Knots, homologies and physics

Dates: Mondays (unless otherwise stated), 11:00
Place: Institute of Mathematics, Polish Academy of Sciences (Śniadeckich 8, Warsaw), lecture room 403

March 4
The first welcome and organizational meeting

March 11
Seminar speaker: Sungkyung Kang (Oxford University)
Title: Invariant knot Floer homology of satellites and the invariant splitting principle
Abstract: Involutive knot Floer homology, a refinement of knot Floer theory, is a powerful knot invariant which was used to solve several long-standing problems, including the one-is-not-enough result for 4-manifolds with boundary. In this talk, we show that if the involutive knot Floer homology of a knot K admits an invariant splitting, then the induced splitting of the knot Floer homology of P(K), for any pattern P, can be made invariant under its \iota_K involution. As an application of this “invariant splitting principle”, we construct an infinite family of examples of pairs of exotic contractible 4-manifolds which survive one stabilization, and observe that some of them are potential candidates for surviving two stabilizations. This is an ongoing work with Gary Guth.

April 8
Seminar speaker: Alessio di Prisa (Scuola Normale Superiore)
Title: Solvability of concordance groups and Milnor invariants
Abstract: In this talk I will introduce the notions of concordance for several knotted objects in the 3-space, namely strongly invertible knots, theta curves and string links, and I will explain how the respective concordance groups of these objects are related to one another. Using Milnor invariants, I will prove that the concordance group of 2-string links is not solvable. Exploiting the close relation between the objects mentioned above, it follows that the equivariant concordance group of strongly invertible knots and the cobordism group of theta curves are also not solvable. Finally, I will show how these results can be used to answer a conjecture due to Kuzbary regarding the solvability of the quotient of the concordance group of n-string links modulo pure braids. Join work with G. Framba.

April 29
Seminar speaker: Alberto Cavallo (IMPAN)
Title: Transverse links in Stein fillable contact 3-manifolds
Abstract: We study the behavior of different versions of the Ozsváth-Szabó tau-invariant for holomorphically fillable links in Stein domains. More specifically, we relate the Hedden’s version of the invariant, which needs the assumption that our links live in a contact 3-manifold with non-vanishing contact invariant, with the one introduced by Grigsby, Ruberman and Strle, which on the other hand only depends on the pair link-Spin^c 3-manifold and is then a purely topological invariant. This is joint work with Antonio Alfieri. The main goal of the talk is to describe how our work allows us to recover results about properly embedded holomorphic curves, such as the slice-Bennequin inequality and the relative Thom conjecture, and to find new restrictions on the topology of Stein fillings of certain 3-manifolds. In particular, we show that a Brieskorn 3-sphere, with its canonical orientation, never bounds a rational homology 4-ball Stein filling; confirming a conjecture of Gompf.

Dates: March 18-22, 2024
Place: University of Warsaw – Faculty of Mathematics, Informatics, and Mechanics (Banacha 2), lecture room 2180
Deadline for registration: February 15, 2024

Speakers: Sachin Chauhan, Miranda Cheng, Dusan Djordjevic, Tobias Ekholm, Sergei Gukov, Irena Matkovic, Sunghyuk Park, Davide Passaro, Ramadevi Pichai, Mark Powell, Markus Reineke, Radmila Sazdanovic, Paul Wedrich

Informal workshop dinner: Wednesday, March 20, 18:00, restaurant Shuk (address: Grójecka 107)

Titles and abstracts

Sachin Chauhan (Indian Institute of Technology, Bombay)
Title: Refined Gukov-Pei-Putrov-Vafa (GPPV) Conjecture
Abstract: GPPV conjecture relates the two quantum invariants of closed 3-manifolds. On one side we have Witten-Reshetikhin-Turaev (WRT) invariant and on the other side, the q-series valued invariant (also known as Z-hat). In this talk, I will present a more refined version of this conjecture by studying it for SU(N)/Z_m group, for N divisible by m. 

Dusan Djordjevic (University of Belgrade)
Title: A gentle introduction to knots and physics: from early days to knots-quivers correspondence and generalizations
Abstract: We will give a highly biased review of the relation between knots and physics, culminating our discussion with the knots-quivers correspondence. By working out examples, we will review some basic facts about knot theory, as well as tackle some original ongoing research. 

