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Timetable (HTLW01)

Winter school workshop

Monday 16th January 2017 to Friday 20th January 2017

Monday 16th January 2017
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from Christie Marr (INI Deputy Director)
10:00 to 11:00 Adam Levine (Princeton University)
Topics in Heegaard Floer homology I
We will introduce the basics of Heegaard Floer homology, with a view toward results on the classification of L-spaces.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Cameron Gordon (University of Texas at Austin)
Left-orders on 3-manifolds I
INI 1
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Problem Session INI 1
14:30 to 15:30 Cameron Gordon (University of Texas at Austin)
Left-orders on 3-manifolds II
INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Andy Wand (University of Glasgow)
Contact structures on 3-manifolds I
INI 1
17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 17th January 2017
09:00 to 10:00 Adam Levine (Princeton University)
Topics in Heegaard Floer homology II
After continuing with some basics of Heegaard Floer homology, we will discuss knot Floer homology and its relation to surgeries on knots.
INI 1
10:00 to 11:00 Adam Levine (Princeton University)
Topics in Heegaard Floer homology II, cont.
After continuing with some basics of Heegaard Floer homology, we will discuss knot Floer homology and its relation to the Heegaard Floer homology of surgeries on knots.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Cameron Gordon (University of Texas at Austin)
Left-orders on 3-manifolds III
INI 1
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Problem Session INI 1
14:30 to 15:30 Anthony Licata (Australian National University)
Braid groups and their 2-representations
The main topic of this lecture series will be the Artin braid group B_n.  We study this group using two main tools:
1) linear 2-representations of B_n, and
2) annular Khovanov homology (see Eli Grigsby's lectures).
The first lecture will introduce the braid group, explain what it means for B_n to act on a category, and motivate the study of 2-representations of braid groups.
In later lectures, we will study a particular example of a (faithful) 2-representation of B_n - the categorified Burau representation - in detail, and explain how various structures of interest in the study of the braid group can be studied through the lens of this 2-representation.  Time permitting, we will discuss the relationship between linear 2-representations of B_n and geometric representation theory.


INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Andy Wand (University of Glasgow)
Contact structures on 3-manifolds II
INI 1
Wednesday 18th January 2017
09:00 to 10:00 Adam Levine (Princeton University)
Topics in Heegaard Floer homology III
We will introduce bordered Floer homology, including some of the underlying algebraic concepts (e.g. A_\infty modules).
INI 1
10:00 to 11:00 Adam Levine (Princeton University)
Topics in Heegaard Floer homology III, cont.
We will introduce bordered Floer homology, including some of the underlying algebraic concepts (e.g. A_\infty modules).
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Eli Grigsby (Boston College)
Trivial braid detection via Khovanov homology
In this 3-lecture series we will explore the many ways in which key  representation-theoretic features of the annular Khovanov-Lee homology of braid closures give information about the surfaces they bound in the 4-ball as well as their dynamics when viewed as mapping classes of the punctured disk.

In the first lecture, we will introduce Plamenevskaya's transverse invariant in Khovanov homology and show how it can be used to detect the trivial braid (j. work with J. Baldwin). The proof uses fundamental properties of the left-invariant order on the braid group as well as algebraic properties of the Khovanov complex of braid closures.
INI 1
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Problem Session INI 1
14:30 to 15:30 Anthony Licata (Australian National University)
The categorified Burau representation
The goal of this talk will be to introduce a particularly important 2-representation of B_n - the categorified Burau representation - which appears prominantly in symplectic geometry, algebraic geometry, and in representation theory.  We will give several proofs of the faithfulness of this 2-representation, and explain how it allows us to study various structures on interest in braid theory (e.g. Garside structures, word-length metrics, left-invariant orders, etc.) using tools of linear and homological algebra.
INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Rachel Roberts (Washington University in St. Louis)
Foliations on 3-manifolds I
INI 1
19:30 to 22:00 Formal Dinner at Trinity College INI 1
Thursday 19th January 2017
09:00 to 10:00 Adam Levine (Princeton University)
Topics in Heegaard Floer homology IV
We will continue the discussion of bordered Heegaard Floer homology, including two different perspectives on its relation to knot Floer homology.
INI 1
10:00 to 11:00 Adam Levine (Princeton University)
Topics in Heegaard Floer homology IV, cont.
We will continue the discussion of bordered Heegaard Floer homology, including two different perspectives on its relation to knot Floer homology.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Eli Grigsby (Boston College)
Annular Khovanov-Lee theory and representation theory
After reminding you of the main properties of Khovanov-Lee theory underlying Rasmussen's proof of the topological Milnor conjecture, we will discuss the representation theory underlying an annular version of Khovanov-Lee theory (j. work with T. Licata and S. Wehrli), focusing on how geometric features of the braid interact with representation-theoretic features of the Khovanov-Lee complex.
INI 1
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Problem Session INI 1
14:30 to 15:30 Eli Grigsby (Boston College)
Annular Khovanov-Lee theory of braid closures and braided surfaces
By endowing the annular Khovanov-Lee complex with the structure of a bifiltered complex, we define a family of annular Rasmussen invariants that gives information both about the positivity of braids viewed as mapping classes and the complexity of braided surfaces bounded by their closures (j. work with T. Licata and S. Wehrli).
INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Rachel Roberts (Washington University in St. Louis)
Foliations on 3-manifolds II
INI 1
Friday 20th January 2017
09:00 to 10:00 Liam Watson
Bordered via curves and train tracks I
INI 1
10:00 to 11:00 Liam Watson
Bordered via curves and train tracks II
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Jacob Rasmussen (University of Cambridge)
HOMFLY-PT homology
I'll discuss the HOMFLY-PT polynomial and its categorification (due to Khovanov and Rozansky) and explain how it ties in with some of the things Tony and Eli have been discussing.
INI 1
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Problem Session INI 1
14:30 to 15:30 Anthony Licata (Australian National University)
2-representations of braid groups, continued
We will continue the study of the braid group using the categorified Burau representation, as well as describe other appearances of 2-representations of Braid groups in geometric representation theory.
INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Rachel Roberts (Washington University in St. Louis)
Foliations on 3-manifolds III
INI 1
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons