## About Me

I am a postdoctoral fellow at Harvard University. Starting Fall 2022 I will be a Veblen Research Instructor and NSF Postdoctoral Fellow at Princeton University/IAS. You may also find me at the MSRI program on Floer Homotopy Theory in Berkeley, CA.

As a mathematician, I am interested in symplectic geometry, and I think about how the many pseudoholomorphic curves in a symplectic manifold are organized into elaborate and useful algebraic structures. Lately I've been finishing a project regarding several problems problems in higher-dimensional Hamiltonian dynamics, and am excited about the Fueter equations and about certain relations between symplectic invariants and number theory.

Beyond that, I am interested in applications of mathematics, broadly construed. It seems to me that the intersection of machine learning, physics, geometry, and probability theory is both rich and important.

I got my Ph.D from Columbia University in 2021 under the direction of Mohammed Abouzaid.

My favorite book is Edmund Spenser's Faerie Queene.

### Contact

Email: rezchikov at math dot harvard dot edu

Snail mail:

Department of Mathematics

Harvard University

1 Oxford St, Cambridge, MA 02138

### Mathematics

- 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**

- Renormalization Group Flow as Optimal Transport

*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 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.*

### Teaching

In Spring 2020, I taught Calculus II (UN1102). Previously:

- Fall 2019: TA for Graduate Algebraic Topoogy I (taught by John Morgan).
- Summer 2019: 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!
- Spring 2019: TA for Calculus II (taught by Henri Roesch)
- Fall 2018: TA for Calculus I (taught by Henri Roesch)
- Spring 2018: TA for Graduate Algebraic Topology II (taught by Akram Alishahi)
- Fall 2017: TA for Graduate Algebraic Topology I (taught by Mohammed Abouzaid).

### Talks

- 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
- Bonn,
- 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)