Winter school workshop

Workshop theme

The first event of the INI program Homology theories in low-dimension will be a week-long winter school, running January 16-20 2017, targeted at graduate students and early career researchers. There will be three distinct streams, organized to reflect the focus of the program as a whole. The anticipated streams are as follows:

Topics in Floer homology
Stream leader: Adam Levine, (Princeton University)
Additional lectures by: Liam Watson (University of Glasgow)

This lecture stream will begin with an overview of Heegaard Floer homology, before turning to the main focus: bordered Floer homology. This relatively new suite of invariants gives rise to powerful cut-and-paste tools for studying the Heegaard Floer homology of closed three-manifolds. Particular focus will be given to the case of manifolds with torus boundary, towards describing applications to knot Floer homology and to the study of L-spaces — those manifolds with simplest possible Heegaard Floer homology.

Structures on three-manifolds
Stream leader: Cameron Gordon, (University of Texas at Austin)
Additional lectures by: Rachel Roberts (Washington University in St-Louis), Andy Wand (University of Glasgow)

This lecture stream will give an overview of foliations and left-orderable three-manifold groups, with a view to describing the state of the art in an emerging relationship: for a closed, orientable, prime three-manifold Y, it is conjectured that the existence of a taut foliation foliation on Y, the existence of a left-order on the fundamental group of Y, and the condition that Y not be an L-space are equivalent. This latter condition comes from Heegaard Floer homology, and this will result in an important point of interaction with the stream Topics in Floer homology.

Aspects of geometric representation theory
Stream leader: Anthony Licata, (Australian National University)
Additional lectures by: Eli Grigsby (Boston College), Jacob Rasmussen (University of Cambridge)

This lecture stream aims to describe some of the representation theory which motivates and is used in constructions of knot homology theories. We will focus on the representation theory of sl(2) and its quantized enveloping algebra, and the first lectures will recall the structure theory of finite dimensional Lie algebras, quantum groups and skew Howe duality. We will then explain how in the case of sl(2), this representation theory gives rise to the Jones polynomial, and how constructions in geometric and categorical representation theory can be used to upgrade constructions of the Jones polynomial to constructions of Khovanov homology. The latter part of the lecture series will describe how sl(2) appears again as an internal structure in the annular version of Khovanov homology.

Applications for this workshop are now closed

The Institute kindly requests that any papers published as a result of this programme’s activities are credited as such. Please acknowledge the support of the Institute in your paper using the following text:

The author(s) would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Winter school workshop, where work on this paper was undertaken. This work was supported by EPSRC grant EP/K032208/1.

Timetable

Monday 16th January 2017
09:00 to 09:50	Registration	No Room Required
09:50 to 10:00	Welcome from Christie Marr (INI Deputy Director)	No Room Required
10:00 to 11:00	Adam Levine Princeton University Topics in Heegaard Floer homology I	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Cameron Gordon University of Texas at Austin Left-orders on 3-manifolds I	Room 1
12:30 to 13:30	Lunch @ Wolfson Court	No Room Required
13:30 to 14:30	Problem Session	Room 1
14:30 to 15:30	Cameron Gordon University of Texas at Austin Left-orders on 3-manifolds II	Room 1
15:30 to 16:00	Afternoon Tea	No Room Required
16:00 to 17:00	Andy Wand University of Glasgow Contact structures on 3-manifolds I	Room 1
17:00 to 18:00	Welcome Wine Reception at INI	No Room Required

Tuesday 17th January 2017
09:00 to 10:00	Adam Levine Princeton University Topics in Heegaard Floer homology II	Room 1
10:00 to 11:00	Adam Levine Princeton University Topics in Heegaard Floer homology II, cont.	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Cameron Gordon University of Texas at Austin Left-orders on 3-manifolds III	Room 1
12:30 to 13:30	Lunch @ Wolfson Court	No Room Required
13:30 to 14:30	Problem Session	Room 1
14:30 to 15:30	Anthony Licata Australian National University Braid groups and their 2-representations	Room 1
15:30 to 16:00	Afternoon Tea	No Room Required
16:00 to 17:00	Andy Wand University of Glasgow Contact structures on 3-manifolds II	Room 1