Tobias Ekholm (Uppsala University)
Title: Knot invariants and skein valued open Gromov-Witten invariants
Abstract: We describe how skein valued counts of holomorphic curves lead to deformation invariant open Gromov-Witten theory. As an application we derive rather simple recursion relations for all colored HOMFLYPT of the unknot and Hopf link. 

Sergei Gukov (Caltech)
Title: What do we want from categorification and non-perturbative complex Chern-Simons, and why?
Abstract: This will be a gentle introduction – guided by “Why?” questions – into several topics of recent work that quickly converge toward a common goal. The big overarching goal is to build a bridge between low-dimensional topology (think of the Poincare conjecture, the slice-ribbon conjecture, etc.) and quantum topology (think of many quantum invariants of knots and 3-manifolds). I will explain why one of the most promising paths towards this Big goal requires categorification of quantum 3-manifold invariants and will offer a candidate for “Khovanov homology of 3-manifolds” that is rooted in non-perturbative formulation of complex Chern-Simons theory and in integrality of DT-type invariants.

Irena Matkovic (Uppsala University)
Title: Heegaard Floer homology and contact structures
Abstract: I will review both the applications of Heegaard Floer homology to 3-dimensional contact topology, and contact topology to Heegaard Floer homology. The talk wishes to give an overview of the subject, including results and arguments of many researchers. 

Sunghyuk Park (Harvard University)
Title: Towards invariants of 3-manifolds from quantum groups at generic q
Abstract: Motivated by the categorification problem of quantum invariants of 3-manifolds, S. Gukov, D. Pei, P. Putrov and C. Vafa conjectured the existence of a q-series-valued invariant of 3-manifolds. This program was further developed by S. Gukov and C. Manolescu who extended this conjecture to knot complements and gave conjectural surgery formulas. I will give an overview of this program and discuss how this invariant can be defined for a large class of link complements, using representations of quantum groups at generic q.

Ramadevi Pichai (Indian Institute of Technology, Bombay)
Title: Computational status of knot polynomials
Abstract: First, I will give a pedagogical introduction to the elegant tools giving knot polynomials. Then, I will present how these polynomials can address `knot classification’ as well as obtaining three-manifold invariants. The extent to which explicit computation is possible will be highlighted.

Mark Powell (University of Glasgow)
Title: Simple spines of knot traces
Abstract: A knot trace is a 4-manifold obtained by attaching a single 2-handle to the 4-ball along a knot. A simple spine for a knot trace is a locally flat embedded 2-sphere such that the embedding is a homotopy equivalence, and the fundamental group of the complement is abelian. With Feller-Miller Nagel-Orson-Ray, and in two projects with Orson, we characterised exactly when knot traces admit simple spines, and showed that any two are isotopic. I will explain these results and an outline of how they are proven.

Markus Reineke (Ruhr-Universität Bochum)
Title: (Refined) Donaldson-Thomas invariants of symmetric quiver
Abstract: We define (refined) Donaldson-Thomas invariants of symmetric quiver and discuss several of their interpretations in terms of moduli spaces and Cohomological Hall algebras.

Paul Wedrich (Universität Hamburg)
Title: From link homology to TQFTs
Abstract: Skein theory offers several plausible strategies for extending link homology theories, such as Khovanov homology, to topological quantum field theories in 4 or 5 dimensions. In this talk, I will focus on a categorified analog of a TQFT of Turaev-Viro type. Joint work with Matthew Hogancamp and David Rose.

Dates: March 25-29, 2024
Place:
– Monday and Tuesday: IMPAN = Institute of Mathematics, Polish Academy of Sciences (Śniadeckich 8, Warsaw), lecture room 321
– Wednesday and Thursday: MIMUW = Faculty of Mathematics, Informatics and Mechanics (Banacha 2, Warsaw), lecture room 2180

Lisa Lokteva (Uppsala University)
Title: New Examples of Graph Manifolds Bounding Rational Homology Balls

Ramadevi Pichai (Indian Institute of Technology Bombay)
Title: Chern-Simons Field Theory Invariants: Knots, Links and Three-Manifolds

Mark Powell (University of Glasgow)
Title: Symmetries of 4-manifolds

Sayyed Rassouli (University of Nottingham)
Title: Topological evolution of time in Unimodular Gravity

Vivek Kumar Singh (NYU, Abu Dhabi)
Title: Asymptotics in the invariants of weaving knots W(3,n)

Dates: April 22-26, 2024 (no lectures on Thursday!)
Place: IMPAN = Institute of Mathematics, Polish Academy of Sciences (Śniadeckich 8, Warsaw), lecture room 321

