Thesis Defenses

2025

• Evan Chen

  Date: Wednesday, December 11, 2024 | 12:30pm | Room: 4-237 | Zoom Link

  Committee: Wei Zhang, Zhiwei Yun, Ben Howard

  Explicit formulas for weighted orbital integrals for the inhomogeneous and semi-Lie arithmetic fundamental lemmas conjectured for the full spherical Hecke algebra

  As an analog to the Jacquet-Rallis fundamental lemma that appears in the relative trace formula approach to the Gan-Gross-Prasad conjectures, the arithmetic fundamental lemma was proposed by W. Zhang and used in an approach to the arithmetic Gan-Gross-Prasad conjectures. The Jacquet-Rallis fundamental lemma was recently generalized by S. Leslie to a statement holding for the full spherical Hecke algebra. In the same spirit, there is a recent conjectural generalization of the arithmetic fundamental lemma to the full spherical Hecke algebra. This paper formulates another analogous conjecture for the semi-Lie version of the arithmetic fundamental lemma proposed by Y. Liu. Then this paper produces explicit formulas for particular cases of the weighted orbital integrals in the two conjectures mentioned above.

• Davis Evans

  Date: Thursday, December 12, 2024 | 10:30am | Room: 2-135

  Committee: John Bush, Jörn Dunkel, Ruben Rosales, Bauyrzhan Primkulov

  Ponderomotive Forces in Pilot-Wave Hydrodynamics

  Droplets bouncing on a vibrating bath may self-propel ('walk') via a resonant interaction with their own wavefield. In this hydrodynamic pilot-wave system, the spontaneous emergence of coherent 'wave-like' statistics from chaotic trajectories has been reported in several settings. Among the most compelling is the hydrodynamic analog of the quantum corral: when walking droplets are confined to a circular corral, a coherent statistical pattern emerges, marked by peaks in the positional histogram coincident with extrema of the cavity eigenmode.
  
  In this thesis, we present a joint experimental and theoretical study of chaotic pilot-wave dynamics in confined geometries. First, we present new experiments that quantitatively measure the height of the pilot-wave in the circular corral, as well as experiments that measure the droplet's times of impact. We showcase novel experimental methods for simultaneously tracking the droplet's position and pilot wave height.
  
  We propose a new theoretical model of droplet dynamics for this regime that treats the pilot-wave force as a stochastic process consistent with our experimental observations in the corral. This statistical model predicts a emergent mean "ponderomotive" force that is responsible for generating the observed statistical signature. The "ponderomotive" potential is found to be proportional to the squared gradient of the observed pilot wave. We also show that this modeling framework can be applied to some related experimental observations in elliptical corrals and droplets interacting with a steady standing wave.
  
  Finally, we situate the present theory alongside other mechanisms that generate coherent statistical patterns in pilot-wave hydrodynamics: I. chaotic switching between semistable orbitals and II. coherent dynamical speed oscillations. We propose that present theory, III. Ponderomotive forces emerging from stochastic dynamics, is a generic mechanism that arises upon coarse-graining the various chaotic dynamical systems present in models of hydrodynamic pilot wave dynamics.

• Andrey Khesin

  Date: Thursday, December 5, 2024 | 10:00am | Room: 2-449 | Zoom Link

  Committee: Peter Shor (chair and advisor), Isaac Chuang, Aram Harrow, Jonathan Kelner

  Quantum Computing from Graphs

  Many are familiar with the notion that quantum computers are fundamentally different to classical ones. One of these differences is the fact that performing quantum measurements can change the underlying quantum state. Additionally, quantum information is difficult to transmit and store, so algorithms for quantum error-correction and fault-tolerance are of much interest. While the most common representations of error-correcting codes have proven exceptionally useful as a descriptive tool, they otherwise offer little guidance for concrete constructions or coding algorithm analysis. We introduce a representation of certain quantum error-correcting codes as graphs with certain structures. With these graphs we can convert efficiently between various code representations, gain insight into how the codes propagate information, and discuss properties of codes by examining analogous properties in the codes' graphs. In particular, we show that one such graph property puts lower bounds on its code's distance, as well as gives us a simple and efficient decoding procedure for the code. This procedure is very similar to playing a quantum version of the children's game Lights Out. This change in perspective has already led to discovering several new codes and proving general results about typical graph codes, extending results on best known bounds. This defense will include a general introduction to quantum error-correction, a showcase of various graph codes, both old and new, as well as an explanation of the quantum Lights Out game and its relationship to decoding.