Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008. / Includes bibliographical references (p. 161-173). / The classical transport theory cannot explain the experimental behavior of electrochemical systems in the extreme operating conditions required by modern microfluidics devices. Some experimental puzzles include strange behavior of colloidal particles, high-frequency flow reversal in microfluidic ACEO pumps, and concentration dependence of electrokinetic slip. Theoretical developments would help not only in exploiting poorly understood effects favorably, but also in building more efficient microfluidics devices. The goal of this thesis is to explore possible mechanisms and modifications of the current theory that would enable us to interpret the experimental data. The following is a brief summary of the contributions of this thesis to the subject: Colloidal Particles. A new invention in colloidal science is the Janus particle, which is a two-faced spherical particle where one face is polarizable, and the other non-polarizable. These particles have potential applications in drug delivery, building of nanowires and solar energy. Experiments show that Janus particles strongly interact with boundaries: they approach walls, swim along walls, or sometimes jump away from walls. We show, by conducting numerical simulations of this truly 3D problem, that at least some of those observations can be explained within the classical linear theory. Finite Size Effects in Electrolytes. The classical Poisson-Boltzmann (PB) theory of electrolytes assumes a dilute solution of point charges with mean-field electrostatic forces. Even for very dilute solutions, however, it predicts absurdly large ion concentrations (exceeding close packing) for surface potentials of only a few tenths of a volt, which are often exceeded, e.g., in microfluidic pumps and electrochemical sensors. Since the 1950s, there have been numerous attempts in the literature to incorporate steric effects into the standard models. / (cont.) Most of those theories are complex, and require non-trivial numerical methods even for a simple problem. For this reason, they have not found applications in other contexts such as electrokinetics. In contrast, we focus on qualitative finite size effects, and incorporate only the essential elements of ion-crowding, using a lattice-gas model based statistical mechanical approach. Nonetheless, we are able to reach many conclusions about how steric effects play a role in electrochemical systems at large applied voltages. While dilute solution theory predicts that the differential double layer capacitance is exponentially increasing, steric effects predict that it varies non-monotonically: the differential capacitance always decays to zero after an initial increase. In addition, the net salt adsorption by the double layers in response to the applied voltage is greatly reduced, and so is the tangential "surface conduction" in the diffuse layer, to the point that it can often be neglected, compared to bulk conduction. This explains why, contrary to PB theory, limiting current is rarely attained in experiments. It has been shown that an asymmetric array of electrodes can be used to pump fluids in micro devices. These pumps operate only with AC voltage. Experiments have demonstrated that, when the AC frequency is high enough, the fluid flow supplied by such pumps reverses. This reversal, while not predicted by any of the standard theories, can be explained using our steric theory. We also generalize our steric models to the time-dependent case, deriving the first modified Poisson-Nernst-Planck (MPNP) equations, which incorporate finite size effects into the PNP equations. The modified equations should be used instead of PNP when the thin double layer approximation fails and the ion concentrations are high enough to make steric effects important. / by Mustafa Sabri Kilic. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008. / Includes bibliographical references (p. 81-82). / This dissertation is concerned with the foundations of a new approach to Floer theory. As opposed to the traditional approach, which can be viewed as a generalization of Morse theory to an infinite dimensional setting, our approach is a generalization of bordism to infinite dimensions. The key new insight, based on unpublished work of Tom Mrowka and Peter Ozsvath, is an understanding of how to axiomatize compactness in the infinite dimensional setting. We describe a general axiomatic framework for setting up a Floer theory of a polarized Hilbert space equipped with a functional. The resulting bordism theory can be seen as a refinement of the traditional Floer theory. By introducing cycles with corners, we demonstrate how the bordism theory leads to a geometric description of homology. We relate our geometric construction to the Morse-theoretic approach by indicating how one might compute the Floer homology of the space, if the associated functional is Morse. The general theory is illustrated in two examples: Seiberg-Witten-Floer homology and symplectic Floer theory for loops in Cn. We end by indicating various generalizations of the theory. / by Maksim Lipyanskiy. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995. / Includes bibliographical references (leaves 194-195). / by Glenn Paul Tesler. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 123-136). / Neural networks underpin both biological intelligence and modern Al systems, yet there is relatively little theory for how the observed behavior of these networks arises. Even the connectivity of neurons within the brain remains largely unknown, and popular deep learning algorithms lack theoretical justification or reliability guarantees. This thesis aims towards a more rigorous understanding of neural networks. We characterize and, where possible, prove essential properties of neural algorithms: expressivity, learning, and robustness. We show how observed emergent behavior can arise from network dynamics, and we develop algorithms for learning more about the network structure of the brain. / by David Rolnick. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1989. / Includes bibliographical references (leaves 238-245). / by Irvin C. Schick. / Ph.D.

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996. / Includes bibliographical references (leaves 40-42). / by Chunyan Jennifer Huang. / M.S.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994. / Includes bibliographical references (p. 81-85). / by Yiqun Yin. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017. / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / Cataloged from student-submitted PDF version of thesis. / Includes bibliographical references (pages 289-302). / In this thesis, we build connections between classic methods from convex optimization and the modern toolkit from the fast Laplacian solver literature, in order to make progress on a number of fundamental algorithmic problems: *-- We develop a faster algorithm for the unit capacity minimum cost flow problem, which encompasses the shortest path with negative weights and minimum cost bipartite perfect matching problems. In the case of sparse graphs, this provides the first running time improvement for these problems in over 25 years. *-- We initiate the study of solving linear systems involving directed Laplacian matrices, and devise an almost-linear time algorithm for this task. This primitive enables us to also obtain almost-linear time algorithms for computing an entire host of quantities associated with Markov chains, such as stationary distributions, personalized PageRank vectors, hitting times, or escape probabilities. This significantly improves over the previous state-of-the-art, which was based on simulating random walks, or applying fast matrix multiplication. *-- We develop faster algorithms for scaling and balancing nonnegative matrices, two fundamental problems in scientific computing, significantly improving over the previously known best running times. In particular, if the optimal scalings/balancings have polynomially bounded condition numbers, our algorithms run in nearly-linear time. Beyond that, we leverage and extend tools from convex geometry in order to design an algorithm for online pricing with nearly-optimal regret. We also use convex optimization to shed a new light on the approximate Caratheodory problem, for which we give a deterministic nearly-linear time algorithm, as well as matching lower bounds. / by Adrian Valentin Vladu. / Ph. D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 57-59). / Incidence geometry is the study of the intersection patterns of simple geometric objects. One of the breakthroughs in this field is the polynomial partitioning technique introduced by Guth and Katz. In this thesis, I will present two results on incidence problems with high-dimensional objects: an almost tight bound on the number of joints formed by varieties in Rn and a tight bound on the number of flags in Rn. The proofs are based on the polynomial partitioning technique and its variations.. / by Ben Yang. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1963. / Vita. / Includes bibliographical references (leaf 52). / by Hung-Hsi Wu. / Ph.D.