Jarosław Duda (Jagiellonian University)
Title: Exploring resemblance between topological charges/vortices/knots and particle physics

Wei Li (Chinese Academy of Sciences)
Title: Introduction to quiver BPS algebras

Dmitry Noshchenko (Dublin Institute for Advanced Studies)
Title: Quivers, Nahm sums and hidden symmetries in knot homologies
Note: these lectures will be broadcasted online in the lecture hall

Dates: May 6-10, 2024
Place: Institute of Mathematics, Polish Academy of Sciences (Śniadeckich 8, Warsaw)
Deadline for registration: March 31, 2024

Speakers include: Irving Dai, Helder Larraguivel, Wei Li, Pietro Longhi, Abhishek Mallick, Dmitry Noshchenko, Sunghyuk Park, Ramadevi Pichai, Józef Przytycki, Marko Stosic, Ian Zemke

Dates: May 13-17, 2024
Place: Institute of Mathematics, Polish Academy of Sciences (Śniadeckich 8, Warsaw)
Deadline for registration: April 14

Irving Dai (Stanford University)
Title: Floer homology and symmetries of knots and manifolds

Sergei Gukov (Caltech and Dublin IAS)
Title: Categorification of quantum invariants: from knots to 3-manifolds

Józef Przytycki (George Washington University)
Title: Introduction to Skein Modules and their Khovanov type categorification

Marko Stosic (Technical University of Lisbon)
Title: Knot quantum invariants, graphs, and combinatorics

Dates: June 3-7, 2024
Place: Institute of Mathematics, Polish Academy of Sciences (Śniadeckich 8, Warsaw)

Vladimir Dotsenko (University of Strasbourg)
Title: Introduction to Koszul duality

Paul Wedrich (Hamburg University)
Online course: Master level course on knot homology 
Wednesdays, 16:15-17:45, April 3 – July 10
Fridays, 12:15-13:45, April 5 – July 12
Lecture course website

In order to register and attend some part of the semester, or to attend one of the workshops listed above, please fill in this form. Please fill it in only once (and mark relevant events), even if you are going to attend more than one event!

There is a limited funding to support participation of younger participants in lecture series and advanced schools (see the dates in the program!) – if you need some support for your visit please mention this upon registration.

Note that during the semestr there will be a few periods with holidays, when the academic institutions are closed and there will be no lectures or other activities:
– Easter break: March 29 – April 1
– May holidays: May 1-5 (with Labour Day on May 1st and Constitution Day on May 3rd)
– Corpus Christi: May 30 – June 2 (the official holiday on Thursday is usually followed by the long weekend)

Various activities during the semester (lectures, workshops, seminars) will be held:
– at the Institute of Mathematics, Polish Academy of Sciences (street address: Śniadeckich 8)
– University of Warsaw, Faculty of Mathematics, Informatics, and Mechanics (Banacha 2)
– Faculty of Physics, University of Warsaw (Pasteura 5)

Contact: knot_simons@impan.pl

Warsaw Chopin Airport – the main airport serving Warsaw. It is located within the city limits, 10 km from the city center, which you can easily reach by a few city buses, a train line, and a taxi.

Warsaw Modlin Airport – another airport, mainly serving budget airlines, located 35 km from the city . It takes around 50 minutes to get to the city center, either by bus and train or a taxi.

Warszawa Centralna Railway Station – the main railway station located in the city center, also serving international trains.

Public transport in Warsaw – the city has a vast transportation network, which includes buses, metro, and trams. You can buy tickets for a single trip (up to either 20 or 75 minutes, with transfers allowed), or multi-day tickets.

Warsaw Public Bike “Veturilo” – one of the largest urban bike systems in Europe. It is a good complement to public transport in Warsaw and allows quick navigation through our city, with over 300 stations and more than 4500 bicycles available. You can register and use at little or (for short enough rides) no cost.

Here are some (relatively) nearby hotels in which participants might choose to stay. For more options you can check Google maps, booking.com, etc.

Guest/hotel rooms at Institute of Mathematics, Polish Academy of Sciences (contact us directly)

Hera – guesthouse of the University of Warsaw (budget option)

Campanile Warszawa (***)

MDM Hotel (***)

Radisson Blue Sobieski (****)

Polonia Palace (****)

Participants requiring a visa to visit Poland are advised to apply for one at Polish consulate well in advance. Information about Polish embassies and consulates can be found here.