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Seminars (HTL)

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Event When Speaker Title Presentation Material
HTLW01 16th January 2017
10:00 to 11:00
Adam Levine 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.
HTLW01 16th January 2017
11:30 to 12:30
Cameron Gordon Left-orders on 3-manifolds I
HTLW01 16th January 2017
14:30 to 15:30
Cameron Gordon Left-orders on 3-manifolds II
HTLW01 16th January 2017
16:00 to 17:00
Andy Wand Contact structures on 3-manifolds I
HTLW01 17th January 2017
09:00 to 10:00
Adam Levine 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.
HTLW01 17th January 2017
10:00 to 11:00
Adam Levine 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.
HTLW01 17th January 2017
11:30 to 12:30
Cameron Gordon Left-orders on 3-manifolds III
HTLW01 17th January 2017
14:30 to 15:30
Anthony Licata 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.


HTLW01 17th January 2017
16:00 to 17:00
Andy Wand Contact structures on 3-manifolds II
HTLW01 18th January 2017
09:00 to 10:00
Adam Levine Topics in Heegaard Floer homology III
We will introduce bordered Floer homology, including some of the underlying algebraic concepts (e.g. A_\infty modules).
HTLW01 18th January 2017
10:00 to 11:00
Adam Levine 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).
HTLW01 18th January 2017
11:30 to 12:30
Eli Grigsby 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.
HTLW01 18th January 2017
14:30 to 15:30
Anthony Licata 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.
HTLW01 18th January 2017
16:00 to 17:00
Rachel Roberts Foliations on 3-manifolds I
HTLW01 19th January 2017
09:00 to 10:00
Adam Levine 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.
HTLW01 19th January 2017
10:00 to 11:00
Adam Levine 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.
HTLW01 19th January 2017
11:30 to 12:30
Eli Grigsby 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.
HTLW01 19th January 2017
14:30 to 15:30
Eli Grigsby 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).
HTLW01 19th January 2017
16:00 to 17:00
Rachel Roberts Foliations on 3-manifolds II
HTLW01 20th January 2017
09:00 to 10:00
Liam Watson Bordered via curves and train tracks I
HTLW01 20th January 2017
10:00 to 11:00
Liam Watson Bordered via curves and train tracks II
HTLW01 20th January 2017
11:30 to 12:30
Jacob Rasmussen 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.
HTLW01 20th January 2017
14:30 to 15:30
Anthony Licata 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.
HTLW01 20th January 2017
16:00 to 17:00
Rachel Roberts Foliations on 3-manifolds III
HTLW02 30th January 2017
10:00 to 11:00
Stefan Friedl Polytope invariants of groups and manifolds
Co-authors: Wolfgang L\"uck (University of Bonn), Kevin Schreve (University of Michigan), Stephan Tillmann (University of Sydney)

We will associate to L^2-acyclic groups and manifolds, modulo some not overly restrictive technical hypothesis, a formal difference of polytopes. We will relate it to the Thurston norm and compute it if the fundamental group has a 2-generator 1-relator presentation
HTLW02 30th January 2017
11:30 to 12:30
Marc Lackenby The complexity of unknot recognition
HTLW02 30th January 2017
14:00 to 15:00
Nathan Dunfield Floer homology, group orders, and taut foliations of hyperbolic 3-manifolds
A bold conjecture of Boyer-Gorden-Watson and others posit that for any irreducible rational homology 3-sphere M the following three conditions are equivalent: (1) the fundamental group of M is left-orderable, (2) M has non-minimal Heegaard Floer homology, and (3) M admits a co-orientable taut foliation. Very recently, this conjecture was established for all graph manifolds by the combined work of Boyer-Clay and Hanselman-Rasmussen-Rasmussen-Watson. I will discuss a computational survey of these properties involving half a million hyperbolic 3-manifolds, including new or at least improved techniques for computing each of these properties. 
HTLW02 30th January 2017
15:30 to 16:30
Sarah Rasmussen L-space surgeries on iterated satellites by torus links
Satellites by torus links provide multi-component versions of cables.  I will describe the region of L-space surgeries on any torus-link satellite of any L-space knot, with a result that precisely extends Hedden’s and Hom’s analogous result for cables.  More generally, I will characterize the region of L-space surgeries for iterated torus-link satellites, including the case of algebraic links.  This is joint work with Maciej Borodzik.
HTLW02 31st January 2017
09:00 to 10:00
Ana Lecuona Slopes, colored links and Kojima's eta concordance invariant
In this talk we will introduce an invariant, the slope, for a colored link in a homology sphere together with a suitable multiplicative character defined on the link group. The slope takes values in the complex numbers union infinity and it is real for finite order characters. It is a generalization of Kojima's eta-invariant and can be expressed as a quotient of Conway polynomials. It is also related to the correction term in Wall’s non-additivity formula for the signatures of 4-manifolds, and as such it appears naturally as a correction term in the expression of the signature formula for the splice of two colored links. This is a work in progress with Alex Degtyarev and Vincent Florens.
HTLW02 31st January 2017
10:00 to 11:00
Yoav Moriah Diagram Uniqueness for Highly Twisted Plats
Co-author: Jessica Purcell (Monash U. Melbourne Australia)

