A part of our bijection is based on the Robinson-Schensted-Knuth correspondence. We also show that our bijection preserves additional parameters. Next, motivated by these results, we study the distribution of fixed points and excedances in permutations avoiding subsets of patterns of length 3. We solve all the cases of simultaneous avoidance of more than one pattern, giving generating functions which enumerate them. Some cases are generalized to patterns of arbitrary length. For avoidance of one single pattern we give partial results. We also describe the distribution of these statistics in involutions avoiding any subset of patterns of length 3. The main technique consists in using bijections between pattern-avoiding permutations and certain kinds of Dyck paths, in such a way that the statistics in permutations that we consider correspond to statistics on Dyck paths which are easier to enumerate. Finally, we study another kind of restricted permutations, counted by the Motzkin numbers. By constructing a bijection into Motzkin paths, we enumerate them with respect to some parameters, including the length of the longest increasing and decreasing subsequences and the number of ascents. / by Sergi Elizalde. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 109-112). / Let G be a connected reductive algebraic group over an algebraically closed field of characteristic p, g the Lie algebra of G and g* the dual vector space of g. This thesis is concerned with nilpotent orbits in g and g* and the Springer correspondence for g and g* when p is a bad prime. Denote W the set of isomorphism classes of irreducible representations of the Weyl group W of G. Fix a prime number 1 7 p. We denote ... the set of all pairs (c, F), where c is a nilpotent G-orbit in g (resp. g*) and F is an irreducible G-equivariant Q1-local system on c (up to isomorphism). In chapter 1, we study the Springer correspondence for g when G is of type B, C or D (p = 2). The correspondence is a bijective map from W to 2t.. In particular, we classify nilpotent G-orbits in g (type B, D) over finite fields of characteristic 2. In chapter 2, we study the Springer correspondence for g* when G is of type B, C or D (p = 2). The correspondence is a bijective map from ... . In particular, we classify nilpotent G-orbits in g* over algebraically closed and finite fields of characteristic 2. In chapter 3, we give a combinatorial description of the Springer correspondence constructed in chapter 1 and chapter 2 for 8 and g*. In chapter 4, we study the nilpotent orbits in 8* and the Weyl group representations that correspond to the pairs ... under Springer correspondence when G is of an exceptional type. Chapters 1, 2 and 3 are based on the papers [X1, X2, X3]. Chapter 4 is based on some unpublished work. / by Ting Xue. / Ph.D.

Thesis (M.S.) Massachusetts Institute of Technology. Dept. of Mathematics, 1947. / Bibliography: leaves 37-38. / by C. W. Murchison, Jr. / M.S.