**Wehrheim/Fish/Hofer (9 lectures):** *Polyfolds and the construction of Symplectic Field Theory.* Polyfold theory provides a language and a large body of results in an area, which is a mixture of a generalized differential geometry, nonlinear functional analysis, and category theory. It includes a very general concept of a Fredholm section accompanied by Sard-Smale type transversality theory over the rational numbers, i.e. perturbing through rationally weighted multi-sections. As such, it is a powerful tool to address complicated transversality issues as they arise in constructing compact moduli spaces in symplectic geometry. In the first week, Fish and Wehrheim will introduce the analytic foundations of polyfold theory, including the notions of sc-calculus, retracts, M-polyfolds, strong bundles, Fredholm sections and regularization theorems. In the second week, Hofer will build on these foundations, aiming to construct the moduli spaces needed for general Symplectic Field Theory.

**Lecture 1 -- (Wehrheim)***Introduction to Polyfold Regularization.*This lecture will discuss the overall ideas and challenges in regularizing moduli spaces, and introduce the two basic ideas behind polyfold theory: Making reparametrization actions "smooth" and making pregluing a "chart map". [related literature: Sections 2.1 and 3.3 of Polyfolds: A First and Second Look. Related videos: Lecture 8 and Lecture 20 from Wehrheim's special topics course.]**Lecture 2 -- (Fish)***sc-Banach spaces and the sc-calculus.*(preliminary lecture notes) Polyfold theory is built on two new fundamental analysis concepts, and this talk is focused of the first: the sc-calculus. We discuss how in general the action of a finite dimensional smooth reparametrization group on typical Banach spaces of maps is not smooth (in fact, not even differentiable), and then introduce sc-Banach spaces and the notion of sc-differentiability. Two key results of this talk are that the action of reparametrization is sc-smooth, and for sc-differentiable functions the chain rule holds, so that many constructions in classical differential geometry functorially extend to sc-differentiable geometry. [Related literature: Sections 2.2 and 4.2 of Polyfolds: A First and Second Look. Related videos: Lecture 20 from Katrin's special topics course.**Lecture 3 -- (Fish)***The rise of sc-retracts.*(preliminary lecture notes) In this talk, we discuss the second of two fundamental analysis concepts polyfold theory is built on: sc-retracts. In particular, we discuss how they arise naturally as a means of using pre-gluing maps to parametrize a neighborhood of nodal and non-nodal (or broken and unbroken) maps near a nodal (or broken) map. Despite locally varying dimensions, such retracts support a version of the sc-calculus on which the chain rule holds, and we define M-polyfolds (manifold-like polyfolds) to be those topological spaces locally modeled on such retracts. [Related literature: Sections 2.3 and 5.1 of Polyfolds: A First and Second Look. Related videos: Lecture 22 from Wehrheim's special topics course.**Discussion I. -- (Bottman)**Homework problems on sc-calculus and sc-smooth retracts will be discussed. Provided there is sufficient interest and time, we will sketch out the definition of the the local charts for polyfolds for Gromov-Witten.**Lecture 4 -- (Fish)***Boundary, corners, strong bundles, and implicit function theorems.*(lecture notes) In this talk, we generalize the notion of sc-retracts to include cases with boundary and corner structure. In addition, we develop the notion of a strong bundle (of which the Cauchy-Riemann operator is a section) and state an implicit function theorem for transverse Fredholm sections with compact zero-set, which guarantees the zero set of the section is a manifold with boundary and corners, with boundary/corner structure induced from the ambient M-polyfold. [Related literature: Sections 5.2, 5.3, and 6.1 of Polyfolds: A First and Second Look. Related videos: Lecture 23 and Lecture 24 from Wehrheim's special topics course.**Lecture 5 -- (Wehrheim)***Regularization theorem for Fredholm sections of M-polyfold bundles.*This lecture will state a rigorous version of this theorem, and explain the notion of a (sc-)Fredholm section. [related literature: Sections 6.2 and 6.3 of Polyfolds: A First and Second Look. Related videos: Lecture 24 and Lecture 25 from Wehrheim's special topics course.]**Lecture 6 -- (Hofer)***Polyfolds and the construction of Symplectic Field Theory.*Topics:- Short overview.
- The category of stable maps.
- I/O structures.
- Covering structures.
- Additional structural constraints.

**Discussion II. -- (Hofer)****Lecture 7 -- (Hofer)***Polyfolds and the construction of Symplectic Field Theory.*Topics:- Polyfold structures.
- Consequences of polyfold structures.
- Weighted categories and their smooth versions.
- The polyfold of stable maps.
- Bundle category.

**Lecture 8 -- (Hofer)***Polyfolds and the construction of Symplectic Field Theory.*Topics:- Strong bundle structure and the CR-section as Fredholm functor.
- Polyfold packaging of the SFT problem.
- Smooth Multisection functors and smooth weighted subcategories.
- Construction of sc^+ multisection functors.
- Auxiliary norms and compactness control.

**Lecture 9 -- (Hofer)***Polyfolds and the construction of Symplectic Field Theory.*Topics:- Perturbation algorithm in the homogeneous case.
- Perturbation algorithm for relating two perturbations.
- Remarks on orientations.
- Representation theory and SFT.

**Discussion III. -- (Hofer)**

- Homework Problems -- Suggested homework problems for the first week of the polyfold mini-course. (Solutions to be discussed during Bottman's discussion on Wednesday.)
- Extra Homework Problems -- Some extra homework problems that were given during Wehrheim's lecture course. Solutions.

- Wehrheim's lecture course
- Polyfolds: A First and Second look
- Polyfolds and Fredholm Theory
- Polyfold and Fredholm Theory I: Basic Theory in M-Polyfolds
- Applications of Polyfold Theory I: The Polyfolds of Gromov-Witten Theory
- Sc-Smoothness, Retractions and New Models for Smooth Spaces
- A General Fredholm Theory III: Fredholm Functors and Polyfolds
- Polyfolds And A General Fredholm Theory
- Integration Theory for Zero Sets of Polyfold Fredholm Sections
- A General Fredholm Theory II: Implicit Function Theorems
- A General Fredholm Theory I: A Splicing-Based Differential Geometry
- A General Fredholm Theory and Applications
- Compactness results in Symplectic Field Theory
- Introduction to Symplectic Field Theory