Lecture: Higher structures in geometry and moduli spaces

4 hours lecture + 2 hours exercise session

(In der Prüfungsordnung erscheint diese Vorlesung unter Modul M40: Ausgewählte Themen der Mathematik)

Dingyu Yang

Winter term 2019/2020



Lecture: Tuesdays 11:00-12:30 in room 1.012 (RUD 25) and Fridays 9:30-11:00 in room 1.012 (RUD 25)
Veranstaltungsnummer: 3314514
First lecture: 15.10.2019

Exercise session: Fridays 11:00-13:00 in room 1.012 (RUD 25)
Veranstaltungsnummer: 33145141
First exercise session: 18.10.2019



Announcements:

Content: The term "higher structure" refers to a phenomenon in which natural algebraic identities hold only "up to homotopy". For example, in topology, addition in the fundamental group is associative thanks to a homotopy that reparametrizes the domain when concatenating three loops. This is a special case of a product operation on the chain complex of a based loop space (or analogously on the free loop space) being only associative up to homotopy; the higher structure mainly comes from the domain reparametrization. Aside from this reparametrization issue, the operation of cutting up loops is strictly co-associative on the nose, but to make things parametrized by manifolds feasible for a homology theory, one has to achieve transversality, which makes the co-associativity identity hold only up to homotopy. Higher structures can also naturally arise in moduli spaces with corners for elliptic geometric PDEs: for example, the Floer cohomology ring acting on a Fukaya algebra at the chain level should be an E_2-algebra acting on a cyclic A-infinity algebra. The combinatorial objects underlying the infinity-structures are also very interesting: these include Stasheff associahedra, permutohedra and secondary polytopes.

Higher structures are much richer than numerical invariants and can in themselves be regarded as geometric objects, which are sometimes even conjectured to faithfully represent the original geometric objects, e.g. in mirror symmetry and noncommutative geometry. At a more basic level, they provide a framework for understanding what invariant information one has captured, and for parametrizing the inductive constructions one hopes to carry out. Understanding and constructing higher structures is an active field of research and under rapid development. I hope to cover some aspects of the above story.

The following will provide some concrete topics and key words: I hope to cover the classical higher Massey product (as a better way to capture the ring structure on cohomology), and its "quantum" analogue in the Fukaya A-infinity algebra/category, the latter being an important symplectic invariant that provides bridges to other vastly different subjects. As an A-infinity structure is the first basic infinity structure, I will provide various viewpoints on it. I hope to cover the interaction of product and coproduct in the involutive bi-Lie infinity algebra of Cieliebak-Fukaya-Latschev, which connects string topology with symplectic field theory and the higher genus Fukaya category (and all three pictures can co-exist). Supplementary to this concrete description, I also hope to cover the recent conceptual treatment by Campos-Merkulov-Willwacher of a similar topic: the Frobenius properad satisfies Koszul duality. As a last topic, when a symplectic manifold is Kähler and equipped with a holomorphic Morse function, there is also a deep conjectural infinity-structure uncovered by physicists Gaiotto-Moore-Witten and mathematicians Kapranov-Kontsevich-Soibelman on a Fukaya category.

Prerequisites and Exam: I will provide background motivations and necessary definitions gradually, with useful results carefully stated and packaged into independent units. Prerequisites should include Differential Geometry I+II and Topology I+II, but will otherwise be consciously kept to a minimum, aside from curiosity and a certain willingness to favor "global and scenery pictures" over "complete details from axioms" (the latter can still be furnished for motivated individuals with guided follow-up reading). The knowledge assumed in each lecture will be summarized at the beginning of that lecture. Hopefully, the course should appeal to an audience with a variety of different interests and tastes in geometry, topology and/or algebra. The examination format will be flexible, e.g, it may consist in explaining or following up on a specific topic or result from the lecture.



Exercise sheets: (available after lectures on Tuesdays)
Exercise 1 (pdf)
Exercise 2 (pdf)
Exercise 3 (pdf)
Exercise 4 (pdf)



Table of Contents: (updated after the lectures, lecture notes are expository mostly borrowing from places freely with credits (borrowing from Banksy: everything here is recycled including ideas), typed version may appear optimistically with a lag of one week or two.)