In this paper we prove that if a knot or link has a sufficiently complicated plat projection, then that plat projection is unique. More precisely, if a knot or link has a 2m-plat projection, where m is at least 3, each twist region of the plat contains at least three crossings, and n, the length of the plat, satisfies n > 4m(m − 2), then such a projection is unique up to obvious rotations. In particular, this projection gives a canonical form for such knots and links, and thus provides a classification of these links.
HTLW02 31st January 2017
11:30 to 12:30
Masakazu Teragaito Generalized torsion elements and bi-orderability of 3-manifold groups
Co-author: Kimihiko Motegi (Nihon University)
 
It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds, and verify the conjecture for non-hyperbolic, geometric 3-manifolds. We also confirm the conjecture for some infinite families of closed hyperbolic 3-manifolds. In the course of the proof, we prove that each standard generator of the Fibonacci group F(2, m) (m > 2) is a generalized torsion element.   
HTLW02 31st January 2017
14:00 to 15:00
Andras Juhasz Cobordism maps in knot Floer homology
Decorated knot cobordisms functorially induce maps on knot Floer homology.
We compute these maps for elementary cobordisms, and hence give a formula for 
the Alexander and Maslov grading shifts. We also show a non-vanishing result in the case of
concordances and present some applications to invertible concordances. 
This is joint work with Marco Marengon.
HTLW02 31st January 2017
15:30 to 16:30
Scott Morrison The Temperley-Lieb category in operator algebras and in link homology
The Temperley-Lieb category appears in a fundamental way in both the study of subfactors and in link homology theories. Indeed, the discovery of the importance of the Temperley-Lieb category for subfactors led to the creation of the Jones polynomial, and thence, after a long gestation, Khovanov homology.
HTLW02 1st February 2017
09:00 to 10:00
Yi Ni Null surgery on knots in L-spaces
Co-author: Faramarz Vafaee (Caltech)
 
Let   be a knot in a rational homology sphere  . Then there is a unique surgery on   which results a manifold with    . We call this surgery the null surgery. When   is an L-space, the null surgery remembers the information about the genus and fiberedness of the knot. A special case of our theorem is that if the resulting manifold is      , then the dual knot is a spherical braid. This is joint work with Faramarz Vafaee.
HTLW02 1st February 2017
10:00 to 11:00
Ken Baker Constructions of asymmetric L-space knots
Until July 2014, all known L-spaces admitted an involution.  Then, through a clever search of the SnapPy census, Dunfield-Hoffman-Licata found examples of asymmetric one-cusped hyperbolic manifolds with two lens space fillings and consequently many asymmetric L-space fillings. Yet since none of these lens space fillings were $S^3$, so still stood the conjecture that L-space knots in $S^3$ are strongly invertible.

In this talk we present 
(1) a `natural' realization and vast generalization of the Dunfield-Hoffman-Licata examples (joint work with Hoffman and Licata) and
(2) the first construction of asymmetric L-space knots in $S^3$ (joint work with Luecke).

Both of these constructions produce asymmetric one-cusped hyperbolic manifolds with two fillings that are double branched covers of alternating links, though the approaches are different.
HTLW02 1st February 2017
11:30 to 12:30
John Luecke Boundary-reducing surgeries and bridge number
Let M be a 3–dimensional handlebody of genus g > 1. We give examples of hyperbolic knots in M with arbitrarily large genus g bridge number which admit Dehn surgeries which are boundary- reducible manifolds.
HTLW02 2nd February 2017
09:00 to 10:00
Tali Pinsky On tunnel number one knots with lens space surgeries
HTLW02 2nd February 2017
10:00 to 11:00
Steven Boyer Branched covers of quasipositive links and L-spaces
Co-authors: Michel Boileau (Université Aix-Marseille), Cameron McA. Gordon (University of Texas at Austin)
 
We show that if L is an oriented non-trivial strongly quasipositive link or an oriented quasipositive link which does not bound a smooth planar surface in the 4-ball, then the Alexander polynomial and signature function of L determine an integer n(L) such that \Sigma_n(L), the n-fold cyclic cover of S^3 branched over L, is not an L-space for n > n(L). If K is a strongly quasipositive knot with monic Alexander polynomial such as an L-space knot, we show that \Sigma_n(K) is not an L-space for n \geq 6 and that the Alexander polynomial of K is a non-trivial product of cyclotomic polynomials if \Sigma_n(K) is an L-space for some n = 2, 3, 4, 5. Our results allow us to calculate the smooth and topological 4-ball genera of, for instance, quasi-alternating oriented quasipositive links. They also allow us to classify strongly quasipositive 3-strand pretzel knots. 
HTLW02 2nd February 2017
11:30 to 12:30
Michel Boileau 3-manifold groups epimorphisms and rigidity
Co-Author Stefan Friedl (Universität Regensburg)

