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.

• Sarah Greer

  Date: Monday, February 10, 2025 | 10:00am | Room: 2-255

  Committee: Laurent Demanet, Alan Edelman, John Urschel

  Geometrically-informed methods of wave-based imaging

  In this thesis, we are interested in understanding and advancing wave-based imaging techniques defined by the adjoint-state method. Wave-based imaging uses wavefield data from receivers on the boundary of a domain to produce an image of the underlying structure in the domain of interest. These images are defined by the imaging condition, derived from the first-order adjoint-state method, which corresponds to the gradient and maps recorded data to their reflection points in the domain. In the first part, we introduce a nonlinear modification to the standard imaging condition that can produce images with resolutions greater than that ordinarily expected using the standard imaging condition. We show that the phase of the integrand of the imaging condition, in the Fourier domain, has a special significance in some settings that can be exploited to derive a super-resolved modification of the imaging condition. Whereas standard imaging techniques can resolve features of a length scale of $\lambda$, our technique allows for resolution level $R < \lambda$, where the super-resolution factor (SRF) is typically $\lambda/R$. We show that, in the presence of noise, $R \sim \sigma$. In the second part, we investigate the Hessian operator, which arises from the second-order adjoint-state method, in the context of full-waveform inversion, a non-linear least-squares problem for estimating material properties within the domain of interest. We analyze the contributions of reflected and transmitted waves to the linearized Hessian operator, demonstrating that reflected waves generally produce a high-rank component with well-distributed eigenvalues, while transmitted waves contribute to a low-rank component with poorly distributed eigenvalues. This decomposition of the Hessian, motivated by the underlying physical system, provides insights that can be used to improve inversion strategies. The advancements in both parts of this thesis leverage the underlying structure and geometry of the domain of interest, providing the foundation for the zero-phase imaging condition in the first part and informing the decomposition of the Hessian operator in the second part.

• Mitchell Harris

  Date: Wednesday, February 5, 2025 | 2:00pm | Room: 2-132

  Committee: Pablo Parrilo, Steven Johnson, Ankur Moitra

  Computational Tradeoffs and Symmetry in Polynomial Nonnegativity

  Understanding when a polynomial is nonnegative on a region is a fundamental problem in applied mathematics. Although exact conditions for nonnegativity are computationally intractable, there has been a surge of recent work giving sufficient conditions for nonnegativity to address its many practical applications. A major trend in this direction has been the use of convex optimization to characterize polynomials that are sums of squares (SOS); nevertheless, this well-studied condition can be computationally intensive for polynomials of moderate degree and dimension.
  
  This thesis addresses the challenge of balancing computational cost against the strength of sufficient conditions for nonnegativity. We make progress towards bridging the gap between simple but crude sufficient conditions, and the more powerful but expensive SOS approach.
  
  In the first part, we introduce new certificates of nonnegativity that may be used when SOS is too expensive yet cheaper sufficient conditions are too conservative. For this, we leverage different features of the polynomials, including its Bernstein coefficients, a lower-degree interpolant, or its harmonic decomposition.
  
  In the second part, we construct coordinate-invariant sufficient conditions for nonnegativity and study the symmetry properties of the space of Gram matrices. By considering it as a representation of GL(n, R) and combining this module structure with classical invariant theory, we construct an explicit equivariant map for nonnegativity certification. We further introduce an alternative approach using equivariant neural networks, analyzing their benefits and limitations.

• 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.