All talks and activities will be in Skiles 006, 005, or the atrium outside those two rooms.

**Monday**

8:00-8:50 | Morning Refreshments | |

8:50-9:00 | Welcome | |

9:00-10:00 | Cameron Gordon | The L-space conjecture |

10:30-11:30 | Irving Dai | Introduction to Heegaard Floer Homology 1 Dai’s notes 1 |

11:30-1:00 | Lunch | |

1:00-2:00 | Rachel Roberts | Taut foliations in 3-manifolds |

2:00-2:45 | Extended tea | |

2:45-3:45 | Adam Clay | Orderable groups and 3-manifolds 1 Clay’s notes (all) |

4:00-5:00 | Lightning Talks | Slides to lightning talks |

6:00-?? | Social Hour | Cypress Street Pint & Plate |

**Tuesday**

8:00-9:00 | Morning Refreshments | |

9:00-10:00 | Irving Dai | Introduction to Heegaard Floer Homology 2 Dai’s notes 2 |

10:30-11:30 | Rachel Roberts | Taut foliations in 3-manifolds 2 |

11:30-1:00 | Lunch | |

1:00-2:00 | Adam Clay | Orderable groups and 3-manifolds 2 |

2:00-2:45 | Extended tea | |

2:45-3:45 | Nathan Dunfield | A unified Casson-Lin invariant for the real forms of SL(2) |

4:00-5:00 | Office Hours for mini-courses |

**Wednesday **

8:00-9:00 | Morning Refreshments | |

9:00-10:00 | Irving Dai | Introduction to Heegaard Floer Homology 3 Dai’s notes 3 |

10:30-11:30 | Rachel Roberts | Taut foliations in 3-manifolds 3 |

11:30-1:00 | Lunch | |

1:00-2:00 | Adam Clay | Orderable groups and 3-manifolds 3 |

2:00-2:45 | Extended tea | |

2:45-3:45 | Tao Li | Taut foliations of 3-manifolds with Heegaard genus two |

4:00-5:00 | Open Problem Discussion | Open Problem |

6:30 | Banquet | Park Tavern |

**Thursday**

8:00-9:00 | Morning Refreshments | |

9:00-10:00 | Irving Dai | Introduction to Heegaard Floer Homology 4 Dai’s notes 4 |

10:30-11:30 | Rachel Roberts | Taut foliations in 3-manifolds 4 |

11:30-1:00 | Lunch | |

1:00-2:00 | Adam Clay | Orderable groups and 3-manifolds 4 |

2:00-2:45 | Extended tea | |

2:45-3:45 | Hannah Turner | Bi-ordering link complements via braids |

4:00-5:00 | Office Hours for mini-courses |

**Friday**

8:00-9:00 | Morning Refreshments | |

9:00-10:00 | Irving Dai | Introduction to Heegaard Floer Homology 5 |

10:30-11:30 | Rachel Roberts | Taut foliations in 3-manifolds 5 |

11:30-1:00 | Lunch | |

1:00-2:00 | Adam Clay | Orderable groups and 3-manifolds 5 |

2:00-2:30 | Extended tea | |

2:30-3:30 | Jonathan Hanselman | Immersed curve bordered invariants from knot Floer homology |

**Lightning Talks**:

Hugo Zhou | (1,1) almost L-space knots |

Khanh Le | The space of left-orders of groups |

Atzimba Martinez | Taut foliations for Montesinos knots |

Holt Bodish | Three- and Four-Dimensional Invariants of Satellite Knots with (1,1)-Patterns |

Sierra Knavel | A Lefschetz fibration construction with arbitrary fundamental group |

Michele Capovilla-Searle | Birman-Ko-Lee left canonical form and its applications |

Weizhe Shen | Unknotting number and (1,1)-satellites |

Diego Santoro | L-spaces, taut foliations and fibered hyperbolic two-bridge links |

**Titles and Abstracts:**

**Nathan Dunfield**

Title: A unified Casson-Lin invariant for the real forms of SL(2)

Abstract: When M is the exterior of a knot K in the 3-sphere, Lin showed that the signature of K can be viewed as a Casson-style signed count of the SU(2) representations of pi_1(M) where the meridian has trace 0. This was later generalized to the fact that signature function of K on the unit circle counts SU(2) representations as a function of the trace of the meridan. I will define the SL(2, R) analog of these Casson-Lin invariants, and explain how it interacts with the original SU(2) version via a new kind of smooth resolution of the real points of certain SL(2, C) character varieties in which both kinds of representations live. I will use the new invariant to study left-orderability of Dehn fillings on M using the translation extension locus I introduced with Marc Culler, and also give a new proof of a recent theorem of Gordon’s on parabolic SL(2, R) representations of two-bridge knot groups. This is joint work with Jake Rasmussen. Based on https://arxiv.org/abs/2209.03382

**Cameron Gordon**

Title: The L-Space Conjecture

Abstract: The L-Space Conjecture asserts the equivalence of three properties of a 3-manifold with quite different flavors. These involve, respectively, the structure of the fundamental group (algebra), codimension 1 foliations (geometric topology), and Heegaard Floer homology (analysis). We will describe the properties in detail, outline some general connections between them, and survey what is currently known in the direction of the conjecture.