Given a degree-one map  between two aspherical compact orientable 3-manifolds it is a natural question to find out extra conditions to ensure that it is in fact homotopic to a homeomorphism. We will give two criteria in term of virtual ranks and of virtual heegaard genera. We will also discuss the knot spaces case and some open questions.
HTLW02 2nd February 2017
14:00 to 15:00
Duncan McCoy Characterizing slopes for torus knots
We say that   is a characterizing slope for a knot   in the 3-sphere if the oriented homeomorphism type of  -surgery is sufficient to determine the knot   uniquely. I will discuss the problem of determining which slopes are characterizing for torus knots, paying particular attention to non-integer slopes. This problem is related to determining which knots in     have Seifert fibered surgeries.
HTLW02 2nd February 2017
15:30 to 16:30
Brendan Owens A Gordon-Litherland form for ribbon surfaces
Co-authors: Josh Greene and Saso Strle

I will describe work in progress aimed at generalising the Gordon-Litherland form to the case of properly embedded surfaces in the four-ball and the four-sphere.  I will also discuss a motivating non-application involving concordance invariants of links which lie on definite surfaces in S^4.


HTLW02 3rd February 2017
09:00 to 10:00
Ciprian Manolescu Floer homology and covering spaces
Co-author: Tye Lidman (North Carolina State)
 
I will discuss a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer / Heegaard Floer correspondence, it follows that if a 3-manifold Y admits a p^n-sheeted regular cover that is a Z/p-L-space (for p prime), then Y is a Z/p-L-space. Further, we obtain constraints on surgeries on a knot being regular covers over other surgeries on the same knot, and over surgeries on other knots. This is joint work with Tye Lidman.
HTLW02 3rd February 2017
10:00 to 11:00
Matthew Stoffregen Pin(2)-equivariant Floer homology and homology cobordism
We review Manolescu's construction of the  -equivariant Seiberg-Witten Floer stable homotopy type, and apply it to the study of the 3-dimensional homology cobordism group. We introduce the `local equivalence' group, and construct a homomorphism from the homology cobordism group to the local equivalence group. We then apply Manolescu's Floer homotopy type to obstruct cobordisms between Seifert spaces. In particular, we show the existence of integral homology spheres not homology cobordant to any Seifert space. We also introduce connected Floer homology, an invariant of homology cobordism taking values in isomorphism classes of modules.
HTLW02 3rd February 2017
11:30 to 12:30
Jen Hom Knot concordance in homology spheres
The knot concordance group C consists of knots in S^3 modulo knots that bound smooth disks in B^4. We consider C_Z, the group of knots in homology spheres that bound homology balls modulo knots that bound smooth disks in a homology ball. Matsumoto asked if the natural map from C to C_Z is an isomorphism. Adam Levine answered this question in the negative by showing the map is not surjective. We show that the image of C in C_Z is of infinite index. This is joint work with Adam Levine and Tye Lidman.
HTLW02 3rd February 2017
14:00 to 15:00
John Baldwin Stein fillings and SU(2) representations
Co-author: Steven Sivek (Imperial College)

I'll describe recent work with Sivek in which we prove that if a 3-manifold Y admits a Stein filling which is not a homology ball then its fundamental group admits a nontrivial SU(2) representation. Beyond establishing a new connection between contact geometry and the fundamental group, this result allows us to deduce the existence of nontrivial representations where previously existing methods do not appear to suffice. Our proof makes use of a fairly new invariant of contact 3-manifolds which Sivek and I defined in the context of instanton Floer homology.
HTLW02 3rd February 2017
15:30 to 16:30
Gordana Matic Filtering the Heegaard Floer contact invariant
Co-authors: Cagatay Kutluhan ( University at Buffalo), Jeremy Van Horn-Morris (University of Arkansas), Andy Wand (University of Glasgow)

In this joint work with Kutluhan, Van Horn-Morris and Wand, we define the {\it spectral order} invariant of contact structures in dimension three by refining the contact invariant from Heegaard Floer homology. This invariant takes values in the set \mathbb{Z}_{\geq0}\cup\{\infty\}. It is zero for overtwisted contact structures, \infty for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. It gives a criterion for tightness of a contact structure stronger than that given by the Heegaard Floer contact invariant, and an obstruction to existence of Stein cobordisms between contact 3-manifolds. We show this by exhibiting an infinite family of examples with vanishing Heegaard Floer contact invariant on which our invariant assumes an unbounded sequence of finite and non-zero values.
HTL 9th February 2017
15:15 to 16:30
Marco Golla Correction terms and the non-orientable 4-genus
Given a knot K in the 3-sphere, we use Heegaard Floer d-invariants to give lower bounds on the first Betti number of a non-orientable surface in the 4-ball with boundary K, strengthening earlier work of Batson. An amusing feature of our bound is its superadditivity with respect to connected sums. This is joint work with Marco Marengon.



