Title: Complex plane curves and positive braids

Abstract: The easiest example of a complex plane curve is an algebraic embedding of the line in the plane. Our first goal is to sketch Rudolph's proof of the Abhyankar-Moh Theorem: there is only one algebraic embedding of C in C^2, up to algebraic automorphisms. The corresponding statement about embeddings of C in C^3 remains unsettled.

In the second part, we consider singular algebraic curves, which come in countably many types, up to topological equivalence. The complement of an isolated singularity can be described by a knot or link. In the case of simple singularities, classified by Dynkin diagrams, these links even determine the analytic type of the singularity. We will discuss examples of simple and non-simple singularities in detail.

A surprisingly elementary open question about the realisation of simple singularities brings us to the third topic of this course: positive braids. We will focus on a partial order on positive braid links, which allows to study cobordisms between positive braids, their signature invariant, dealternation number and even their Khovanov homology.

Kristen Hendricks (Michigan St.)

Title: Involutions and Floer Homology

Abstract: In this set of lectures, we will discuss equivariant (mostly Z_2, with occasional digressions into other groups) versions of several theories collected on the term "Floer homology." The first two lectures will focus on equivariant Lagrangian Floer cohomology. We will first do a quick review of ordinary equivariant cohomology, give an abbreviated introduction to Lagrangian Floer cohomology, and discuss some of the technical issues involved in constructing equivariant versions of same. We will then go over two major approaches to resolving these technical difficulties, with some examples and applications of each to such theories as Heegaard Floer homology and symplectic Khovanov homology.

The third lecture will take a slightly different tack and focus on equivariant versions of Seiberg-Witten Floer homology (and, eventually, its analog Heegaard Floer homology) with applications to the homology cobordism groups. We will talk briefly about the structure of the integer homology cobordism group, and then discuss Manolescu's use of a Pin(2)-equivariant version of Seiberg-Witten Floer homology to resolve the Triangulations Conjecture. Finally, we will then discuss additional recent applications of the same ideas, including the use of an involution on Heegaard Floer homology to construct an "involutive" theory analogous to Z_4-Seiberg Witten Floer homology.

Lenny Ng (Duke)

Title: Knot contact homology and augmentation varieties

Abstract: Given a submanifold K of a smooth manifold M, the conormal bundle to K is a Lagrangian submanifold of the cotangent bundle T^*M. Studying the symplectic geometry of conormals has proven to be a fruitful technique for studying smooth topology. In this minicourse, we focus on knots in R^3: here counting holomorphic curves with boundary on the conormal yields a knot invariant called knot contact homology.

This minicourse, which I hope will be broadly accessible, will introduce knot contact homology and attempt to place it in some context, including some unexpected relations to other parts of topology as well as mathematical physics. The first lecture will provide background on the conormal construction and the definition of knot contact homology. In the second lecture, we will extract an algebraic variety, the "augmentation variety", from knot contact homology, and discuss a conjectural relation to topological string theory, mirror symmetry, and colored HOMFLY-PT polynomials. In the third lecture, we will discuss a way to understand knot contact homology through "string topology", and how this can be used to prove that the conormal bundle is a complete knot invariant.

Inanc Baykur (UMass)

Title: Small symplectic and exotic 4-manifolds via positive factorizations

Abstract: We will discuss new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes (joint work with Mustafa Korkmaz) which yield novel constructions of interesting symplectic and smooth 4-manifolds, such as small symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz fibrations and pencils.

Peter Feller (Boston College)

Title: Uniqueness of embeddings of the affine line into algebraic groups

Abstract: In the first part of the talk, we present a low-dimensional topology point of view on polynomial embeddings of the complex line into complex n-space. This will include interesting knot-theoretic examples due to Shastri and a surprising result about polynomial embeddings into 3-space due to Kaliman. In the second part of the talk, we discuss joint work with Immanuel Stampfli about uniqueness of polynomial embeddings into affine algebraic groups.

Eugene Gorsky (UC Davis)

Title: Splitting numbers for links and Heegaard Floer homology

Abstract: The splitting number of a link is the minimal number of crossings between different components that should be changed to transform it to a split link. I will describe a lower bound for the splitting number using Heegaard Floer homology. It is especially effective for L-space links, and I will present an infinite family of L-space links with vanishing linking numbers and arbitrary large splitting numbers. The talk is based on a joint work with Maciej Borodzik.

Mark Hughes (BYU)

Title: Recognizing quasipositive braids and links

Abstract: Quasipositive braids are a subclass of braids which can be represented as products of positive twists between pairs of strands. The closures of quasipositive braids give quasipositive links, which are links that arise as the intersections of complex plane plane curves with the 3-sphere. These links also satisfy special properties with respect to their smooth four-ball genus.

A question of Rudolph asks to describe a method for determining whether a given braid or link is quasipositive. Orevkov presented a method for answering this question in the case of 3-strand braids. In this talk we present an algorithm for detecting quasipositive braids of any index. These techniques make use of mapping class orderings of Rourke and Wiest.

The question of detecting quasipositive links is much harder, and remains open. We will however discuss computational methods which in many cases can aid in the recognition of quasipositive links.