**Jonathan Hanselman**

Title: Immersed curve bordered invariants from knot Floer homology

Abstract: Knot Floer homology associates a bigraded chain complex along with a collection of flip maps to a knot in a 3-manifold. Algebraic tools exists to compute from this the Heegaard Floer homology of closed manifolds obtained by Dehn surgery or splicing knot complements. We will explain how the knot Floer data can be interpreted as a decorated immersed curve in the marked torus and the formulas for surgery and splicing are realized by Floer homology of curves in the marked torus. This builds on earlier joint work with J. Rasmussen and L. Watson which proves analogous results for the bordered Heegaard Floer homology of a knot complement; this is equivalent to UV=0 quotient of the the knot Floer data. A key advantage of the curves constructed from knot Floer data is they can recover the stronger minus version of Heegaard Floer homology and they are not restricted to mod 2 coefficients. However, given the theme of the conference we will highlight an application that only requires the earlier hat type results: an L-space gluing criterion that was a key step in verifying the L-space conjecture for graph manifolds.

**Tao Li**

Title: Taut foliations of 3-manifolds with Heegaard genus two

Abstract: Let M be a closed, orientable, and irreducible 3-manifold with Heegaard genus two. We prove that if the fundamental group of M is left-orderable then M admits a co-orientable taut foliation.

**Hannah Turner**

Title: Bi-ordering link complements via braids

Abstract: Any link (or knot) group – the fundamental group of a link complement – is left-orderable. However, not many link groups are bi-orderable – that is, admit an order invariant under *both* left and right multiplication. It is not well understood which link groups are bi-orderable, nor is there is a conjectured topological characterization of links with bi-orderable link groups. I will discuss joint work in progress with Jonathan Johnson and Nancy Scherich to study this problem for braided links – braid closures together with their braid axis. In particular, I will discuss our implementation of an algorithm to decide when braided link groups are bi-orderable.

**Lightning Talks**:

Holt Bodish

Title: Three- and Four-Dimensional Invariants of Satellite Knots with (1,1)-Patterns

Abstract: In this talk I will discuss recent work computing the top and next to top Alexander graded piece of knot Floer homology of satellites with certain (1,1) trefoil patterns. We discuss how this is related to when the pattern is fibered in the solid torus and to the monodromy of the fibration of these satellites. We also use these results to show that many satellites with these trefoil patterns are not Floer thin.

Michele Capovilla-Searle

Title: Birman-Ko-Lee left canonical form and its applications.

Abstract: The dual Garside left canonical form, introduced by Xu (3-braids), Kang-Ko-Lee (4-braids) and Birman-Ko-Lee (n-braids), is used to solve the conjugacy problem in braid theory. The fractional Dehn twist coefficient (FDTC), introduced by Honda-Kazez-Matic, is a powerful tool to detect the tightness or overtwistedness of a given contact structure. In this talk, I will give applications of the left canonical form to the fractional Dehn twist coefficient of braids.

Sierra Knavel

Title: A Lefschetz fibration construction with arbitrary fundamental group.

Abstract: In this talk we will discuss Korkmaz’s proof that every finitely presented group is the fundamental group of a symplectic Lefschetz fibration. This result was originally proved by Amoros-BogomolovKatzarkov-Pantev but re-proven using an explicit monodromy.

Khanh Le

Title: The space of left-orders of groups

Abstract: In this talk, we will introduce the space of left-orders LO(G) for a group G and give some examples of groups with isolated left-orders. Then we will discuss some open questions about LO(G).

Atzimba Martinez

Title: Taut foliations for Montesinos knots

Abstract: In this talk we consider orientable 3-manifolds that arise from Dehn surgery along Montesinos knots. We will discuss the classification of Montesinos knots (in particular those of odd type), described through continued fractions, and study their Seifert surfaces of minimal genus. We then use this information to find persistent foliations.

Diego Santoro

Title: L-spaces, taut foliations and fibered hyperbolic two-bridge links

Abstract: In this lightning talk I will state a result concerning the L-space conjecture for manifolds that are surgeries on fibered hyperbolic two-bridge links. We will also see how to use this result to prove that Whitehead doubles of nontrivial knots (and some more general satellite knots) are persistently foliar.

Weizhe Shen

Title: *Unknotting number and (1,1)-satellites*

Abstract: The unknotting number of a knot, defined as the minimum number of crossing changes needed to untie it, is in general an intractable invariant. Its behavior under cabling is bounded below by recent work of Hom-Lidman-Park using knot Floer homology. A pattern knot P is a (1,1)-pattern if it admits a genus one doubly-pointed bordered Heegaard diagram. I will discuss the unknotting number of satellite knots with such patterns.

Hugo Zhou

Title: (1,1) almost L-space knots

Abstract: We give a diagrammatic characterization of the (1,1) almost L-space knots (knots which admit large Dehn surgeries to manifolds with Heegaard Floer homology of next-to-minimal rank). This is inspired by a corresponding result for (1,1) L-space knots due to Greene-Lewallen-Vafaee.