Research

Student Research Opportunity:

As of fall 2022, I am accepting any undergraduate and graduate student who is interested in research in mathematical physics and quantum mechanics if I have the time. My expectations of any student:

• • Passed Differential Equations with Linear Algebra (Math 81) and preferably Quantum Mechanics (Phys 115) and Linear Algebra (Math 152).

  • Be ready to meet for two hours a week in person or online.

  • Spend about 8 hours each week, reading, taking notes, doing problems, and writing up summaries. Pro-tip: break it up into four two hour sessions throughout the week. Study is a journey, not a sprint.

  • Bring questions and ideas to meetings.

  • Write up your work in LaTeX. (I will teach you. Writing is critical to any success in your studies.)

What I do NOT expect:

• • To figure out everything on your own. Learning is both a social process and an iterative process: we are all trying to figure this out.

  • To know every bit of background I do. We all learn as we go. Don't compare yourself to me, I got a many-years-head-start on you in this material.

  • To commit to years worth of research with me. If you are tired of working on the topic or with me and want to do something else, just let me know. No hard feelings.

  • To work yourself into deep frustration. Feel free to take breaks.

  • To pay for every little thing for research. If you need a resource, I have funding to pay for it from the university.

Please contact me if you are interested.

Current Projects:

• Unruh Effect: Dr. Munoz and I are currently exploring this effect in hopes of finding a cleaner mathematical proof of this phenomenon.

• Modified Commutation Relations: Dr. Singleton, graduate students Erick Aiken, Jaeyeong Lee, Joey Contreras, Peter Martin, and I worked on a potential way of modifying quantum mechanics to accommodate an expected minimum length scale due to effects of gravity.

Past Projects:

• Spectral Gaps in Quantum Spin Systems - With Dr. Nachtergaele and Dr. Young, we proved a geometric dependence of the spectral gap in the single species PVBS models. Subsequently, I was able to prove a similar theorem for the two species PVBS models. I have left this topic because proving a general condition for a system to have a spectral gap is impossible as well as the fact that the gapped and gapless conditions for PVBS models was proven recently for all numbers of species and a wide range of parameters.

• Disordered Quantum systems with Bernoulli potentials- This was the bulk of my graduate research with Dr. Wehr. He presented me with the random Schrodinger operator and asked me to find the distribution for the low-lying energy states given the distribution of the random potential. In my early struggles, I stumbled upon Bernoulli potentials (two-valued) and built a novel method for understanding the ground state and excited states of these systems. This lead to joint work with Dr. Borovyk to provide a new approach to demonstrating Lifschitz tails. We later moved into interacting systems in random potentials with led to joint work with Dr. Lewenstein and his group at ICFO.

Possible Student Projects:

• Basics of quantum mechanics from a mathematical perspective- see me after linear algebra (Math 152) and differential equations (Math 81)

• Advanced topics in quantum mechanics - come to the Functional Analysis and Mathematical Physics Working Group coordinated by emailing Dr. Markin. There is a real chance that we can find a project to jointly supervise you between mathematics and physics faculty.

• Cantor and his developments - One of my favorite mathematicians - see me after proofs and logic (Math 111)

• Entropy - See me after differential equations (Math 81)