This conference is devoted to the study of quantum knot homologies and related invariants. The name of this area of research already indicates that this is an interdisciplinary topic: such homologies are of interest to mathematicians (in particular topologists and representation theorists), while quantum phenomena have in principle their origin in physics. There are communities of both mathematicians and physicists who work on these topics from different perspectives, they use different tools, and sometimes the same tools but described in different languages. The main aim of the conference is to bring researchers from those two communities together, establish a common language and explain important results to each other, summarize the status of the field, and specify goals and set a common program for future research.

Organizers:
Sergei Gukov, California Institute of Technology
Mikhail Khovanov, Columbia University
Andrew Lobb, Durham University
Piotr Sułkowski, University of Warsaw

Deadline to apply: December 15, 2017

Announcement at the Aspen Center for Physics website

Registration through the Aspen Center for Physics website

## Special events

Welcome reception
March 4 (Sunday), Aspen Center for Physics, 5:00 pm - 7:00 pm

Physics café and public lecture
March 7 (Wednesday), Wheeler Opera House
4:30 pm – Physics Café, 5:30 pm – Public Lecture

Conference banquet
March 8 (Thursday), Meadows Resort, 7:00 pm

## Program

SUNDAY, March 4
17:00-19:00 – Welcome reception

MONDAY, March 5

8:30-9:10 – Scott Morrison, "Functoriality in S^3, and 4-d skein modules"
9:10-9:50 – David Rose, "Annular evaluation and link homology"
9:50-10:20 – Recess
10:20-11:00 – Pavel Putrov, "BPS spectra and 3-manifold invariants"
11:00-16:30 – Mid-day break, lunch, discussions, winter sports
16:30-17:10 – Sarah Rasmussen, "L-spaces, Dehn surgery, and topology"
17:10-17:50 – Pedro Vaz, "2-Verma modules and the Khovanov-Rozansky link homologies"
17:50-18:20 – Recess
18:20-19:30 – Special lecture: Philip Argyres, "Superconformal field theories and singular geometries"

TUESDAY, March 6

8:30-9:10 – Masahito Yamazaki, "All-order volume conjecture for closed 3-manifolds"
9:10-9:50 – John Baldwin, "Khovanov homology, instantons, and the trefoil"
9:50-10:20 – Recess
10:20-11:00 – Ciprian Manolescu, "Sheaf-theoretic SL(2,C) Floer homology of 3-manifolds"
11:00-16:30 – Mid-day break, lunch, discussions, winter sports
16:30-17:10 – Matthew Hogancamp, "Curved complexes, Khovanov-Rozansky homology, and Hilbert schemes"
17:10-17:50 – Andrea Brini, "3-manifold and knot invariants from mirror symmetry"
17:50-18:20 – Recess
18:20-19:30 – Special lecture: Robion Kirby, "Trisections"

WEDNESDAY, March 7
8:30-9:10 – Paul Wedrich, "Knots and quivers, HOMFLY-PT and DT"
9:10-9:50 – Du Pei, "2d TQFTs labeled by 3-manifolds"
9:50-10:20 – Recess
10:20-11:00 – Lukas Lewark, "Upsilon-like invariants from Khovanov-Rozansky sl(n) homologies"
11:00-16:30 – Mid-day break, lunch, discussions, winter sports
16:30-17:30 – Physics café: Sarah Rasmussen and Piotr Kucharski (at the Wheeler Opera House)
17:30-18:30 – Public lecture: Louis Kauffman, "Physical knots" (at the Wheeler Opera House)

THURSDAY, March 8

8:30-9:10 – Kevin Walker, "Khovanov homology, 4+\epsilon-dimensional TQFTs, and genus bounds"
9:10-9:50 – Ramadevi Pichai, "Methods for computing colored HOMFLY-PT for arborescent and non-arborescent knots"
9:50-10:20 – Recess
10:20-11:00 – Aaron Lauda, "Towards a quantum construction of Alexander categorifications and related spectral sequences"
11:00-12:10 – Special lecture: Michael Freedman, "Playing with locality"
12:10-19:00 – Mid-day break, lunch, discussions, winter sports
19:00 – Conference banquet (at the Meadows Resort)