HTL 16th February 2017
15:15 to 16:15
Lukas Lewark The Seifert form's optimal bounds for slice genera
What can you say about a knot's slice genus by just looking at its Seifert form?  The talk will recall the (known) answer in the smooth setting and then focus on the topological setting. The talk is based on joint work with Peter Feller.



HTL 23rd February 2017
15:15 to 16:15
Cristina Ana-Maria Palmer-Anghel Towards a homological model for the Colored Jones
HTL 9th March 2017
15:15 to 16:15
Paolo Ghiggini The wrapped Fukaya category of a Weinstein manifold is generated by the cocores of the critical handles
A Weinstein manifold is an open symplectic manifold admitting a handle decomposition adapted to the symplectic structure. It turns out that the handles of such a decomposition have index at most half of the dimension. When the index is half the dimension, they are called critical handles and their cocores are Lagrangian discs.

In a joint work with Baptiste Chantraine, Georgios Dimitroglou Rizell and Roman Golovko, we decompose any object in the wrapped Fukaya category of a Weinstein manifold as a twisted complex built from the cocores of the critical handles in a Weinstein handle decomposition. The main tools used are the Floer homology theories of exact Lagrangian immersions, of exact Lagrangian cobordisms in the SFT sense (i.e. between Legendrians), as well as relations between these theories.

Since most participants of the HTL program are not experts in Fukaya categories (including me, actually) I will try to take it easy.




HTL 16th March 2017
15:15 to 16:15
Olga Plamenevskaya Braid monodromy, orderings, and transverse invariants
HTL 23rd March 2017
15:15 to 16:15
Evgeny Gorsky On the set of L-space surgeries for links
A 3-manifold is called an L-space if its Heegaard Floer homology has minimal possible rank. A link (or knot) is called an L-space link if all sufficiently large surgeries of the three-sphere along its components are L-spaces. It is well known that the set of L-space surgeries for a nontrivial L-space knot is a half-line. Quite surprisingly, even for links with 2 components this set could have a complicated structure. I will prove that for "most" L-space links (in particular, for most algebraic links) this set is bounded from below, and show some nontrivial examples where it is unbounded. This is a joint work with Andras Nemethi.



HTL 6th April 2017
15:30 to 16:30
Tudor Dimofte Vortices and Vermas (and other applications of 3d gauge theory to geometric representation theory
Supersymmetric quantum gauge theories have a long history of relations with low-dimensional geometry and topology. They are also a natural physical home for geometric representation theory. I will review some recent developments in the realm of 3d gauge theories and their representation-theoretic consequences for Category O, symplectic duality, and the AGT correspondence.



HTLW03 10th April 2017
10:00 to 11:00
Cumrun Vafa String Theory and Homological Invariants for 3-Manifolds
In this talk I review the recent progress made in defining homological invariants for 3-manifold using string theory constructions.  This generalizes the constructions of homological invariants for knots using M5 branes, to the case of 3-manifolds.
HTLW03 10th April 2017
11:30 to 12:30
Daniel Roggenkamp Surface operators and categorification of quantum groups
In this talk I will discuss how certain categorifications of quantum groups arise from foams of surface operators in 4-dimensional gauge theories. The talk is based on joint work with Sungbong Chun and Sergei Gukov.
HTLW03 10th April 2017
13:30 to 14:30
Andrea Brini Mirror symmetry, integrable systems and the Gopakumar--Vafa correspondence for Clifford--Klein 3-manifolds
I will report on recent progress on the Gopakumar--Ooguri--Vafa correspondence, relating quantum (Witten--Reshetikhin--Turaev) invariants of 3-manifolds and knots therein with curve-counting theories (Gromov--Witten/Donaldson--Thomas) of local Calabi--Yau threefolds, in the context of Seifert-fibred 3-manifolds. I will describe A- and B- model constructions for the correspondence in the broadest context where the standard form of the duality is expect to hold (spherical space forms), discuss the link with relativistic integrable systems and the Eynard--Orantin topological recursion, and present a rigorous proof of the B-model version of the correspondence via matrix model techniques. Implications for refined invariants, orbifold GW theory, and an allied class of Frobenius manifolds and 2D-Toda reductions will be also discussed time permitting.

Based on joint work with G. Borot and further work in progress.