Wednesday 18th January 2017
09:00 to 10:00	Adam Levine Princeton University Topics in Heegaard Floer homology III	Room 1
10:00 to 11:00	Adam Levine Princeton University Topics in Heegaard Floer homology III, cont.	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Eli Grigsby Boston College Trivial braid detection via Khovanov homology	Room 1
12:30 to 13:30	Lunch @ Wolfson Court	No Room Required
13:30 to 14:30	Problem Session	Room 1
14:30 to 15:30	Anthony Licata Australian National University The categorified Burau representation	Room 1
15:30 to 16:00	Afternoon Tea	No Room Required
16:00 to 17:00	Rachel Roberts Washington University in St. Louis Foliations on 3-manifolds I	Room 1
19:30 to 22:00	Formal Dinner at Trinity College	Room 1

Thursday 19th January 2017
09:00 to 10:00	Adam Levine Princeton University Topics in Heegaard Floer homology IV	Room 1
10:00 to 11:00	Adam Levine Princeton University Topics in Heegaard Floer homology IV, cont.	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Eli Grigsby Boston College Annular Khovanov-Lee theory and representation theory	Room 1
12:30 to 13:30	Lunch @ Wolfson Court	No Room Required
13:30 to 14:30	Problem Session	Room 1
14:30 to 15:30	Eli Grigsby Boston College Annular Khovanov-Lee theory of braid closures and braided surfaces	Room 1
15:30 to 16:00	Afternoon Tea	No Room Required
16:00 to 17:00	Rachel Roberts Washington University in St. Louis Foliations on 3-manifolds II	Room 1

Friday 20th January 2017
09:00 to 10:00	Liam Watson Bordered via curves and train tracks I	Room 1
10:00 to 11:00	Liam Watson Bordered via curves and train tracks II	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Jacob Rasmussen University of Cambridge HOMFLY-PT homology	Room 1
12:30 to 13:30	Lunch @ Wolfson Court	No Room Required
13:30 to 14:30	Problem Session	Room 1
14:30 to 15:30	Anthony Licata Australian National University 2-representations of braid groups, continued	Room 1
15:30 to 16:00	Afternoon Tea	No Room Required
16:00 to 17:00	Rachel Roberts Washington University in St. Louis Foliations on 3-manifolds III	Room 1

Homology theories in low dimensional topology

Organisers

Cameron Gordon University of Texas at Austin
Adam Levine Princeton University
Anthony Licata Australian National University
Liam Watson

Speakers

Cameron Gordon University of Texas at Austin
Eli Grigsby Boston College
Adam Levine Princeton University
Anthony Licata Australian National University
Jacob Rasmussen University of Cambridge
Rachel Roberts Washington University in St. Louis
Andy Wand University of Glasgow
Liam Watson

Participants

Cristina Ana-Maria Anghel Université Paris 7 - Denis-Diderot
Ian Banfield Boston College
Agnese Barbensi Università di Pisa
Mounir Benheddi Université de Genève
Christian Blanchet Université Denis Diderot
Benjamin Bode University of Bristol
Daniele Celoria University of Oxford
Marco De Renzi Université Paris 7 - Denis-Diderot
Viktória Földvári Eötvös Loránd University
Paolo Ghiggini CNRS (Centre national de la recherche scientifique); Université de Nantes
Marco Golla Uppsala Universitet
Cameron Gordon University of Texas at Austin
Jonathan Grant Université Libre de Bruxelles
Thomas Hockenhull Imperial College London
Ben-Michael KOHLI Université de Bourgogne
Danica Kosanovic University of Bonn
Ronja Kuhne University of Warwick
Adam Levine Princeton University
Anthony Licata Australian National University
Beibei Liu University of California, Davis; Chinese University of Hong Kong
Andrew Lobb University of Durham
Daniel López Université Paris 7 - Denis-Diderot
Joseph MacColl University College London; Imperial College London
Maria Marchwicka Adam Mickiewicz University
Marco Marengon Imperial College London
Michael Miller University of California, Los Angeles
Grégoire Naisse Université Catholique de Louvain
Wojciech Politarczyk Uniwersytet Warszawski; Adam Mickiewicz University
Jacob Rasmussen University of Cambridge
Sarah Rasmussen University of Cambridge
Elia Rizzo Université Catholique de Louvain
Rachel Roberts Washington University in St. Louis
William Rushworth Durham University
Michael Snape University of Glasgow
Richard Thomas Imperial College London
Andy Wand University of Glasgow
Liam Watson
Paul Wedrich Imperial College London
Biji Wong Brandeis University
Claudius Zibrowius University of Cambridge

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