I am driven by the information-theoretic aspects of nature. I am especially curious about the nature of space and time and their relationship to quantum information theory. Although motivated largely by questions stemming from physics, I generally adhere to strict mathematical logic in formulating definitions, making statements, and proving theorems. Category theory provides me with the structure to do this, and it appears heavily in my thinking.

More specifically, my research program primarily focuses on extending inference, retrodiction, Bayes’ rule, entropy, and causality to quantum systems and dynamics.

Videos on a categorical introduction to (classical) Bayes’ theorem can be watched here: https://www.youtube.com/playlist?list=PLSx1kJDjrLRQksb7H9fqRE8GVMJdkX-4A.

I have also spent some time thinking about K-theory, higher gauge theory, and 2-dimensional algebra. The latter describes a way of associating algebraic data to 2-dimensional surfaces that is compatible with the way that 2-dimensional surfaces can be glued (i.e., “sewn”) together.

During the summer of 2009, while still an undergraduate in Queens College (CUNY), I visited Steven G. Johnson at MIT. There, we worked on a project in condensed matter to prove the existence of bound states for certain periodic potentials using tools from functional analysis and measure theory. The summer internship resulted in a publication in *Phys. Rev. B.*

Below, I include some preprints and publications, organized first by category and then from most recent to older works.

### Quantum Bayes, Probability, and Retrodiction

- (with James Fullwood) “On quantum states over time”
*Proc. R. Soc. A.***478**: 20220104. DOI: 10.1098/rspa.2022.0104, arXiv: 2202.03607 [quant-ph] - “Conditional distributions for quantum systems,” EPTCS 343, Proceedings 18th International Conference on Quantum Physics and Logic (2021), pp 1–13. DOI: 10.4204/EPTCS.343.1, arXiv: 2102.01529 [quant-ph]
- (with Benjamin P. Russo) “A non-commutative Bayes’ theorem” Linear Algebra Appl. 644 (2022), pp 28–94. DOI: 10.1016/j.laa.2022.02.030, arXiv: 2005.03886 [quant-ph]
- “Inverses, disintegrations, and Bayesian inversion in quantum Markov categories” (91 pages) arXiv: 2001.08375 [quant-ph]
- (with Benjamin P. Russo) “Non-commutative disintegration: existence and uniqueness in finite dimensions” (51 pages) Accepted in
*Journal of Noncommutative Geometry*, arXiv: 1907.09689 [quant-ph]

### Entropy

- “A functorial characterization of von Neumann entropy”
*Cah. Topol. Géom. Différ. Catég.***LXIII**, 1 (2022), 89–128, arXiv: 2009.07125 [quant-ph] - (with James Fullwood) “The information loss of a stochastic map,”
*Entropy***23**, no. 8 (2021). DOI: 10.3390/e23081021, arXiv: 2107.01975 [cs.IT] - “Towards a functorial description of quantum relative entropy,” in: Nielsen F., Barbaresco F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science, vol
**12829**. Springer, Cham. (2021). DOI: 10.1007/978-3-030-80209-7_60, arXiv: 2105.04059 [quant-ph]

### Algebraic quantum theory and operator algebras

- (with Luca Giorgetti, Alessio Ranallo, and Benjamin P. Russo)

“Bayesian inversion and the Tomita–Takesaki modular group” (38 pages) arXiv: 2112.03129 [math.OA] - “Stinespring’s construction as an adjunction,”
*Compositionality*1, 2 (2019). DOI: 10.32408/compositionality-1-2, arXiv: 1807.02533 [math.OA] - “From observables and states to Hilbert space and back: a 2-categorical adjunction,”
*Appl. Categorical Struct.*, Vol.**26**, Issue 6, pp 1123–1157 (2018). DOI: 10.1007/s10485-018-9522-6, arXiv: 1609.08975 [math-ph]

### K-theory

- (with Byungdo Park, Corbett Redden, and Augusto Stoffel) “Noncommutative differential K-theory”
*J. Geom. Phys.*174 (2022), 104446. DOI: 10.1016/j.geomphys.2021.104446, arXiv: 2106.12073 [math.KT]

### Analysis

- “Discrete probabilistic and algebraic dynamics: a stochastic Gelfand-Naimark Theorem,”

https://arxiv.org/abs/1708.00091

### Higher gauge theory

- “Two-dimensional algebra in lattice gauge theory,”
*Journal of Mathematical Physics***60**, 043506 (2019); https://doi.org/10.1063/1.5078532. Also available as

https://arxiv.org/abs/1802.01139 - “Gauge invariant surface holonomy and monopoles,” Theory and Applications of Categories, Vol.
**30**, No. 42, pp 1319-1428 (2015)

http://www.tac.mta.ca/tac/volumes/30/42/30-42abs.html

### Condensed matter

- “Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps” with Karen K. Y. Lee, Yehuda Avniel, and Steven G. Johnson, Phys. Rev. B
**81**, 155324 (2010)

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.81.155324