Mathematics - Ph.D. / Sc.D.
http://hdl.handle.net/1721.1/7680
2014-10-21T10:41:42ZThe affine Yangian of gl₁, and the infinitesimal Cherednik algebras
http://hdl.handle.net/1721.1/90192
The affine Yangian of gl₁, and the infinitesimal Cherednik algebras
Tsymbaliuk, Oleksandr
In the first part of this thesis, we obtain some new results about infinitesimal Cherednik algebras. They have been introduced by Etingof-Gan-Ginzburg in [EGG] as appropriate analogues of the classical Cherednik algebras, corresponding to the reductive groups, rather than the finite ones. Our main result is the realization of those algebras as particular finite W-algebras of associated semisimple Lie algebras with nilpotent 1-block elements. To achieve this, we prove its Poisson counterpart first, which identifies the Poisson infinitesimal Cherednik algebras introduced in [DT] with the Poisson algebras of regular functions on the corresponding Slodowy slices. As a consequence, we obtain some new results about those algebras. We also generalize the classification results of [EGG] from the cases GL, and SP2n to SOl. In the second part of the thesis, we discuss the loop realization of the affine Yangian of gl₁. Similar objects were recently considered in the work of Maulik-Okounkov on the quantum cohomology theory, see [MO]. We present a purely algebraic realization of these algebras by generators and relations. We discuss some families of their representations. A similarity with the representation theory of the quantum toroidal algebra of gl₁ is explained by adapting a recent result of Gautam-Toledano Laredo, see [GTL], to the local setting. We also discuss some aspects of those two algebras such as the degeneration isomorphism, a shuffle presentation, and a geometric construction of the Whittaker vectors.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 183-186).
2014-01-01T00:00:00ZA trajectory equation for walking droplets : hydrodynamic pilot-wave theory
http://hdl.handle.net/1721.1/90191
A trajectory equation for walking droplets : hydrodynamic pilot-wave theory
Oza, Anand Uttam
Yves Couder and coworkers have demonstrated that millimetric droplets walking on a vibrating fluid bath exhibit several features previously thought to be peculiar to the microscopic quantum realm, including single-particle diffraction, tunneling, quantized orbits, and wave-like statistics in a corral. We here develop an integro-differential trajectory equation for these walking droplets with a view to gaining insight into their subtle dynamics. The orbital quantization is rationalized by assessing the stability of the orbital solutions. The stability analysis also predicts the existence of wobbling orbital states reported in recent experiments, and the absence of stable orbits in the limit of large vibrational forcing. In this limit, the complex walker dynamics give rise to a coherent statistical behavior with wave-like features. We characterize the progression from quantized orbits to chaotic dynamics as the vibrational forcing is increased progressively. We then describe the dynamics of a weakly-accelerating walker in terms of its wave-induced added mass, which provides rationale for the anomalously large orbital radii observed in experiments.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014.; 65; Cataloged from student-submitted PDF version of thesis.; Includes bibliographical references (pages 185-189).
2014-01-01T00:00:00ZOn planar rational cuspidal curves
http://hdl.handle.net/1721.1/90190
On planar rational cuspidal curves
Liu, Tiankai
This thesis studies rational curves in the complex projective plane that are homeomorphic to their normalizations. We derive some combinatorial constraints on such curves from a result of Borodzik-Livingston in Heegaard-Floer homology. Using these constraints and other tools from algebraic geometry, we offer a solution to certain cases of the Coolidge-Nagata problem on the rectifiability of planar rational cuspidal curves, that is, their equivalence to lines under the Cremona group of birational automorphisms of the plane.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014.; 18; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 145-146).
2014-01-01T00:00:00ZEffective Chabauty for symmetric powers of curves
http://hdl.handle.net/1721.1/90189
Effective Chabauty for symmetric powers of curves
Park, Jennifer Mun Young
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.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 75-76).
2014-01-01T00:00:00Z