HTLW03 10th April 2017
16:00 to 17:00
Anna Beliakova Unified invariants of homology 3-spheres
In 2015 K. Habiro and T. Le defined unified quantum invariants of integral homology 3-spheres associated with simple Lie algebras. These invariants dominate Witten-Reshetikhin-Turaev  invariants and belong to the Habiro ring of analytic functions at roots of unity. In the talk I will review the construction of unified invariants, discuss their properties and give few applications. Then I will mention our generalisations of the unified invariants for rational homology 3-spheres. Joint work with T. Le, C. Blanchet and I. Buehler.
HTLW03 11th April 2017
10:00 to 11:00
Jørgen Andersen The Verlinde formula for Higgs bundle moduli spaces
In this talk we will present a Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. We further present a Verlinde formula for the quantization of parabolic Higgs bundle moduli spaces and stacks. We will explain how all these dimensions fit into a one parameter family of 2D TQFT's, encoded in a one parameter family of Frobenius algebras, which we will construct.​
HTLW03 11th April 2017
11:30 to 12:30
Matthew Hogancamp Khovanov-Rozansky homology and q,t Catalan numbers
I will discuss a recent proof of the Gorsky-Oblomkov-Rasmussen-Shende conjecture for (n,nm+1) torus knots, which generally expresses the Khovanov-Rozansky homology of torus knots in terms of representations of rational DAHA.  The proof is based off of a computational technique introduced by myself and Ben Elias, using complexes of Soergel bimodules which categorify certain Young symmetrizers.  We will summarize this technique and indicate how it results in a remarkably simple recursion which computes the knot homologies in question.
HTLW03 11th April 2017
13:30 to 14:30
Piotr Kucharski Knots, (extremal) A-polynomials, and BPS invariants
Co-author: Piotr Sulkowski (University of Warsaw, Caltech)

In this talk I will introduce a new class of algebraic curves called extremal A-polynomials of a knot and use it to describe BPS invariants introduced by Labastida, Marino, Ooguri, and Vafa.

I will present results obtained from the analysis of both classical and quantum extremal A-polynomials. The first lead to exact formulas for BPS invariants imposing nontrivial integrality statements in number theory. The latter enabled us to construct the combinatorial model for calculating BPS invariants.

I will also indicate how these results relate to the formalism of quivers introduced in the talk by Piotr Sulkowski.
HTLW03 11th April 2017
14:30 to 15:30
Ramadevi Pichai Arborescent knots, mutants - current status on their invariants
Computation of colored HOMFLY-PT polynomials for knots carrying arbitrary representations is still a challenging problem. First I will recapitulate the necessary tools for determining the colored knot invariant within Chern-Simons theory. Then, I will present our results on quantum Wigner 6j useful for writing polynomial form of the knot invariant. Further, we will discuss our results for mutant knot pairs. Finally, we summarize the current status on these polynomials which we periodically update on the website http://knotebook.org
HTLW03 11th April 2017
16:00 to 17:00
Paul Wedrich On colored link homologies
I will talk about recent progress in understanding the structure of type A link homologies. This includes the definition of integral, equivariant, colored sl(N) Khovanov-Rozansky link homologies, which are functorial under link cobordisms, as well as a study of their deformations and stability properties. I will finish by discussing some implications for colored, triply-graded HOMFLY-PT homologies, including an exponential growth property conjectured by Gorsky, Gukov and Stosic.
HTLW03 12th April 2017
10:00 to 11:00
Pavel Putrov Integrality in analytically continued Chern-Simons theory
Physics predicts existence of homological invariants of closed oriented 3-manifolds similar to Khovanov-Rozansky homology of knots in a 3-sphere. The decategorified version of such invariants are q-series with integer coefficients. In my talk I will discuss properties of such invariants, how they are related to Chern-Simons partition function (WRT invariant) analytically continued w.r.t. level, and give some examples. If time permits I will also discuss how resurgence theory can be used to construct such invariants and relation to open topological strings.
HTLW03 12th April 2017
11:30 to 12:30
Tobias Ekholm Higher genus knot contact homology and recursion for the colored HOMFLY polynomial
We present a conjectural description of Legendrian symplectic field theory for the conormal of a knot ("higher genus knot contact homology") and discuss its relation to the recursion relation for the colored HOMFLY polynomial. This reports on joint work with Lenny Ng. 
HTLW03 13th April 2017
10:00 to 11:00
Mohammed Abouzaid Towards a symplectic model of odd Khovanov homology
I will report on joint work in progress with Ivan Smith which combines ideas of Lawrence and Bigelow and Gaiotto-Witten with motivation from homological mirror symmetry to propose a symplectic construction of a pair of knot invariants which are expected to correspond to the odd and even Khovanov homologies. I will mostly focus on the only computation which is fully understood: the trivial diagram of the unknot.
HTLW03 13th April 2017
11:30 to 12:30
Piotr Sulkowski BPS states, knots and quivers
I will present a surprising relation between knot invariants and quiver representation theory, motivated by various string theory constructions involving BPS states. Consequences of this relation include the proof of the famous Labastida-Marino-Ooguri-Vafa conjecture, explicit (and unknown before) formulas for colored HOMFLY polynomials for various knots, new viewpoint on knot homologies, new dualities between quivers, and many others.
HTLW03 13th April 2017
13:30 to 14:30
Alexey Sleptsov Colored knot invariants from Reshetikhin-Turaev approach
I will discuss Reshetikhin-Turaev approach for construction of colored quantum link invariants, which are colored HOMFLY polynomials in the case of sl(N). This approach involves quantum R-matrices and inclusive quantum Racah coefficients also known as 6-j symbols and provides a systematic way for calculation of colored invariants. I will present our recent results for three-strand knots and relation with an alternative approach coming from WZW CFT.
HTLW03 13th April 2017
14:30 to 15:30
Amer Iqbal BPS states of M5-brane on T^3
We will discuss a subclass of BPS states in the M5-brane theory on T^3 x R^3 which  are related to little strings and whose degeneracies can be worked out exactly. The generating function of these BPS states has interesting modular properties and seems to have the structure expected of the partition function with target space a symmetric product.
HTL 20th April 2017
15:15 to 16:15
Francesco Lin Bar Natan's deformation of Khovanov homology and involutive monopole Floer homology
We study the conjugation involution in Seiberg-Witten theory in the context of the Ozsvath-Szabo and Bloom's spectral sequence for the branched double cover of a link L in S^3. We show that there exists a spectral sequence of F[Q]/Q^2-modules (where Q has degree −1) which converges to an involutive version of the monopole Floer homology of the branched double cover, and whose E^2-page is a version of Bar Natan's deformation of Khovanov homology in characteristic two of the mirror of L. We conjecture that an analogous result holds in the setting of Pin(2)-monopole Floer homology.