Lecture 1 (15/10/2019): Homotopy transfer via homotopy retract from DGA. Introduction to A infinity algebra. Higher Massey products. Lecture note 1 (pdf)

Lecture 2 (18/10/2019): Loop product and coproduct on geometric chains of a loop space as an example to illustrate situations where associativity up to homotopy naturally arises. Transversality issues and reparametrization discussed. Expected Frobenius infinity compatibility not discussed yet.

Lecture 3 (22/10/2019): Universal property of tensor product. Koszul sign convention in differential graded context. Definition of A infinity algebra with correct signs. Tensor coalgebra (without unit) of the shifted graded (free) module and square 0 coderivation of degree 1 as an alternative definition, which explains the signs in the A infinity relation as an instance of the Koszul convention.

Lecture 4 (25/10/2019): Third interpretation of A infinity algebra, as a self-bracketing 0 (namely, homological) vector field on a non-commutative formal manifold in Kontsevich-Soibelman's viewpoint. A infinity morphisms (in two ways), composition being strictly associative.

Lecture 5 (29/10/2019): A infinity morphism as exponential, homotopy between two A infinity morphisms (in two ways), homological perturbation lemma, transfer by a formal diffeomorphism, quasi-isomorphisms are invertible up to homotopy. Three notions of units and equivalences.

Lecture 6 (1/11/2019): Formality. As an example to illustrate how to work with A infinity algebra and the transfer (by a formal diffeomorphism which is identity up to a certain order), we did a formality theorem due to Seidel that for an A infinity algebra (over field of characteristic 0) equipped with an Euler vector field (which is Hochschild 1-cocycle whose first term acts homologically on degree k homogeneous elements by multiplying by k). The notion of vector field here is different to that in Lecture 4, and will be made precise later. Another classical formality result is left as a guided exercise in Exercise sheet 4, see pdf above.

Lecture 7 (5/11/2019): First 6 lectures are about abstract A infinity algebra (without m_0 term). For next several lectures, the focus will be Fukaya algebra associated to a compact relatively spin Lagrangian in a compact symplectic manifold and it has m_0 term (thus an example of curved A infinity algebra) and defined over universal Novikov ring.

Presented a motivating example on Lagrangian self-Floer theory (cleanly intersecting with itself) based on de Rham model, viewed as half cylinder analogue of Hamiltonian Floer homology with Hamiltonian turned off and based de Rham model, and showed the obstruction for the obvious candidate of the differential to square to 0. Explained the moduli space M_{k+1}(beta) of stable nodal disks, and explained which the domain configurations are admitted (by following Fukaya-Oh-Ohta-Ono's toric survey paper). Introduced de Rham cochains completed over universal Novikov ring and sketched the definition of m_k.

Lecture 8 (8/11/2019): Introduced local models for resolving domain nodes (so the boundary nodal degeneration is codimension 1) and rephrase the more combinatorial description of domain configuration into one that is more suited for analysis (of gluing and deformation). Described in which sense M_{k+1}(beta) is a generalized space by giving a local model of Kuranishi chart, and motivated it as the form of a finite dimensional reduction of a Fredholm section in a Banach bundle over a Banach manifold base.

Lecture 9 (12/11/2019): Looked at how a choice for finite dimensional reduction choice is related to another choice which is more localized with bigger local "obstruction" bundle (see Exercise 3 in Exercise Sheet 4) thus introduced tangent bundle condition and hinted how a coordinate change between different Kuranishi charts should look like. Looked integration along fibers for a Kuranishi chart (ignoring compactness and compatibility across coordinate changes), see Recap 7 in Exercise Sheet 4, which is a bit more detailed. Introduced Stokes theorem for integration along fibers for ev_0 from M_{k+1}(beta) thus explain why include m_{1, 0} term in m_1, see first two paragraphs of Exercise 5 in Exercise sheet 4. Wrote down the projection formula and explain the origin of m_{k_2;beta_2} circ_i m_{k_1;beta_1} in A infinity relation as integration along fibers of ev_0 from one boundary stratum of M_{k+1}(beta), see Exercise 4 in Exercise sheet 4.