FRIDAY, March 9

8:30-9:10 – Matthew Stoffregen, "An odd Khovanov stable homotopy type"
9:10-9:50 – Ingmar Saberi, "Spectral sequences in and around physics"
9:50-10:20 – Recess
10:20-11:00 – Raphael Rouquier, "Heegaard-Floer theory and higher representation theory"
11:00-16:30 – Mid-day break, lunch, discussions, winter sports
16:30-17:10 – Józef Przytycki, "From distributive to Yang-Baxter homology"
17:10-17:50 – Dan Freed, "The two-dimensional Ising model revisited"
17:50-18:20 – Recess
18:20-19:30 – Special lecture: Louis Kauffman, "From the bracket polynomial to Khovanov homology"

SATURDAY, March 10

Informal discussions, winter sports

## (Some) Abstracts

Michael Freedman, "Playing with locality"
Abstract: Electromagnetism is a long ranged force but in condensed matter screening makes "locality" of interactions a reasonable assumption. Topology is (by definition) the study of locality, so it is natural to attempt to move ideas back and forth between topology and condensed matter. I will talk about moving Kirby's "torus trick" into physics (pioneered  by Hastings) and moving Floquet physics and the GNVW index into topology. This is very much work in progress and comments, corrections, and fresh ideas are most welcome.

Matthew Hogancamp, "Curved complexes, Khovanov-Rozansky homology, and Hilbert schemes"
Abstract: I will discuss recent joint work with Eugene Gorsky in which we construct a deformation (which we dub y-ification’') of the triply graded Khovanov-Rozansky link homology. Our invariant is essentially an equivariant HOMFLY-PT version of the Batson-Seed deformation of Khovanov-Rozansky homology and has analogous link splitting properties. Astonishingly, y-ification appears to restore the missing q—> tq^{-1} symmetry of Khovanov-Rozansky homology for links, though this remains conjecture.  I will spend some time discussing our main result, which relates KR homology (both y-ified and the original) to the Hilbert scheme of points in C^2.  This is a first step toward a proof of the conjectures of Gorsky-Neguţ-Rasmussen.

Louis Kauffman, "From the bracket polynomial to Khovanov homology"
Abstract: This talk will be a self-contained introduction to the bracket state sum model for the Jones polynomial and how its structure (in the hands of Michail Khovanov) leads to the construction of Khovanov Homology for knots and links. We will discuss how these constructions extend to virtual knots and links and relationships with simplicial theory.

Louis Kauffman, "Physical knots"
Abstract: This talk is an exploration, using visual ideas, of the role of knots in natural science and mathematics. We will begin with knots and DNA and the production of knots in DNA recombination. Then we will show a movie of knotted vortices in water (courtesy of the work of William Irvine at the University of Chicago). Irvine and his group accomplished a feat that is the equivalent of blowing a knotted smoke ring. We discuss the possibility of knotted gluon fields in relation to the complexity of knots measured by rope length. We discuss the possibility of framed braids as a basis for elementary particles. And, we discuss the remarkable relationships among statistical mechanics, quantum field theory and the construction of invariants of knots, links and three-manifolds.

Aaron Lauda, "Towards a quantum construction of Alexander categorifications and related spectral sequences"
Abstract: Odd Khovanov homology was discovered by Ozsvath, Rasmussen, Szabo in attempt to find an integral version of a spectral sequence defined over Z/2 from even Khovanov homology to the Heegaard-Floer homology of a branched double cover.  Ellis, Khovanov, and I initiated the study of odd categorified quantum groups in attempt to provide a representation theoretic explanation for odd Khovanov homology.  This connection has yet to be realized.  However, in this talk we will provide further evidence to support this connection by explaining recent work with Ilknur Egilmez that equips the odd categorification of sl(2) with the structure of a differential graded super 2-category.  Using ideas of Qi You, we show that the Grothendieck ring of this super DG 2-category gives rise to a categorification of sl(2) at a fourth root of unity as well as a categorification over the integers of an algebra closely connected with gl(1|1) (depending on how the super grading is treated in K_0). Since relatives of the Alexander polynomial can be formulated as quantum invariants associated to the quantum supergroup gl(1|1), or the quantum group sl(2) at a fourth root of unity,  it is tempting to speculate that differentials on odd categorified quantum groups may illuminate the relationship between odd Khovanov homology and Heegaard-Floer theory.