HTL 27th April 2017
15:15 to 16:15
Anthony Licata The 2-linearity of the free group and the topology of the punctured disc
The goal of this talk will be to describe a rather basic linear action of the free group on a finitely-generated triangulated category, and describe two proofs of the faithfulness of this action.  I will also describe some of the geometric group theory and topology that is visible from the lens of this faithful 2-representation.


HTL 18th May 2017
15:15 to 16:15
Claudius Zibrowius Categories of curved complexes for marked surfaces
In 2014, Haiden, Katzarkov and Kontsevich gave a complete algebraic description of the Fukaya category of immersed curves on oriented surfaces with boundary. In this talk, I will introduce dg categories which I suspect to be closely related, if not equivalent, to those Fukaya categories. The objects of these dg categories are curved complexes, which, loosely speaking, are chain complexes whose differentials square to multiples of the identity. As an application, I will mainly focus on two examples of such categories arising from Heegaard Floer theory and discuss why they might be interesting.



HTL 19th May 2017
13:30 to 14:30
Vera Vertesi Combinatorial Tangle Floer homology
Knot Floer homology is an invariant for knots and links defined by Ozsv\'ath and Szab\'o and independently by Rasmussen. It has proven to be a powerful invariant e.g. in computing the genus of a knot, or determining whether a knot is fibered. In this talk I define a generalisation of knot Floer homology for tangles; Tangle Floer homology is an invariant of tangles in D^3, $S^2xI or in S^3. Tangle Floer homology satisfies a gluing theorem and its version in S^3 gives back a stabilisation of knot Floer homology. Finally, I will discuss how to see tangle Floer homology as a categorification of the Reshetikhin-Turaev invariant for gl(1|1).

This is a joint work with I. Petkova and A. P. Ellis.
HTL 25th May 2017
15:15 to 16:15
Andrew Lobb Something about the Khovanov space
I'll say something about the space (really a stable homotopy type, due to Lipshitz-Sarkar) whose cohomology recovers Khovanov cohomology.  I'll tell you why you should care about the Khovanov space if you only care about low-dimensional topology.  I'll talk about one or two generalizations, maybe including work in progress on annular Khovanov cohomology and the Lie ring action on it.  This is all joint work in various combinations with Jones, Orson, and Schuetz.



HTL 1st June 2017
15:15 to 16:15
Matthew Hogancamp How to compute torus link homology
I will outline a technique, introduced by me and Ben Elias, for computing the HOMFLY version of Khovanov-Rozansky homology.  I will show how this technique gives a simple recursion which computes the homology of positive torus links (recently accomplished in special cases by me, and in generality by Anton Mellit).  Time permitting, I would like to use these computations for 3-strand braids to illustrate a connection with recent conjectures of Gorsky-Negut-Rasmussen.



HTL 8th June 2017
15:15 to 16:15
Joan Licata Morse Structures on Open Books
Every contact 3-manifold is locally contactomorphic to the standard contact R^3, but this fact does not necessarily produce large charts that cover the manifold efficiently. I'll describe joint work with Dave Gay and more recently, Dan Mathews, which uses an open  book decomposition of a contact manifold to produce a particularly efficient collection of such contactomorphisms, together with simple combinatorial data describing how to reconstruct the contact 3-manifold from these charts.   We use this construction to define  front projections for Legendrian knots and links in arbitrary contact 3-manifolds, generalising existing constructions of front projections for Legendrian knots in S^3 and universally tight lens spaces.



HTL 9th June 2017
13:30 to 14:30
Adam Levine Khovanov homology and knot Floer homology for pointed links
There are spectral sequences relating reduced Khovanov homology to a variety of other homological link invariants, including the Heegaard Floer homology of the branched double cover and instanton knot homology. However, there is no known relationship between Khovanov homology and knot Floer homology, despite considerable computational evidence and numerous formal similarities. I will describe ongoing efforts to find a spectral sequence relating these two invariants. Specifically, we construct a variant of Khovanov homology for links with one or more basepoints on each component, which more closely parallels the behavior of knot Floer homology and which conjecturally fits into a spectral sequence as required. This is joint work with John Baldwin and Sucharit Sarkar.