Lecture 10 (15/11/2019) Weinstein neighborhood theorem for Lagrangian and maybe a dose of Weinstein's symplectic creed that everything is a Lagrangian submanifold. Index theorem for Cauchy Riemann operator with totally real boundary condition (Olaf and the reading group from his Index Theorem seminar are welcome LOL), following a treatment in the Appendix of now published Abbas-Hofer book, Holomorphic curves and global questions in contac geometry. Some audience will be in Aachen enjoying Rabinowitz Floer theory and its product operation.

Lecture 11 (19/11/2019) We covered a case (when the Lagrangian has no non-constant holomorphic disks) where M_{k+1}(beta) is (transverse and of correct dimension and) a manifold with corners K_{k-2} x L for beta=0 and k>=2, and empty otherwise, and ev_0=pr_2 from it is always a submersion, so the integration along fibers is in the usual sense; fiber products are transverse and manifolds; no transversality issue whatsoever. Note that in this case m_1=d (the moduli space is empty, d is included always in m_1 due to Stokes formula) and m_2 is the wedge product (the moduli space fibers over L with each fiber one point, integration along fibers is identity), both up to signs, and m_k for k greater than 2 is 0, due to definition of integration along fibers (which is reasonable as the wedge product is strictly associative). We took this opportunity to introduce two realizations of Stasheff spaces K_{k-2}=Gr_{k+1}=M_{k+1}. We derived A infinity relation in general situation from projection formula, Stokes formula and Fubini formula. The derivation at the end is:
(ev^2_0)_!(p_2)_!((-1)^* p_2^* eta_2 wedge p_1^* eta_1)
(by projection formula)=(ev^2_0)_!((-1)^* eta_2 wedge (p_2)_! (p_1^* eta_1))
(by Fubini formula)=(ev^2_0)_!((-1)^* eta_2 wedge (ev^2_i)^*((ev^1_0)_! eta_1)))
It seems plausible that another and more non-trivial class of examples can be given by monotone (or semi-positive) Lagrangians, using moduli spaces of stable boundary-marked nodal disks (where each component is simple and without interior nodes), and intersection homology chain of L generated by maps from pseudomanifold (analogous space with boundary to pseudocycle) and its interaction with de Rham cochain version. But we will not go into that now.

Lecture 12 (22/11/2019): Tying up loose ends from last time. Dealing with m_0.

Lecture 13 (26/11/2019): We explained int_L eta wedge (ev_0)_! (ev_1^* eta_1 wedge ... wedge ev_k^1) is int_{M_{k+1}(beta)} ev_0^* eta wedge ev_0^* eta_1 ... wedge ev_k^* eta_k, but cannot express in basis of etas with coefficients open GW counts, nor a closed operation descending to one expressible in basis in homology; thus using (ev_0)_! instead. Also explained conceptually, for e: M to L, e_! eta is PD_L^{-1}(e_*(PD_M(eta))). Explained tensor coalgebra with length 0 factor, how coproduct, coderivation changes, and relation to A infinity algebra with m_0 term. Explained completion with respect to T filtration and valuation of Novikov ring. Recall Maurer-Cartan elements and explained gauge equivalence via a mode of [0,1] times C[1] (more adapted to the de Rham model as supposed to the singular chain model). Posed the question why b is called a Maurer-Cartan elements and why equivalence between them in such a way is called gauge equivalence (as generalization of which familiar object).

Lecture 14 (29/11/2019) Explained pseudo-isotopy (a natural object in the setting of comparison of choices of J), and P-parametrized A infinity algebra, and a construcfion to go from a pseudo-isotopy to [0,1]-parametrized A infinity structure, thus the equivalence of two notions. Show how to get a quasi-isomorphism (using d, not m_1) from a pseudo-isotopy between two A infinity algebras, by using time allocation and exploiting operad structure.

