## About Me

I am a Veblen Research Instructor and NSF Postdoctoral Fellow in the Mathematics Department at Princeton University and the Institute of Advanced Study.

As a mathematician, I am interested in symplectic geometry, which is the geometry of phase spaces of physical systems, or more generally the

My research has focused on foundational aspects of the invariants built from pseudoholomorphic curves, as well as on applications of these "quantum" modifications of topology to dynamical systems theory. I have also recently been working on

Beyond that, I am interested in applications of mathematics. The boundaries between physics, computer science, statistics, and mathematics have never been clear, and their interaction between remains a source of new ideas for me. In particular, I am working on rigorous ways to implement ideas from the renormalization group in machine-learning based models of physical systems.

In the past, you may have seen me at- Berkeley, CA as a member of the research program on Floer Homotopy Theory at the MSRI/SLMath during Fall 2022
- Cambridge, MA as a postdoc at the mathematics department of Harvard University during 2021-2022, or
- New York, NY where I got my Ph.D. from Columbia University the direction of Mohammed Abouzaid in 2021.

My favorite book is Edmund Spenser's Faerie Queene.

### Contact

Email: semonr at princeton dot edu

Snail mail:

Department of Mathematics

Princeton University

Fine Hall, Washington Rd

### Mathematics

- Holomorphic Floer Theory and the Fueter Equation

*(joint with Alexander Doan.) This paper outlines the structure of the Fueter 2-category, a 2-category assigned to a hyperkahler manifold M which categorifies the Fukaya category of M. We introduce new maximum principles for Fueter maps and prove an adiabatic limit theorem which categorifies and complexifies Floer's theorem on pseudoholomorphic curves in cotangent bundles. The Fueter 2-category is the 2-category of boundary conditions of the A-twist of the 3D N=4 sigma nodel, and as such sits at the intersection of several different fields of mathematics and physics which I am excited to explore.* - Integral Arnol'd Conjecture

*This paper constructs global charts for Hamiltonian Floer homology following Abouzaid-McLean-Smith. As a byproduct we prove the Integral Arnol'd conjecture using the Fukaya-Ono-Parker perturbations of Bai-Xu. I see global charts as a flexible tool to be used in future applications.* - Rational Quantum Cohomology of Steenrod Uniruled Manifolds

*This paper closes the gap between the usual formulation of the Chance-McDuff conjecture, which states that the existence of a Hamiltonian diffeomorphism with finitely many periodic points forces nontrivial rational Gromov-Witten invariants, and recent results that use methods in equivariant Floer theory to prove variants of this conjecture via equivariant, positive-characteristic Gromov-Witten invariants. The proof is a combination of a certain amount of `symplectic birational geometry' and a calculation using recently established properties of the quantum Steenrod algebra.* - Generalizations of Hodge-de-Rham degeneration for Fukaya categories, draft version.

*I explain why certain conjectures of Kontsevich about cyclic homology of categories always hold for Fukaya categories, even though Efimov demonstrated that they do not hold in general. The proof gives a TQFT interpretation of the conjectures, and a conceptual perspective on why they turned out to be in error.* -
Floer homology via Twisted Loop Spaces, submitted.

*This is a paper about signs. I explain how to extract an integral variant of Lagrangian Floer homology from pseudoholomorphic curve moduli spaces which may not be orientable. This anwers a question of Witten and lets one prove stronger Lagrangian self-intersection bounds.*

**Applications of Mathematics**

- Renormalizing Diffusion Models, joint with Jordan Cotler
- Diffusion with Forward Models: Solving Stochastic Inverse Problems Without Direct Supervision joint with Ayush Tewari, Tianwei Lin, et al. NeurIPS 2024 Spotlight Paper.
- Renormalization Group Flow as Optimal Transport (PRL D editors pick)

*This paper gives a variational formulation of a very general form of the renormalization group of statistical/quantum field theory in terms of an optimal transport problem. We see some interesting phenomena involving the regularization of the relative entropy in infinite dimensions. Beyond being of theoretical significance, this formalism should allow for the development of novel numerical tools to study physical systems by importing ideas from modern generative models.* - Light Field Networks: Neural Scene Representations with Single-Evaluation Rendering, NeurIPS 2022 spotlight paper.