HTLW04 26th June 2017
10:00 to 11:00
Christian Blanchet Non semisimple TQFTs from quantum sl(2)
We will describe quantum invariants and TQFTs extracted from quantum sl(2) at root of unity, focusing on new features related with the non semisimplicity of the quantum group at root of 1.
HTLW04 26th June 2017
11:30 to 12:30
Zsuzsanna Dancso Lattices and Homological Algebra
Co-author: Anthony Licata (Australian National University)

I'll begin with some dreams and motivation regarding lifting notions of lattice theory to homological algebra. For most of the talk, we'll study a concrete example: the lattices of integer cuts and flows associated to a finite graph. Given a graph G and choice of spanning tree T, we construct an algebra A(G,T) such that the Groethendieck group of the category of finitely generated A(G,T)-modules with the Euler (Ext) pairing contains the cut and flow lattices of G as orthogonal complements. We'll discus many open problems regarding generalisations and possible uses for the extra structure that is present at the category level, such as gradings. Joint work in progress with Anthony Licata.
Navigation:
HTLW04 26th June 2017
13:30 to 14:30
Peter Samuelson Hall algebras and Fukaya categories
(joint with Ben Cooper)

The multiplication in the Hall algebra of an abelian category is defined by "counting extensions of objects," and the representation theory of this algebra tends to be quite interesting (e.g. Ringel showed the Hall algebra of modules over a quiver is the quantum group). Recently, Burban and Schiffmann explicitly described the Hall algebra of coherent sheaves over an elliptic curve, and various authors have connected this algebra to symmetric functions, Hilbert schemes, torus knot homology, the Heisenberg category, and the skein algebra of the torus. Motivated by this last connection and homological mirror symmetry, we discuss some computations in progress involving the Hall algebra of the Fukaya category of a (topological) surface.
HTLW04 26th June 2017
14:30 to 15:30
David Rose Traces, current algebras, and link homologies
We'll show how categorical traces and foam categories can be used to define an invariant of braid conjugacy, which can be viewed as a "universal" type-A braid invariant. Applying various functors, we recover several known link homology theories, both for links in the solid torus, and, more-surprisingly, for links in the 3-sphere. Variations on this theme produce new annular invariants, and, conjecturally, a homology theory for links in the 3-sphere which categorifies the sl(n) link polynomial but is distinct from the Khovanov-Rozansky theory. Lurking in the background of this story is a family of current algebra representations.

This is joint work with Queffelec and Sartori.

HTLW04 26th June 2017
16:00 to 17:00
Hoel Queffelec Around Chebyshev's polynomial and the skein algebra of the torus
The diagrammatic version of the Jones polynomial, based on the Kauffman bracket skein module, extends to knots in any 3-manifold. In the case of thickened surfaces, it can be endowed with the structure of an algebra by stacking. The case of the torus is of particular interest, and C. Frohman and R. Gelca exhibited in 1998 a basis of the skein module for which the multiplication is governed by the particularly simple "product-to-sum" formula.
I'll present a diagrammatic proof of this formula that highlights the role of the Chebyshev's polynomials, before turning to categorification perspectives and their interactions with representation theory.

Joint work with H. Russell, D. Rose and P. Wedrich.

HTLW04 27th June 2017
10:00 to 11:00
Mikhail Khovanov How to categorify the ring of integers localized at two
We construct a triangulated monoidal idempotent complete category with the
Grothendieck ring naturally isomorphic to the ring of integers with two inverted.
This is a joint work with Yin Tian.
HTLW04 27th June 2017
11:30 to 12:30
Vanessa Miemietz Introduction to p-dg 2-representation theory
I will report on joint work with Robert Laugwitz developing an abstract 2-representation theory for p-dg 2-categories, in the spirit of categorification of quantum groups at prime roots of unity by Khovanov, Qi and Elias.
HTLW04 27th June 2017
13:30 to 14:30
Lukas Lewark An Upsilon-like invariant from Khovanov-Rozansky homology
Co-author: Andrew Lobb (Durham University)

Khovanov-Rozansky homology in its most general form (so-called equivariant homology) associates to a knot a chain complex (invariant up to homotopy equivalence) over a certain polynomial ring. Equivariant homology yields various lower bounds to the slice genus, some of them concordance homomorphisms, some not; and also a piecewise linear function which has much resemblance with the recently introduced Upsilon-invariant from Heegaard-Floer homology.
HTLW04 27th June 2017
14:30 to 15:30
Ben Webster Representation theory and the Coulomb branch
For many years, my collaborators and I tried to understand the Coulomb branches of certain field theories from physics and failed miserably. Luckily, recent work of Braverman-Finkelberg-Nakajima gives a mathematical construction of these spaces, and algebras quantizing them. I'll discuss an approach to the representation theory of these algebras (building on joint work with Braden-Licata-Proudfoot and many other authors). Applications include a version of the Koszul duality between the Higgs and Coulomb branches of such a theory, a new perspective on category O for Cherednik algebras, and a new understanding of coherent sheaves on Coulomb branches in terms of KLR algebras.