Lecture 15 (3/12/2019): Convergence in C^odd hat otimes Lambda_0 to define Maurer-Cartan elements, using forgetful map. Also explain why strict unit is achievable. Weak Maurer-Cartan using a strict/homotopy unit. Its geometric/obstruction theoretic interpretation. Potential function vs superpotential. Possibly cyclic symmetry and another criterion. H^1 subset weak MC/gauge equiv for toric case.
We will focus on next series of topics on M_{k+1}(beta), and they will fit into the following slots and spill over a few slots in January. After these, we have covered in depth algebraic aspect of A infinity structure, and a detailed example from moduli spaces. We will study L infinity, IBL infinity, Frobenius properad being Koszul, algebra of infrared, among the topics mentioned in the synopsis above.

Lecture 16 (6/12/2019): Explained strict unit for de Rham model for all cases where forg_i exists, the remaining case gives rise to the usual wedge. Explain the weak MC on (last time) extended H^odd otimes Lambda_0, and explain why the deformed d squares to 0.

Explained Gromov compactness of (higher genus) marked nodal J holomorphic curves with boundary on Lagrangians: Explain three types of stable Riemann surface (codimension-1) degenerations, corresponding to a separating curve connecting same boundary component, a non-separating curve connecting same boundary component, and a curve connecting two boundary components (necessarily non-separating), respectively degenerating. The fact that these possibilities are the only cases and realizable can be viewed via doubling using hyperbolic metric with the middle geodesic circle in thin part degenerating to a cylinder with a cusp in the middle. Compactness being local, if Lagrangian L is real analytic and J is integrable near L, then exists an anti holomorphic involution on W near L fixing L, and map pieces in that region can be doubled and reduced to pieces in the closed case. In general, proceed as in SFT compactness paper by Bourgeois-Eliashberg-Hofer-Wysocki-Zehnder. Explained why bubbling off occures, as only W^{1,2} bound is a priori available, which is not enough for applying Arzela-Ascoli which requires C^{0,alpha}. Apply Hofer's lemma to bubble off spheres or disks depending on how bubbling sequence and blowing up gradient interact relative to the boundary. Adding marked points to get gradient bound and domain stability; get domain convergence as above (via doubling), and reparametrization to bring domains to the same footing; obtain C infinity loc convergence away from geodesic circles on thin parts; use long cylinder of small area in the thin parts to connect, and to address the parts near the punctures; remove added marked points and collapse unstable components in the limit to obtain the true limit. References given.

Lecture 17 (10/12/2019) We followed the last chapter of FOOO book II to explain the signs in the A infinity operations in the Lagrangian Floer theory. We explain the canonical orientation of moduli space of J disks if L is equipped with a relative spin structure; the main argument involves a choice of trivialization of totally real subbundle along the boundary gives orientations of two linear operators glued together at one point to which the original operator is connected (forced to degenerate) to, where one is surjective and has Maslov index zero with the kernel identified, and the other admits complex orientation, and fiber product is transverse; then one glues back to original linearizaed operator. Use the data of relative spin structure and show path independence of disk boundary trivialization connecting the linearized operator at a general curve with a chosen curve in each component of the space of maps. This together with some convention determines the orientation of the space unparametrized marked disk (we explained the orientation for a local chart in a Kuranishi structure and indicated compatibility of coordinate change). Convention of fiber product orientation (including the case where one map is embedding). Convention of orientation of fiber product of evaluations of moduli space of disks and include of smooth singular chains into the Lagrangians. Explain the result that underlies the sign in front of m_{k_2} circ_i m_{k_1}. (One needs to switch the order of fiber products of two disk moduli spaces to confirm the conventions of the book and de Rham formalism, and also need to switch from smooth singular chain to de Rham cochain. Things are consistent after translating.)

Lecture 18 (13/10/2019) (Lazzarini and geometric transversality) We explained the second paper of Lazzarini on structure theorem non-constant J-holomorphic disk (c.f. Kwon-Oh), especially his notions of relation via limits of pairs of points with overlapping basis, of frame, of having relatively simple frame. We outline his Theorem A that away from frame (connected or not, where in the second case a simple J sphere can be constructed by removing deadends in the frame and gluing along boundaries as a tool to complete the argument), the original J disk restricts to each component factorizes into multiple covers of simple J-disks. We also explained the reason for the dimension constraint for his Theorem B. We explained the original formulation (not Deligne's perverse sheaf theoretic reformulation) of the intersection chain complex of Goresky-MacPherson for a pseudomanifold which admits a chain-level intersection theory if given a perversity function p(.) between 0 perversity and top perversity (extra dimension p(k) of leniency for failure of transversality of a chain intersecting with codimension k strata of the pseudomanifold), and use this, one should be able to bypass virtual techniques to construct what we have covered so far, by showing the moduli space of nodal disks each of whose component is simple and with boundary on a monotone (maybe even semi-positive) Lagrangian of dimension 3 or greater is a pseudomanifold. It remains an interesting question whether Theorems A and B of Lazzarini are true for general bordered Riemann surfaces, and locate where new arguments are needed and supply new arguments.