*This paper proposes to represent 3d scenes by using a neural network to parameterize the colors of every ray hitting the scene. One uses meta-learning to learn a prior over consitent light fields, thus making rendering very cheap. It is implausible that the brain does ray-tracing! Amusingly, the space of oriented rays is a symplectic manifold, and the subspace of rays along which the light field changes discontinuously is a singular Lagrangian submanifold.* *Stochastic natural gradient descent draws posterior samples in function space, NeurIPS workshop paper.*

*This discusses the (rather popular) connection between stochastic gradient descent and Bayesian inference, and looks at the connection from the perspective of natural gradients. I helped fix some errors regarding how changes of variables work in stochastic calculus.*

### Invited Talks

- IAIFI Summer Workshop, Renormalizing Diffusion Models, 08/16/2023
- Stanford Symplectic Geometry Seminar, "Equivariant Structures in Symplectic Floer Homotopy", 05/2022/2023
- Symplectic Monday Seminar, "Equivariant Structures in Symplectic Floer Homotopy", IBS Center For Geometry and Physics (online talk), 04/24/2023
- Optimal Transport Theory and Applications to Physics, "Renormalization Group Flow as Optimal Transport, Ecole de Physique des Houches, 03/15/2023
- Kansas State M-Seminar, "Categorical Aspects of the Fueter Equation", 02/09/2023
- Symplectic Zoominar, "Hyperbolicity of Periodic Points of Hamiltonian Maps", 01/27/2023
- String-Data 2022, "Renormalization Group Flow as Optimal Transport", University of Cambridge, 12/09/2022
- Berkeley Topology Seminar, "Complex Morse Theory and the Fueter Equation", 11/30/2022
- Floer Homotopical Methods in Low Dimensional and Symplectic Topology, , SLMath, Berkeley, CA, 11/14/2022
- Workshop on Sampling, Transport, and Diffusions, "Renormalization Group Flow as Optimal Transport", Flatiron Institute,New York City, 11/17/2022
- Institute of Advanced Study, "Floer Theory on Complex-Symplectic Manifolds", Sept 22, 2022
- Floer Homotopy Foundations Seminar, "A pedagogical introduction to global charts for Hamiltonian Floer trajectories", SLMath, Berkeley, CA, 09/27/2023
- Max Plank Institute for Mathematics Topology Seminar, Miniseries "Local methods in categorical symplectic geometry", "Noncommutative and Symplectic Geometry", 07/02/2022
- EPFL Mathematical Physics and Probability Seminar, "Renormalization Group Flow as Optimal Transport, 05/11/2022,
- Physics meets ML Online Seminar, "Renormalization Group Flow as Optimal Transport", 05/04/2022
- Perimeter Institute Mathematical Physics Seminar, "Holomorphic Floer Theory and the Fueter Equation
- University of Southern California Topology Seminar, "Holomorphic Floer Theory and the Fueter Equation" , 3/21/2022
- Recent Developments in Lagrangian Floer Theory, Simons Center for Geometry and Physics @ Stony Brook, "Holomorphic Floer Theory and the Fueter Equation", 3/17/2022
- Freemath seminar, "Holomorphic Floer Theory and the Fueter Equation", 3/15/2022
- Meta AI Reading Group, "Renormalization Group Flow as Optimal Transport", 3/15/2022
- Mahadevan Group Meeting, Harvard University, "Renormalization Group Flow as Optimal Transport", 3/9/2022
- Geometry & Topology Seminar, Einstein Institute of Mathematics at Hebrew University, "Holomorphic Floer Theory and the Fueter Equation", 01/04/2022
- MIT Informal Symplectic Seminar, "Rational Quantum Cohomology of Steenrod Uniruled Manifolds", 10/22/2021
- Western Hemisphere Symplectic Geometry Seminar, "Generalizations of Hodge-de-Rham degeneration for Fukaya categories", 5/8/2020
- Harvard Gauge Theory Seminar, delayed due to coronavirus
- ``Structural Aspects of Fukaya Categories'', conference delayed due to coronavirus
- Stony Brook/Simons Center Symplectic Geometry Seminar, "Floer homology via Twisted Loop Spaces", 10/03/2019

### Symplectic geometry seminars that I have been involved with:

- Fall 2019: Global Homotopy Theory and Orbifolds, Atiyah-Bott Seminar
- Spring 2019: Floer Homotopy Theory (organized by me)
- Fall 2018: Monotone Lagrangians and wall crossing (organized by Alex Pieloch), Localization in Floer Theories Seminar (organized by Melissa Zhang)
- Fall 2017: Seiberg Witten Theory (organized by me)
- Spring 2016: Hendricks-Sarkar and Cohen-Jones-Segal seminar (organized by Abouzaid)

### Teaching

In 2020, I taught Calculus II (UN1102) at Columbia University. In 2019 I co-organized with Kyler Siegel a Columbia REU program for undergraduate students. The students used numerical optimization methods, tools from symplectic integration, and Haim-Kislev's recent work to find new symplectic embeddings. I have also TAed for graduate and undergraduate courses.