Paul Seidel: Symplectic Floer cohomology with mod p coefficients
The theory of pseudo-holomorphic curves works most naturally if one uses cycles with rational coefficients. Still, there are reasons to use mod p coefficients, even if that is technically awkward and restricts one's generality. The reasons can be roughly speaking grouped as follows.
(1) There are specific topological applications where using rational coefficients is either impossible or yields a trivial result. This includes quite simple Lagrangian submanifolds, such as the real part of CP^n.
(2) There are algebraic structures which only work well with mod p coefficients, and that can again be useful for specific geometric applications. Localisation theorems for cyclic group actions, and quantum Steenrod operations, are examples of such structures.
(3) One wants to derive results about enumerative geometry with classical characteristic 0 coefficients from mod p arguments. This is a familiar idea in number theory, but its entry into symplectic topology is more recent (and so far, its impact is limited to a specific class of questions).
A preliminary plan would be spend one lecture on (1) and the beginning of (2); the second lecture on the rest of (2); and the last lecture on (3). Looking at the list of references below, it seems that a thorough discussion would require the exposition to accellerate to warp speed at the start of the second lecture, and then to ludicrous speed for the third; given realistic speed limitations, we'll see how far we actually get. At least for (1), I intend to keep things to very classical and basic questions in symplectic topology.
References for (1):
Y.-G. Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Int. Math. Res. Notices (1996), 305-346.
J. Evans and Y. Lekili, Floer cohomology of the Chiang Lagrangian, arXiv:1401.4073 G. Alston, Lagrangian Floer homology of the Clifford torus and real projective space in odd dimensions, J. Symplectic Geom. 9 (2011), 83-106.
References for (2):
P. Seidel and I. Smith, Localization for involutions in Floer cohomology, arXiv 1002.2648.
P. Seidel, The equivariant pairs-of-pants product in fixed point Floer cohomology, arXiv:1411.2639.
K. Hendricks, A spectral sequence for the Floer cohomology of symplectomorphisms of trivial polarization class. IMRN 2017, 509-528.
P. Seidel and N. Wilkins, Covariant constancy of quantum Steenrod operations, arXiv:2102.06432.
E. Shelukhin, Pseudorotations and Steenrod squares revisited. Math. Res. Lett. 28 (2021), 1255-1261.
E. Shelukhin and J. Zhao, The Z/p-equivariant product-isomorphism in fixed point Floer cohomology. arXiv:1905.03666.
E. Cineli, V. Ginzburg and B. Gurel. From pseudo-rotations to holomorphic curves via quantum Steenrod squares. IMRN 2022, 2274-2297.
References for (3):
N. Katz, Nilpotent Connections and the Monodromy Theorem. Publ. Math. IHES 39 (1970), 175--232.
Z. Chen, On the exponential type conjecture. arXiv:2409.03922 D. Pomerleano and P. Seidel, The quantum connection and its mod p reduction. To appear
Stretch goal (probably not): A. Petrov and V. Vologodsky, On the periodic topological cyclic homology of DG categories in characteristic p, arXiv:1905.03666
Jae Lee: Arithmetic aspects of quantum cohomology
Lecture (1) Quantum cohomology and the quantum connection
We’ll review the basics of (genus 0) Gromov—Witten invariants and the algebraic structures (quantum cohomology and the quantum connection), and discuss the situation in which their classical definitions yield integer invariants. The monotone and the Calabi—Yau case behave considerably differently.
Lecture (2) Quantum cohomology over F_p
Differential equations in characteristic p behave very differently from their ordinary, characteristic zero counterparts, as differential operators become linear over a much larger ring. In symplectic topology, this phenomenon is reflected by the new structures that arise in (equivariant) quantum cohomology: the quantum Steenrod operation, and the p-curvature of the quantum connection.
Lecture (3) Quantum cohomology over Z_p
We will study the p-adic lifts of the quantum connection, situating them in the context of p-adic differential equations. We review the notion of a Frobenius structure and how it is related to the contents of the previous lectures under “reduction mod p”. We will end by introducing the key conjectures regarding the structure of the quantum connection in the p-adic setting.