Lecture 19 (17/12/2019 last lecture of 2019) Involutive Bi Lie infinity algebra structure by Cieliebak-Fukaya-Latschev, with motivation from higher genus multiple boundary Lagrangian Floer theory.

No lecture (20/12/2019). Merry Christmas and Happy New Year!

No lecture (07/01/2020). Notification mails sent. (If you have not received email, please let me know).

Lecture 20 (10/01/2020). A bit of spectral sequence, central statement of calculation of Lagrangian Floer homology and some applications. We recalled an SES of cochain complexes induces the LES of the cohomology groups, then truncated back to SES from 0 to coker psi to cohomology of middle chain to ker psi to 0, where psi is connecting homomorphism. Following a note of Hutchings, regarding initial SES as an inclusion of filtration with graded piece quotient, we recast both nontrivial ends as cohomology of connecting homorphism extending by 0 into complex; which can be regarded as a map between cohomology of graded pieces, whereas graded pieces of cohomology are precisely the two nontrivial ends, which can determine cohomology of the middle (up to extension or when split). Thus the usual story becomes trying to calculate graded cohomology pieces using cohomology of graded pieces. This generalizes to spectral sequence for a filtered cochain complex. Introduce FOOO's definition and convention, compared with Hutchings and Voisin's definitions converted in FOOO's convention. Described completed filtered free module and gapped connection which allows filtration to be indexed by nonnegative integers. Explain E_1 with classical differential, E_2, and why E_infinity exists. Explain weak finite, which is true for Lagrangian Floer due to transfer, as a criterion for spectral sequence to stabilize and converges to graded pieces of Floer cohomology. Explain the central calculation result Theorem D in Book 1, and stated a deep non-vanishing result in the (homotopy) unital setting, and several symplectic geometric results regarding Maslov index conjecture, (monotone and full) Audin conjecture with citations.

Lecture 21 (14/01/2020) Explained a criterion of vanishing of obstruction for existence of deformed weak Maurer-Cartan elements in terms of Lagrangian embedding. Explained FOOO's q_k operator, which is similar to m_k except the 0th marked point is in the interior evaluating in M. Try to connect back with obstruction but time runs out before a comprehensive explanation can be given. There will be a note explaining this soon, to be appeared at this location.

