Semon Rezchikov

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.

I am on the academic job market this year.

As a mathematician, I am interested in symplectic geometry, which is the geometry of phase spaces of physical systems, or more generally the geometry of the principle of least action. While the importance of symplectic geometry was known since Lagrange, the subject remained underdeveloped until the 80s, when Gromov's introduction of pseudoholomorphic curve theory, a nonlinear version of complex analysis, led to a continual transformation of the field. It turns out that the pseudoholomorphic curves in a symplectic manifold are organized into elaborate algebraic structures, which can be interpreted as a supersymmetric field theory which mirrors simpler structures related to polynomials, and incidentally gives rise to a quantum generalization of topology. Remarkably, these exotic connections between disparate areas of mathematics and physics can be used to solve concrete problems in chaotic dynamics and even children's problems about drawing rectangles inside curves.

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 complexifying symplectic geometry, which turns out to be connected to a quaternionic version of pseudoholomorphic curve theory and to the mysterious subject of 3D mirror symmetry.

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.

• 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

• Cyclotomic Structures in Symplectic Topology
  A video of a talk on this paper can be found here.
  This paper defines a genuine p-cyclotomic spectrum which lifts symplectic cohomology. This spectrum should agree with Topological Hochschild Homology of the Fukaya category of the ambient symplectic manifold (with coefficients in the sphere), and should explain various arithmetic phenomena in mirror symmetry. The underlying methods developed in the paper, which allow the use of genuine equivariant Floer homology, should have many other applications.
• 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

• Computational Dynamical Systems, (FOCS 2024), joint with Jordan Cotler We build a bridge bewteen computational complexity theory and smooth dynamical systems theory. We give upper bounds on the computational complexity of Axiom A systems and quantitative complexity bounds for one-dimensional maps, using highly nontrivial structure theory from known results in differentiable dynamical systems.
• Renormalizing Diffusion Models, joint with Jordan Cotler
  We give a precise correspondence between multiscale diffusion models from machine learning and exact renormalization group flows. We explain how to design models which rigorously learn the RG flow / effective action of a lacctice field theory, and do an initial numerical investigation of this method..
• 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.