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: Wednesday, January 8, 2025 | 10:30am | Room: 2-105 | Zoom Link

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

  Ponderomotive Forces in Pilot-Wave Hydrodynamics

  Droplets bouncing on a vibrating bath may self-propel (or 'walk') via a resonant interaction with their self-induced pilot wave. In pilot-wave hydrodynamics (PWH), the spontaneous emergence of coherent, wave-like statistics from chaotic trajectories has been reported in several settings. Owing to the similarity of PWH to Louis de Broglie's realist picture of quantum mechanics, the question of how such statistics emerge has received considerable recent attention. A compelling setting where coherent statistics emerge in PWH is the hydrodynamic analog of the quantum corral. When walking droplets are confined to a circular cavity or 'corral', a coherent statistical pattern emerges, marked by peaks in the positional histogram coincident with extrema of the cavity eigenmode. Stroboscopic models that idealize the drop's bouncing dynamics as being perfectly resonant with their Faraday wave field have proven incapable of capturing the emergent statistics.
  
  In this thesis, we present new experimental and theoretical findings in a variety of pilot-wave hydrodynamical settings where non-resonant bouncing plays a key role in the droplet dynamics and emergent statistics. We present an integrated experimental and theoretical study of the hydrodynamic corral, highlighting the role of non-resonant bouncing in the emergent statistics.
  Our experimental findings motivate a new theoretical framework that predicts that modulations in the histogram emerge as a consequence of ponderomotive effects induced by non-resonant bouncing. We then connect the ponderomotive drift observed in hydrodynamic corrals to extant theories of quantum mechanics.

• 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.