Raphael Rouquier, "Heegaard-Floer theory and higher representation theory"
Abstract: I will discuss a program with Andy Manion towards reinterpreting Heegaard-Floer constructions of Ozsvath-Szabo, Lipshitz-Ozsvath-Thurston, Douglas-Manolescu and others as 2-representations of gl(1|1).

## Poster

Click on the image below to get a PDF file of the poster:

## Spacetime details

Dates & Time: March 4-10, 2018
Venue: Aspen Center for Physics, Aspen, USA

## Participants

1. Argyres, Philip (University of Cincinnati)
2. Baldwin, John (Boston College)
3. Belfiore, Laurence A. (Colorado State University)
4. Brini, Andrea (Imperial College London)
5. Chun, Sungbong (California Institute of Technology)
6. Clingempeel, Joel (Rutgers University)
7. Dedushenko, Mykola (California Institute of Technology)
8. Ferrari, Francesca (University of Amsterdam)
9. Francis, John (Northwestern University)
10. Freed, Daniel (University of Texas at Austin)
11. Freedman, Michael (Microsoft Research)
12. Fuji, Hiroyuki (Kagawa University)
13. Gang, Dongmin (Seoul National University)
14. Grigsby, Julia (Boston College)
15. Gu, Jie (Ecole Normale Superieure)
16. Gukov, Sergei (California Institute of Technology)
17. Harrison, Sarah (McGill University)
18. Hedden, Matthew (Michigan State University)
19. Hogancamp, Matthew (University of Southern California)
20. Kauffman, Louis H. (University of Illinois at Chicago)
21. Khovanov, Mikhail (Columbia University)
22. Kirby Robion (University of California,  Berkeley)
23. Klug, Michael (University of California,  Berkeley)
24. Kucharski, Piotr (Uppsala University)
25. Lauda, Aaron (University of Southern California)
26. Lewark, Lukas (University of Bern)
27. Lobb, Andrew (Durham University)
28. Longhi, Pietro (Uppsala University)
29. Manion Andrew (University of California Los Angeles)
30. Manolescu Ciprian (UCLA)
31. Morrison, Scott (Australian National University)
32. Nawata, Satoshi (Fudan University)
33. Paquette, Natalie (California Institute of Technology)
34. Pei, Du (California Institute of Technology)
35. Pichai, Ramadevi (Indian Institute of Technology Bombay)
36. Przytycki, Józef H. (George Washington University)
37. Putrov, Pavel (Institute for Advanced Study)
38. Rasmussen, Sarah Dean (University of Cambridge)
39. Roggenkamp, Daniel (University of Mannheim)
40. Romo, Mauricio (IAS, Princeton)
41. Rose, David (University of North Carolina)
42. Rouquier, Raphael (University of California Los Angeles)
43. Saberi, Ingmar (University of Heidelberg)
44. Sarkar, Sucharit (University of California Los Angeles)
45. Stoffregen, Matthew (Massachusetts Institute of Technology)
46. Sułkowski, Piotr (University of Warsaw)
47. Sun, Haoyu (University of California Berkeley)
48. Tripathy, Arnav (Harvard University)
49. Vaz, Pedro (Universite Catholique de Louvain)
50. Walker, Kevin (Microsoft Station Q)
51. Watson, Liam (Université de Sherbrooke)
52. Wedrich, Paul (The Australian National University)
53. Yamazaki, Masahito (University of Tokyo)
54. Ye, Weicheng (Stony Brook University)