Xin Jin (Northwestern)

Title: Nadler-Zaslow correspondence without Floer theory

Abstract: The Nadler-Zaslow correspondence assigns every exact Lagrangian in the cotangent bundle of a manifold M a constructible sheaf on M. The construction follows from Floer theory. I will present an alternative sheaf-theoretic way to realize the correspondence, which has the benefit that it can be generalized to coefficients in a ring spectrum. Along the way, we will see that this approach gives a natural way of understanding "brane obstructions" in the usual Floer theory, which turns out to be closely related to the J-homomorphism. This is joint work with David Treumann.

Caitlin Leverson (GA Tech)

Title: Invariants of Legendrian knots

Abstract: Given the plane field ker(dz-xdy) in R^3, a Legendrian knot is a knot which at every point is tangent to the plane at that point. One can similarly define a Legendrian knot in any contact 3-manifold (manifold with a plane field satisfying some conditions). In this talk, we will explore Legendrian knots in R^3 and the connect sum of finitely many copies of (S^1x S^2) as well as a few Legendrian knot invariants. We will also look at the relationships between two of these knot invariants. No knowledge of Legendrian knots will be assumed though some knowledge of basic knot theory would be useful.

Ina Petkova (Dartmouth)

Title: Tangle Floer homology and quantum gl(1|1)

Abstract: The Reshetikhin-Turaev construction for the standard representation of the quantum group gl(1|1) sends tangles to C(q)-linear maps in such a way that a knot is sent to its Alexander polynomial. After a brief review of this construction, I will give an introduction to tangle Floer homology — a combinatorial generalization of knot Floer homology which sends tangles to (homotopy equivalence classes of) bigraded dg bimodules. Finally, I will discuss how to see tangle Floer homology as a categorification of the Reshetikhin-Turaev invariant. This is joint work with Alexander Ellis and Vera Vertesi.

Jacob Rasmussen (Cambridge)

Title: Loops and L-space intervals

Abstract: Let Y be an oriented three-manifold with torus boundary. If the bordered Floer homology of Y has a particularly simple form - known as loop-type - it can be represented by an immersed curve in the solid torus. I'll describe how this idea can be used to characterize the set of L-space Dehn fillings of Y. This is joint work with Jonathan Hanselman and Liam Watson.

Peter Samuelson (Iowa)

Title: The elliptic Hall and Homfly skein algebras

Abstract: The Homfly skein algebra H(F) is spanned by "links modulo skein relations" in a (thickened) surface F, where the product comes from stacking. The elliptic Hall algebra E has structure constants that count extensions of sheaves over elliptic curves in finite characteristic. We show H(T^2) is isomorphic to E by describing H(T^2) explicitly and comparing this to a description of E given by Burban and Schiffmann. We also show a natural action of H(T^2) on the space of symmetric functions agrees with an action of E constructed by Schiffmann and Vasserot using Hilbert schemes. (This is joint work with H. Morton.)

Steven Sivek (Princeton)

Title: Stein fillings and SU(2) representations

Abstract: In recent work, Baldwin and I defined invariants of contact 3-manifolds with boundary in sutured instanton Floer homology. I will sketch the proof of a theorem about these invariants which is analogous to a result of Plamenevskaya in Heegaard Floer homology: if a 4-manifold admits several Stein structures with distinct Chern classes, then the invariants of the induced contact structures on its boundary are linearly independent. As a corollary, we conclude that if a homology sphere Y admits a Stein filling which is not a homology ball, then its fundamental group admits a nontrivial representation to SU(2). This is joint work with John Baldwin.

Laura Starkston (Stanford)

Title: Line arrangements in symplectic topology

Abstract: We will discuss topological and symplectic versions of a classical topic in combinatorics and algebraic geometry: arrangements of lines in the plane. We will give an overview of some interesting phenomena that occur in the algebraic setting, and discuss how this compares to the topological and symplectic settings. We will also discuss what consequences these results can have in contact and symplectic topology. Knowledge about line arrangements could potentially weigh in on some long standing difficult open problems in symplectic topology, and there are also many accessible interesting open questions. This talk is based on joint work with Danny Ruberman.

Lisa Traynor (Bryn Mawr)

Title: A Quantitative Look at Lagrangian Cobordisms

Abstract: Lagrangian cobordisms between Legendrian submanifolds arise in Relative Symplectic Field Theory. In recent years, there has been much progress on answering qualitative questions such as: For a fixed pair of Legendrians, does there exist a Lagrangian cobordism? I will address two quantitative questions about Lagrangian cobordisms: For a fixed pair of Legendrians, what is the minimal “length” of a Lagrangian cobordism? What is the relative Gromov width of a Lagrangian cobordism? Regarding length, I will give examples of pairs of Legendrians where Lagrangian cobordisms are flexible in that the non-cylindrical region can be arbitrarily short; I will also give examples of other pairs of Legendrians where Lagrangian cobordisms are rigid in that there is a positive lower bound to their length. For the second quantitative measure, I will give some calculations and estimates of the relative Gromov width of particular Lagrangian cobordisms. This is joint work with Joshua M. Sabloff.