Jeff Hicks

I'm an EPSRC postdoctoral fellow at the University of Edinburgh, working with Nick Sheridan. I grew up in Cardiff-by-the-Sea, California before getting my MA at UCLA. At UC Berkeley I received my PhD under the supervision of Denis Auroux.

I study the relation between symplectic, algebraic, and tropical geometry via mirror symmetry.

Outside of mathematics, my hobbies include playing piano, partner dancing, board games, and puzzles.

An outline of my research and teaching experience can be found in my CV.

Research

Applications of Mirror Symmetry

Homological mirror symmetry predicts that symplectic geometry should be related to complex geometry on a mirror space via mutual comparison to tropical geometry on the base of an SYZ fibration. My research focuses on applying this correspondence to study problems in complex geometry. I'm currently looking at how to interpret realizability criteria in tropical geometry from a Lagrangian perspective; and how Morse theory in symplectic geometry relates to resolutions of coherent sheaves on a mirror space.

Lagrangian Cobordisms

Monotone Lagrangian cobordisms provide equivalences in the Fukaya category. My current research is extending this equivalence to unobstructed Lagrangian cobordisms, with the goal of algorithmically constructing these equivalences in terms of the surgery handle decompositions for Lagrangian cobordisms. I'm also interested in quantiative aspects of Lagrangian cobordisms.

Publications and preprints

  1. Homological mirror symmetry for Batyrev mirror pairs. (with Sheel Ganatra, Andrew Hanlon,Daniel Pomerleano, and Nick Sheridan ). arXiv:2406.05272
  2. Integrality of mirror maps and arithmetic homological mirror symmetry for Greene--Plesser mirrors. (with Sheel Ganatra, Andrew Hanlon,Daniel Pomerleano, and Nick Sheridan ). Submitted. arXiv:2312.01949
  3. A short computation of the Rouquier dimension for a cycle of projective lines. (with Andrew Hanlon). Submitted. arXiv:2311.05753
  4. Relating categorical dimension in topology and symplectic geometry. (with Andrew Hanlon, Oleg Lazarev). Submitted. arXiv:2308.13677
  5. Reverse isoperimetric ineq. for Lagrangian intersection Floer theory. (with Jean-Philippe Chassé, Yoon Jae Nick Nho). Submitted. arxiv:2306.04761
  6. Resolutions of toric subvarieties. by line bundles and applications. (with Andrew Hanlon, Oleg Lazarev). Accepted. Forum of Mathematics, Pi
  7. Some cute applications of Lagrangian cobordisms towards examples in quantative symplectic geometry. (with Cheuk Yu Mak). Submitted. arXiv:2204.06432
  8. Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds. To appear. Geometry and Topology
  9. Lagrangian cobordisms and Lagrangian surgery. Commentarii Mathematici Helvetici 98 no. 3
  10. Aspects of functoriality in HMS for toric varieties. (with Andrew Hanlon). Advances in Mathematics 401
  11. Observations on Disks with tropical Lagrangian boundary. MATRIX Annals 2019
  12. Wall-crossing from Lagrangian Cobordisms. To appear. Algebraic and Geometric Topology
  13. Tropical Lagrangian Hypersurfaces are Unobstructed. Journal of Topology 13
  14. Tropical Lagrangians in toric del-Pezzo surfaces. Selecta Mathematica 27
  15. Tropical Lagrangians and Homological Mirror Symmetry. PhD Thesis, UC Berkeley

Undergraduate Research Students

Teaching and Outreach

  • Teaching at University of Edinburgh

    I'm currently supervising master's students and reading projects at the University of Edinburgh. Previously, I ran tutorial sessions for Several Variable Calculus and Differential Equations and cotaught "Interactions in Algebra, Geometry, and Topology".

  • Teaching at University of Cambridge

    I've been an ASCR with Kings College, where I supervised students in analysis and algebraic topology. I've additionally helped out with examples courses for symplectic geometry.

  • Teaching at UC Berkeley

    At UC Berkeley, I have been supported by both GSI, GSR and teaching fellowship funding. Courses include second semester calculus in Fall 13; Linear algebra in Spring 14 ; and Multivariable calculus in Fall 14 , Spring 16 , and Fall 17 . I was also the instructor for Math 32 in Spring 19. In addition to my GSI work, I've taught and developed course material for an undergraduate topics and introductory research course during Spring 15 and Spring 17. If you are interested in using some of my course materials, please contact me and I can provide you with the source material.

  • Mentoring and Other Outreach

    In 2014, I started up the Berkeley Mathematics Directed Reading Program along with Nick Ramsey and Alex Youcis. In addition to being a program organizer through 2017, I have run several mentored several students through the DRP program. I also have had several research undergrads.

    I have mentored for several summers at Canada-USA Mathcamp and was the 2018 academic program coordinator with Apurva Nakade. I've also taught for the Prison University Project, the Los Angeles Math Circle, and organized Splash at UCLA.

    • Notes

  • Symplectic Snippets

    Symplectic snippets is a proof-of-concept for organizing some of my mathematical notes on symplectic geometry. It was initially put together for the PIMS Summer School on Floer Homotopy theory. Notes are written as whole articles in LaTeX and then processed into mathematical tags -- snippets -- which can be individually referenced by other articles. The site is available here.

  • Assorted Notes

    Realizability and Obstruction in Tropical Geometry (2021)

    Lagrangian Mutation and Cobordism (2021)

    Lagrangian Cobordism and Lagrangian Surgery (2021)

    Generating the Mirror to a Toric Variety (Stanford 2020)

    Counting Disks on Lagrangian Cobordisms (ETH 2019)

    Tropical Lagrangian Submanifolds (Montreal 2019)

    Homological Algebra (Course from Mathcamp 2019)

    Combinatorial Topology (Being Updated!)

  • Seminars

    EDGE Seminar (2022-2023) co-organizer

    Hodge Seminar (2021-2023) co-organizer

    Student Symplectic Seminar (2016) organized with Morgan Weiler

    SFT Seminar (2016) with M. Weiler, C. Cannizzo, and B. Gammage

  • Seminar Notes

    Here are a few notes I've put together for seminar talks I've given. As these are seminar notes assembled primarily for my own benefit, they are all in draft form, and all mathematical errors are my own.

  • Floer theory and Lagrangian cobordisms(Simons 2022)

    Exact Sequences in Symplectic Geometry(PIMS 2022)

    A sketch of Symplectic cohomology(PIMS 2022)

    A sketch of Heegaard Floer(PIMS 2022)

    Exposition on Family Floer Theory (HMSS 2018)

    Continuation map for the Fukaya Morse Algebra (HMSS 2018)

    Polyfolds School (Summer 2017)

    Graph Cohomology and Perturbative Methods(STQFTS 2016)

    Classical Mirror Symmetry (SSS 2016)

    Stability Structures for Categories (MSS 2016)

    Sheaf-approaches to the Fukaya Category (MSS 2016)

    Knot Contact Homology (SDGT, 2016)

  • Berkeley Mirror Symmetry Seminar (Fall 2015)

    Abouzaid's Generation Criterion (MSS 2015)

    Hochschild Homology (MSS 2015)

    Moduli Space in Hamiltonian Floer Theory (SSS 2015)

    Donaldson's Divisor Theorem (SaQG 2014)

    Cobordism Map in ECH (ECHP 2014)

    Categorification (SOTS 2014)

    Functoriality of Chromatic Complex (Math 249 2013)

    Morse Homology (SOTS 2013)

    Module and Filtered Khovanov Homology (UCLA VIGRE 2012)

    Some equivalences in Extremal combinatorics (PUMC 2012)

    • Contact

  • Current Location

    office: 5622 James Clerk Maxwell Building

    email:

    office hours: by appointment

  • Upcoming Locations

    London, 18 nov 2024

    MPIM Bonn, 25 nov 2024

    University of Geneva, 6 dec 2024

    Oberwolfach, Toric Geometry, 6 apr 2025