Lecture 22 (17/01/2020) Explain the set up of Lagrangian Floer theory for a pair of (relatively spin) cleanly intersecting embedded Lagrangians L_1 and L_0; this Morse-Bott setting at two extremes either reproduces the version (Fukaya algebra) we have covered at length, or containing the case of two Lagrangian transversely intersecting at points (including the case of Hamiltonian Floer theory). We motivate it as a (semi-infinite where relative index between two critical points/Lagrangian intersection points is defined) Morse theory of closed action 1-form on path spaces. We explained the abelian Novikov covering where action 1-form lifts to a action functional. We explained generalized Maslov index in this setting which allows turning the relative bundle over a square strip into a bundle pair where the Maslov index can be assigned (after choosing the connected component near a linear Lagrangian corresponding to positive definite quadratic form). We explained the underlying chain complex and its grading. Explained why L_1 comes first in notation. We explain the moduli space of broken marked strips of several levels fiber producted together, where each level is a non-boundary nodal strip with boundary marked points with fiber product with ev_0 M_{k+1}(beta) to L_i, i=1, 0 before over some of the marked points. An operation can be cooked up using this moduli space. Codimensinal-1 boundary strata gives the bimodule identity which describes the higher-level failure of two sided module multiplication by Fukaya algebra of L_1 and L_0 being associative. We wrote down bimodule identity as a square 0 of an operation, commented on the similarity to A infinity structure by declaring on input as being special. This higher structure at chain level can be used to get a deformed differential via two-sided insertion of MC elements of each Lagrangians to give the differential of the chain complex, giving rise to Lagrangian intersection Floer homology of two cleanly intersecting Lagrangians. I quickly commented Hamiltonian moving boundary condition gives a parametrized A infinity bimodule structure (we covered parametrized A infintity structure before), which gives a pseudo-isotopy which in turn gives a A infinity homotopy equivalence/quasi isomorphism. Using A: Hamiltonian invariance in the previous sentence, B: this Floer homology reproducing single Lagrangian case for a pair of same Lagrangians, we applied this Lagrangian Floer homology to a symplectic manifold X Lagrangian embedded into the anti diagonal and consider its Hamiltonin deformation which is the graph of a non-degeneration Hamitonian diffeomorphism of X; then using homological injectivity of Lagrangian embedding in this case which implies C: (weakly deformed) unobstructedness of differential, and D: the spectral sequence convergence/collapsing theorem we covered one week ago, we arrived the homological Arnold conjecture classically proved using Hamiltonian Floer homology, and now proved via Lagrangian Floer theory. The last part was covered more quickly than I wished, a short note about it will appear here. The content of this talk is due to FOOO and can be found in their book 1 on Lagrangian Floer theory.

Lecture 23 (21/01/2020) In the honor of our visitor Zhengyi, we covered BV algebra. We followed Getzler, define Delta (which looks like an order 2 operator Laplacian) as a squaring 0, unit killing, degree +1 (opposite sign to the differential degree in the dg setting), which is a differential operator of degree (at least) 2. The last condition is a 7-term identity, which is equal to the 'failure of Delta being a derivation with respect to the product' satisfying the Poisson relation (the latter being that [a, ] satisfies derivation wrt product b.c with sign deg b(deg a-1) for all a; notice there is a typo of sign in definition of braid/Gerstenharber algebra in loc. cit. but the proof there uses the correct sign as specified in earlier part of this sentence). Moreover [,] is a Lie bracket compatible with Delta. We explained Cieliebak-Latschev's BV infinity notion, which decodes both differential and BV and BV squaring 0 up to homotopy. We connect IBL with BV. To do this first note sum m_{k,l,g} h^{g-1} square 0, and due to uncertainty relation [p_gamma ,q_gamma]=h (e.g. visible from SFT moduli space), we can represent p_gamma as h partial_{q_gamma} to reproduce this relation. p_gamma corresponds to positive puncture/boundary, thus hat m_{k,l,g} is order k with h^{k+g-1}. h coefficient corresponds to BV, thus Delta:= hat m_{2, 1, 0}+ h hat m_{1,2,0} has h^1 if regarded as a differential operator, and is order 2, and both have h^1 factor. Delta squaring 0 is precisely the IBL identity. Delta is a BV operation due to the aforementioned observations. So we have a BV from a IBL. If we have a dg BV algebra, we can get associated [ , ], and differential d+u Delta with formal variable h of degree -2, then we have (A[[u]], d+u Delta, [,]) is a dg Lie algebra which is an IBL with trivial cobracket. We explain homotopy S^1 action induces BV. To paint a big picture and connect with what we did before (bulk deformation from q was mentioned with some motivation before), we explain chains of framed little disk (operad) gives rise to BV_infinity. Using this with k+1 boundary marked points around the outer boundary, with inner circles with markers corresponding to orbits in symplectic (co)homology, and using chains of framed little disk, Floer data and pullback and pushforward of moduli spaces, we have conjecturally the E_2 action of SC (chain underlying symplectic (co)homology) on the cyclic Fukaya A infinity algebra, alluded to in the synopsis. In the closed case, BV operation degenerates as SC is just chain level genus 0 GW/chain of Quantum homology, and this is the generalized q operator of FOOO with compatibilify of product structures. Some short note will appear here.