References
Z. Chen, On the exponential type conjecture, arXiv:2409.03922
McDuff—Salamon, J-holomorphic curves and symplectic topology, Amer. Math. Soc. Colloq. Publ., 52 (2004)
Lee, Quantum Steenrod operations of symplectic resolutions, Geom. Topol. 29 (2025)
Seidel—Wilkins, Covariant constancy of quantum Steenrod operations, J. Fixed Point Theory Appl. 24 (2022)
Bai—Pomerleano—Seidel, P-adic Gamma classes and overconvergent Frobenius structures for quantum connections, arXiv:2509.26295
Bai—Lee—Pomerleano, Arithmetic geometry of quantum connections on Calabi-Yau 3-folds, arXiv:2601.01654
Katz, On the differential equations satisfied by period matrices, Inst. Hautes Études sci. Publ. Math. 35 (1968)
Seidel, P-adic splittings of the quantum connection, arXiv:2503.00500
Chen, The Getzler-Gauss-Manin connection and Kontsevich-Soibelman operations on the periodic cyclic homology, arXiv:2601.16437
Chen, Quantum Steenrod operations and Fukaya categories, arXiv:2405:05242
Ganatra, Cyclic homology, S1-equivariant Floer cohomology, and Calabi-Yau structures, Geom. Topol. 27 (2023)
Petrov—Vologodsky, On the periodic topological cyclic homology of DG categories in characteristic p, arXiv:1912.03246
Soham Chanda: Global Kuranishi charts and applications
Moduli spaces of pseudoholomorphic curves have proved to be a monumental tool in symplectic topology, but unfortunately they are not always as well behaved as one would like. Under suitable constraints, one can achieve transversality and ensure certain moduli spaces are manifolds, but one cannot avoid virtual techniques when dealing with a general setup. Virtual techniques often carry substantial technical overhead, but global Kuranishi charts, introduced in AMS21, have made the toolkit much more accessible. In the course of three lectures, we will start from the basics of the local structure of moduli spaces of pseudoholomorphic curves and then cover the construction of global charts along with the necessary complex geometry background.
Lecture I : Local structure of moduli spaces of pseudoholomorphic curves
After a quick recap of moduli spaces of pseudoholomorphic curves, we will show that the said moduli spaces can be locally presented as the zero locus of a section of a finite dimensional problem - aka they admit local Kuranishi charts.
Lecture II : Global Kuranishi Charts : Background and Construction for genus 0
A key step in constructing the global Kuranishi chart uses input from moduli spaces of curves in projective space. We will cover the necessary background and start the construction of global chart for genus-zero curves in closed symplectic manifolds.
Lecture III: Global Kuranishi Charts : Technical bits and more constructions
We will finish the construction of the global Kuranishi chart for genus-zero curves and, time permitting, discuss further constructions, such as curves in Hamiltonian Floer theory, open curves with Lagrangian boundary conditions, and curves in rational SFT.
Egor Shelukhin: Quantum Steenrod powers and Hamiltonian maps
Recent years have seen new developments in Hamiltonian dynamics, as new methods related to characteristic p methods in Floer theory have become available. In particular, quantum deformations of Steenrod powers, first suggested by Fukaya, as well as Smith theory in Floer cohomology via the equivariant pants product introduced by Seidel, play an important role in these new advances. We survey these developments, in particular presenting a recent joint work with Shaoyun Bai, Nicholas Wilkins, and Guangbo Xu obtaining a number of definitive results in this direction.
Tentative lecture plan:
Lecture 1
Basic notions related to Floer theoretic methods in dynamics, including spectral invariants, local Floer homology and related index theory, as well as barcodes of Hamiltonian diffeomorphisms; sketch of a proof of the Conley conjecture for symplectically aspherical manifolds.
Lecture 2
Equivariant Steenrod operations and pants products; applications to the existence of infinitely many periodic points under various conditions: Hofer-Zehnder conjecture, non-torsion orbits, symplectically degenerate maxima.
Lecture 3
Obstructions to the existence of pseudorotations and finite order maps. Elements of C^0 symplectic topology, Polterovich's question, the Hilbert-Smith conjecture in the symplectic setting.
References:
Bai, Shelukhin, Wilkins, Xu: Quantum Steenrod powers and Hamiltonian maps, https://arxiv.org/abs/2607.25960
Shelukhin: A symplectic Hilbert-Smith conjecture, https://arxiv.org/abs/2403.07195
Shelukhin: On the Hofer-Zehnder conjecture, https://annals.math.princeton.edu/2022/195-3/p01
Shaoyun Bai: Integral Curve Counting and Applications
Due to the presence of automorphisms of J-holomorphic maps, Gromov-Witten--type invariants and Floer-theoretic invariants are only defined over the rationals in general using classical methods. In this minicourse, we will give a brief introduction to the Fukaya-Ono-Parker perturbation method, which allows us to define integral counts that lead to concrete applications in the topology of algebraic varieties and Hamiltonian dynamics. We will use the construction of quantum Steenrod operations as an example illustrating how to deploy the modern machinery to go much beyond the traditional technical restriction.
Lecture 1: Equivariant Transversality and FOP perturbations
Lecture 2: Cohomological Splitting in Positive Characteristics
Lecture 3: Quantum Steenrod Operations for General Symplectic Manifolds
References:
Bai--Xu, A new transversality condition on orbifolds and integer-valued Gromov-Witten type invariants, arxiv: 2201.02688
Bai-Pomerleano-Xu, Cohomological splitting over rationally connected bases, arxiv: 2406.00931
Bai--Xu, Integral Hamiltonian Floer theory: foundations, arxiv: 2606.15847