Related Links
HTLW04 27th June 2017
16:00 to 17:00
Paul Wedrich On colored link homologies
I will talk about recent progress in understanding the structure of type A link homologies. This includes the definition of integral, equivariant, colored sl(N) Khovanov-Rozansky link homologies, which are functorial under link cobordisms, a study of their deformations and stability properties and the question of how these invariants extend to links in thickened surfaces. 

Joint work with M. Ehrig, H. Queffelec, D. Rose and D. Tubbenhauer.
HTLW04 28th June 2017
10:00 to 11:00
Andrei Negut Categorified knot invariants and algebraic geometry
In this talk, we will survey some recent progress in a broad and developing framework that seeks to connect categorified knot invariants, geometric representation theory, Hilbert schemes and matrix factorizations. The contributions discussed come from the work of many mathematicians, and I hope to also convey some interesting future perspectives on the topic.
HTLW04 28th June 2017
11:30 to 12:30
Matthew Hogancamp Categorical diagonalization
It goes without saying that diagonalization is an important tool in linear algebra and representation theory.  In this talk I will discuss joint work with Ben Elias in which we develop a theory of diagonalization of functors, which has relevance both to higher representation theory and to categorified quantum invariants.  For most of the talk I will use small examples to illustrate of components of the theory, as well as subtleties which are not visible on the linear algebra level.  I will also state our Diagonalization Theorem which, informally, asserts that an object in a monoidal category is diagonalizable if it has enough ``eigenmaps''.  Time allowing, I will also mention our main application, which is a diagonalization of the full-twist Rouquier complexes acting on Soergel bimodules in type A.  The resulting categorical eigenprojections categorify q-deformed Young idempotents in Hecke algebras, and are also important for constructing colored link homology theories which, conjecturally, are functorial under 4-d cobordisms.  
HTLW04 29th June 2017
10:00 to 11:00
Pedro Vaz 2-Verma modules and the Khovanov-Rozansky link homologies
In this talk I will describe a construction of Khovanov and Rozansky's HOMFLY-PT and sl(N)-link homologies using a categorification of certain parabolic Verma modules for gl(2k), the latter based on a generalization of Khovanov-Lauda-Rouquier algebras.I will also explain how to prove a conjecture about the spectral sequence from the HOMFLY-PT-link homology to the sl(N)-link homology (due to Dunfield, Gukov and Rasmussen) using our version of these link homologies.This is joint work with G. Naisse.
HTLW04 29th June 2017
11:30 to 12:30
Radmila Sazdanovic Non-multiplicative TQFTs and diagrammatic categorifications of the polynomial ring
Co-author: Mikhail Khovanov (Columbia University)

We focus on non-multiplicative TQFTs via representations of non-unital rings built out of cobordism categories modulo various relations. Examples include diagrammatic categorifications of the polynomial ring Z[x] and several classes of orthogonal polynomials.
HTLW04 29th June 2017
13:30 to 14:30
Daniel Tubbenhauer Some webs and q-Howe dualities in types BCD
Co-author: Antonio Sartori (University of Freiburg)

In seminal work, Cautis-Kamnitzer-Morrison gave a diagrammatic presentation, via a so-called web calculus, of the category of quantum gln modules tensor generated by the exterior powers. Their novel observation was that a classical tool from representation and invariant theory, known as skew Howe duality, can be quantized and used to construct the corresponding diagrammatic presentation.

The work of Cautis-Kamnitzer-Morrison was then extended to various other instance (and even categorified), but all of these have in common that they stay in type A.

In this talk I will describe the first steps towards generalizing Cautis-Kamnitzer-Morrison's results to other types, where several new features (or flaws?) appear.
HTLW04 29th June 2017
14:30 to 15:30
Yian Tian Towards a categorical boson-fermion correspondence
The celebrated boson-fermion correspondence is an isomorphism between the bosonic Fock space and the fermionic Fock space. We present  categorification of the bosonic Fock space and the Heisenberg algebra which is a modification of Khovanov's Heisenberg category. The categorifcation of the fermionic Fock space is based on Honda's category studying contact topology in dimension three. This is a joint work with Mikhail Khovanov.
HTLW04 29th June 2017
16:00 to 17:00
Anna Beliakova Quantum Annular Link Homology via Trace Functor
In this talk I will construct  a new triply graded
invariant of links in a solid torus, which is functorial with respect to
annular link cobordisms and admits a quantum sl(2) action.
We discuss several properties of this invariant.
This is a joint work with Kris Putyra and Stephan Wehrli.
HTLW04 30th June 2017
15:00 to 16:00
Cameron Gordon Left-orderability and 3-manifold groups: Rothschild Lecture
The fundamental group is a more or less complete invariant of a 3-dimensional manifold. We will discuss how the purely algebraic question of whether or not this group has a left-invariant total order appears to be related to two other, seemingly quite different, properties of the manifold, one geometric and the other essentially analytic.



University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons