Abstract: Non-semisimple invariants became a hot topic in quantum topology
after physicists predicted their categorification. In the talk I will give a gentle introduction to
the theory of quantum link and 3-manifold invariants and then focus on the relationship
between non-semisimple (ADO) and universal 𝖘𝖑(2) link invariants as well as
Costantino–Geer–Patureau (CGP) and Witten–Reshetikhin–Turaev (WRT) 3-manifold ones.
Title: Mapping class group actions from Hopf monoids.
Abstract: We show that pivotal Hopf monoids in symmetric monoidal categories give
rise to actions of mapping class groups of oriented surfaces with
boundary components. These mapping class group actions are associated
with edge slides in embedded ribbon graphs that generalise chord slides
in chord diagrams. They can be described simply and concretely in terms
of generating Dehn twists. Under certain assumptions on the symmetric
monoidal category and the Hopf monoid, they induce actions of mapping
class groups of closed surfaces. Based on 2002.04089.
Title: About reversing surgery in immersed Lagrangian fillings of Legendrian knots.
Abstract: From an immersed filling of a knot, one can performed surgery on the double
points to get an embedded filling of the same knot. Each solved double point
increases the genus of the filling by one. A natural question is then: is the
converse true? can we trade genus for double points? We are interested in the
symplectic-contact version of this question. In this talk, we will see how a
typical Legendrian invariant, namely the set of augmentations, can help giving a
negative answer in some cases. This is work in progress joint with
Capovilla-Searle, Legout, Murphy, Pan and Traynor.
Abstract: Due to work of Poincaré and Thurston, a generic surface or 3-manifold
admits a hyperbolic structure. The role of hyperbolic geometry in dimension
4 or higher is still mysterious. In this talk we expose what is currently
known in dimension 4; we then focus on the construction of convex
hyperbolic manifolds, some objects that play a fundamental role in
dimension 3. We show in particular that many plumbings of surfaces admit
such a convex hyperbolic structure, and how this leads to the first
construction of closed hyperbolic manifolds without spin structures.
(joint with Stefano Riolo and Leone Slavich)
Abstract: I will look at some algebras based on braids
with a quadratic relation, which are natural candidates
for Homfly skein theory models.
The main theme will be the use of braids in thickened
surfaces to construct algebras. These readily produce
the Hecke algebras and the affine Hecke algebras of type
A. I will point out a problem in doing the same for the
double affine Hecke algebras, along with a way round it,
which also allows for extensions to the algebras by the
use of closed curves and tangles in addition to
braids. This is part of recent work with Peter
Title: (Almost)-crystallographic quotients of Artin and surface braid groups and their finite subgroups.
Abstract: We discuss some recent results regarding quotients of Artin
and surface braid groups by elements of the lower central series of
the corresponding pure braid group, and the embedding of finite groups
in these quotients. This is joint work with D. L. Gonçalves, O. Ocampo
and C. Pereiro.
Title: The Volume Conjecture for Turaev–Viro invariants.
Abstract: Several connected "Volume Conjectures"
inquire about the relationship between various quantum invariants of
links, or 3-manifolds, and their geometric properties. A recent
version of Chen–Yang, in particular, relates the asymptotic growth of
the Turaev–Viro invariants of compact 3-manifolds and their hyperbolic
volume. I will give an overview of the known results and techniques
used to attack this conjecture, and then focus on a recent result that
proves it for an infinite family of hyperbolic manifolds built from
certain right-angled ideal polyhedra. The main new tool used in the
proof is a Fourier Transform for quantum invariants first introduced
by Barrett. The talk will only assume basic knowledge of
low-dimensional topology (in particular, no knowledge of the
Turaev–Viro invariants is needed).
Abstract: The n-trace of a knot is a 4-manifold, homotopy equivalent to the
2-sphere, obtained by attaching a 2-handle to the 4-ball along the knot
with framing n. The knot is said to be n shake slice if the generator of
second homology of the n-trace can be represented by a locally flat
embedded 2-sphere. There is also a smooth version. A slice knot is
n-shake slice for every n, but many other statements of this kind that
you can think of are either false or open. I will discuss some
background, some of these other statements, and my results on this
question from work with Peter Feller, Allison Miller, Matthias Nagel,
Patrick Orson, and Arunima Ray.
Title: A full homological model for quantum Verma modules and their
representations of braid groups.
Abstract: Categories of modules on quantum groups
produce strong topological invariants (knots, braids, 3-manifolds,
TQFTs...), such as the famous Jones polynomial for knots. They rely
essentially on this purely algebraic tool, so that their topological
content remains often mysterious. In this work we will build
relative homology modules from configuration spaces of points. We will
endow them with an action of the quantum sl(2) algebra and we will
recognize a tensor product of Verma modules. They are provided with
(an extension of) Lawrence representations of braid groups which turns
out to be the quantum representation of braid groups (given by the
R-matrix). We work over the ring of Laurent polynomials – suitable for
evaluating variables – and we preserve this structure all along the
construction; it yields a full homological model for integral versions
of quantum Verma modules. If time allows, we will apply this model to
give an interpretation for colored Jones polynomials dealing with
Title: The meridional rank conjecture: an attack with crayons.
Abstract: The meridional rank conjecture posits equality between the bridge
number β and meridional rank μ of a link L⊂𝕊3. I will
describe a diagrammatic technique – relying on "coloring" knot diagrams –
by which we establish the conjecture for new infinite classes of links. We
obtain upper bounds for β via the Wirtinger number of L, which is a
combinatorial equivalent of the bridge number. Matching lower bounds on
μ are found using Coxeter quotients of π1(𝕊3∖L). As a
corollary, we derive formulas for the bridge numbers for the links in
question. Based on joint works with Ryan Blair, Sebastian Baader, Filip
Abstract: We will compare various metrics on the set
of knots. These metrics are defined in terms of crossing changes, or
of genera of certain cobordisms. The distance of a knot K to the set
of knots with Alexander polynomial 1 can be shown to be the same in
all those metrics. This distance is thus a knot invariant with
four-dimensional, three-dimensional, algebraic characterizations. For
example, it can be defined in terms of Seifert matrices or the
Blanchfield pairing; or as minimum genus of a Seifert surfaces with
boundary K union a Alexander-polynomial-1 knot; or as minimum genus of
a topological slice surface of K whose complement has cyclic
fundamental group. It also equals the topological super slice genus
(minimum genus of a locally flat slice surface for K whose double is
an unknotted surface in the 4-sphere) and the ℤ-stabilization number
(minimum n such that K is boundary of a locally flat disk whose
complement has cyclic fundamental group in the connected sum of the
four-ball with n copies of ℂℙ2 # -ℂℙ2). Joint work with Peter
Feller, see arXiv:1905.08305.
Title: Automorphisms of the character variety of a surface.
Abstract: The character variety of a surface group into SL2(ℂ) is an
affine algebraic variety. We will show that its automorphism group is (up
to a finite group) the mapping class group of the surface. To that aim, we
will study certain valuations among which we will recognize Thurston's
boundary of the Teichmüller space. Joint work with Christopher-Lloyd Simon.
Title: Almost equivalence for transitive Anosov flows.
Abstract: The world of 3-dimensional Anosov flows is still mysterious: for example we do not know which 3-manifold support Anosov flows, or whether some 3-manifold support infinitely many non-equivalent Anosov flows.
Fried asked whether all transitive Anosov flows admit genus-one Birkhoff sections and Ghys asked whether all transitive Anosov flows are almost-equivalent (two 3-dimensional flows are almost equivalent if one can go from one to the other by Dehn surgering finitely many periodic orbits).
We will motivate these two questions, show that they are equivalent, and give a positive answer in some special cases.
This is a joint work with Mario Shannon.