(Just applicable to lecture 23: If you can, please go to the Floer homology seminar class immediately before this class (9:20am in Room 1.315), Alex Fauck will explain compactness (no escaping lemma) of symplectic homology among other things, and it would be helpful (but not strictly necessary) as an example where BV (chain level) structure natutally arises, where we use marked cylinder which twists to get a non-translation invariant and 1-parameter family operation, which is of opposite degree to rigid Floer cylinder.)

Lecture 24: (24/01/2020 notice the coincidence of Lecture number and the date). Since symplectic (co)homology is referred to or explained in SG seminar or Klaus's seminar class, we will cover the open analogue which is wrapped Fukaya algebra/wrapped Fukaya category. We will (not assume any SH background and) follow Abouzaid-Seidel's chain level homotopy direct limit treatment for the higher structure in the case of Liouville domains (we will also mention Fukaya's version on the complement in a compact Khler manifold of a possibly non-positive divisor). This will be useful for mirror symmetry and Atiyah-Floer conjecture, to each of which we will dedicate lecture(s). Since this Friday is not BMS Friday (and there is upcoming conference next month, certain subset of audience will be away), we are running low on classes and to reach enough areas I want to cover, I will teach 2 classes this Friday. I will try to make it easy and self-contained within our lecture course as much as possible.

Lecture 25: (28/01/2020) We recapped what we had covered last time in the construction of wrapped Fukaya algebra via homotopy limit of integer constant slope of Hamiltonian at the conic end, via the compactified moduli space of stable nodal/broken popsicle maps with weights as lengths of chords. This made being present on last Friday not a strict prerequisite. We explained (achievable) conditions ensuring that everything stays inside the domain (away from the end in the completion) thanks to genericity and convexity and by construction (choice of H, and subclosed consistent 1-form which interpolates dt with weights at strip-like ends). We explain abstruct d bar operator on half plane asymptotic to a X_H chord and getting degree and orientation line from it. We explained the construction of A infinity structure, using geometric perturbation, dimension 0 smooth moduli, and admissible cut; we described the homotopy direct limit construction which is convenient for chain level (A infinity) structure. This construction is adapted to show Viterbo functoriality restricting to an inner Liouville subdomain is an A infinity morphism. Which we sketched quickly by the analogy to pseudo-isotopy. We mentioned the alternative of using quadratic Hamiltonian and rescaling trick, also due to Abouzaid and Seidel, which is convenient for other purposes. This completely echoes with Alex's lecture on closed case of symplectic (co)homology in the other seminar/course.

Lecture 26: (31/01/2020) We plan to do a bit of homological mirror symmetry of Calabi-Yau hypersurface following Sheridan with lots of statement/black boxes, to show some logic flow and flavor of this fast developing area. Only one lecture due to BMS Friday event.

The plan for remianing lectures of this course is the following:

Lecture 29: (04/02/2020) We will talk a bit about Atiyah-Floer conjecture which features a nice application of infinity structure where algebra seems much easier than analysis.

(07/02/2020) Conference for a subset of the audience. No lecture.

Lecture 30: (11/02/2020) We say a bit about quadratic relation, operad, Koszulness of Frobenius properad, and relevance to our discussion of involutive bi-Lie infinity algebra, we covered before. (This fulfills a part of synopsis.)

Lecture 31: (14/02/2020) For this Friday, due to the interplay of no BMS event and not wanting the last official lecture to be too hardcore, we will do one lecture worth of material, but no speeding towards speed of light at the end. So we will take our time and freely spill over into the second session and can end early in that session if material coverage reaches its conclusion. Namely, Russian seminar style but with a cap or 3 hours. Also good news is that non-mathematical popsicles will be present as well.

In this romantic date for the last lecture (or lectures if no BMS Friday), we will study transversality via virtual technique. We will explain several issues that show up and need to be addressed. This will be an abridged version of below, 'game of thrones season 8 style':

Transversaity I: Kuranishi structures, good coordinate systems, how to switch between, corners, fiber products.
Transversality II: Iterative construction of Kuranishi structures with boundary strata being fiber products.
Transversality III: Iterative perturbation and choices, integration along fibers.
Transversality IV: Respecting the forgetful map forgetting boundary marked points: homotopy unit and cyclic symmetry.

If you intend to take exams for this course, please choose a date and sign up a slot before the deadlines pass. Because I was notified just now the deadline for the exam date 25/02/20 has passed. The exam format will be a presentation of some topic in or relevant to this course in a friendly setting with some discussion. Please feel free to email me or talk to me about details after the lecture or in my office. See you on Friday for the grand finale.

An extra session on gluing will happen due to request, with the date to be confirmed later, after Feb 22 my return from a conference.

I also want to cover:
Topic 1: gluing of maps solving elliptic PDE and smoothness at infinity. Due to demand, this will happen. I think I want to cover this fairly systematically. There are at least four ways of doing the gluing in various contexts, I will try to explain the differences and look more deeply into the mechanism of (2 of) the gluings that guarantee the smoothness at the node/infinty. I will post the date and format here once I get it prepared and will also send an email in advance.
Topic 2: algebraic compatibility of loop product and loop coproduct and infinity structure. (This might just appear as a brief note here, if no one approaches me about listening to this topic.)

Topic 3: I might want to explain a bit on model category, simplicial sets, C infinity rings, quasi-smooth derived manifolds. Intrinsic normal cones, and resp. Riemann-Roch way of getting virtual fundamental classes/chains.

This either makes the course more complete, or highly relevant to the theme of the course.
I could do two more extra sessions (kind of) making up the missed lectures. This is done for some other lectures or seminars in the past. But due to potentially heavy nature of content, I will do this only if the audience do not channel John Snow (who often says, I don't want it).








Literature:

Lecture 1: We borrow from Part 1 of Vallette's https://arxiv.org/abs/1202.3245, but changed into cochain conventions for strict associativity of singular/de Rham cochains. (Remark: The C infinity algebra on cohomology is a complete invariant for simply connected rational homotopy type is a result of Kadeishvili in 2008, https://arxiv.org/abs/0811.1655.)

Lecture 2: Some background of string topology operations and compatibility can be found in Chas-Sullivan's https://arxiv.org/abs/9911159 and Sullivan's status on string topology circa 2007 at https://arxiv.org/abs/0710.4141. (Stasheff polytopes will be dicussed in detail later, as planer rooted trees, as Deligne-Mumford moduli for disks with boundary marked point, as the secondary polytope of a convex polygon.) Typed note for materials covered in the lecture will be provided here soon. (Lecturer's research preprint (with Manuel Rivera) contains a rigorous construction of coproduct that is closest to the geometric picture we discussed and its co-A infinity structure. Might touch upon it later after the preprint is polished and posted.)

Lecture 3: Definition of A infinity algebra can be found in Fukaya-Oh-Ohta-Ono's book 1 on Lagrangian Intersection Floer theory, and Seidel's EMS book on Fukaya categories and Picard-Lefschetz Theory. Some details of tensor coalgebra can be found in 1.2 of Loday-Vallette's Algebraic Operads. (Please read it only after attempting the last exercise in Exercise sheet 2.)

Lecture 4: For A infinity structure as a homological vector field on a non-commutative formal manifold, see Kontsevich-Soibelman's https://arxiv.org/abs/math/0606241 (which elaborates on the functor of points) as well as Cho-Lee's https://arxiv.org/abs/1002.3653.

Lecture 5: Seidel's Fukaya categories and Picard-Lefschetz Theory and Fukaya-Oh-Ohta-Ono's book 1 on Lagrangian Intersection Floer theory.

Lecture 6: Theorem 2.6 in Abouzaid-Smith's paper https://arxiv.org/abs/1311.5535.

Lectures 7 - 9 and more: Fukaya-Oh-Ohta-Ono's (FOOO) book 1 (and 2) on Lagrangian Intersection Floer theory. The virtual techniques based on de Rham cochain models https://arxiv.org/abs/1503.07631 and https://arxiv.org/abs/1704.01848 which should appear together as a book soon. Some lighter reading can be found in their toric survey paper and Ohta's survey. The part of lecture on Fukaya algebra for a Lagrangian hopefully will ease the audience into some ingredients of this picture.

This part needs more updates and some references are given in the content of lectures and will be updated here.

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