- November 16, Khovanov homology and 4-manifolds, Northwestern geometry/physics seminar. (abstract, notes for pre-talk)
Abstract:
I'll describe our attempts to define Khovanov homology for
4-manifolds. We beginning by constructing a `disklike 4-category' from
Khovanov homology. There are certain technical difficulties, and for
now we're forced to work in characteristic 2. From there, we follow
the standard topological field theory recipe to construct vector space
valued invariants of 4-manifolds (possibly with boundary, possibly
with a link in the boundary). Along the way, however, this
construction loses some of the nice properties of Khovanov homology,
in particular the exact triangle for resolving a crossing in a link.
Our recent work on the blob complex began as an attempt to remedy this
defect, and I'll describe the computational tool that the blob complex
provides in this context.
- November 15, Small fusion categories, Vanderbilt colloquium. (abstract, slides)
Abstract:
Over the last two decades our understanding of small index subfactors has improved substantially. We have discovered a slew of examples, some related to finite groups or quantum groups, and other `sporadic' examples. At present we have a complete classification of (hyperfinite) subfactors with index at most 5, and a few results that push past 5. I'll explain the main techniques behind these classification results, and also spend a little time describing how we construct the sporadic examples. (Joint work with many people!)
- November 14, Khovanov homology and 4-manifolds, Vanderbilt topology and group theory seminar. (notes, video)
- November 13, Webs and skew Howe duality, Vanderbilt subfactor seminar. (abstract, notes, video)
Abstract:
The representation category of SL(n) is a pivotal tensor category.
This means that one can draw planar diagrams representing morphisms,
with composition corresponding to vertical stacking, and tensor
products corresponding to horizontal juxtaposition. Any planar isotopy
of such a diagram gives equations between the corresponding morphisms.
For any such category, we'd like to be able to give a presentation via
certain generators modulo local relations. For Rep(SL(n)), we've had a
conjectural presentation for several years, but no good tools for
showing that we have all the relations. With Sabin Cautis and Joel
Kamnitzer, we now have not only a proof that this presentation is
correct, but also a clear conceptual explanation of how the relations
arise. This explanation uses skew Howe duality.
- October 26, Knots and quantum computation, Wollongong colloquium. (abstract, slides, video)
Abstract:
I'll begin with the Jones polynomial, a knot invariant discovered 30
years ago that radically changed our view of topology. From there,
we'll visit the complexity of evaluating the Jones polynomial, the
topological quantum field theories related to the Jones polynomial,
and how all these ideas come together to offer an unorthodox model for
quantum computation.
- October 25, Subfactors as quantum symmetries, Operator Algebra seminar at Wollongong. (abstract, notes, video)
Abstract:
I'll explain how every finite group $G$ can be realized (essentially
uniquely) as outer automorphisms of the hyperfinite $II_1$ factor $R$.
Taking the fixed points we obtain a subfactor $R^G \subset R$. From
the representation theory of $R^G \subset R$ we can recover the group
$G$. A useful perspective on subfactors is to think of any subfactor
$A \subset B$ as arising in this way from a `quantum symmetry group'.
We're still exploring this new world: constructing new examples,
understanding just how `exotic' these are, and proving classification
results in some regimes. I'll tell you what we've seen so far.
- September 27, The search for small fusion categories, Plenary talk at the AustMS annual meeting at Ballarat. (abstract, slides)
Abstract:
Fusion categories are a beautiful generalization of finite groups and quantum
groups. A fusion category is the essential underlying algebraic data of a
topological field theory in 2+1 dimensions.
Over the past decade and a half, we've been trying to understand the small
examples of fusion categories. I'll touch on the connections with group theory,
category theory, condensed matter phyiscs and operator algebra, and show
you some of the most interesting examples. Recent work has revealed that
there are surprisingly few such examples, making the ones we have all the
more precious!
- September 22, Higher categories, lower manifolds, Early Career Workshop at the AustMS annual meeting. (abstract, slides)
Abstract:
I've been interested in the interplay between category
theory and topology, especially through topological field theory. To
tell manifolds apart, we want to compute invariants. Sometime, these
invariants can be computed by cutting the manifold into smaller
pieces, computing something "algebraic" associated to each piece, then
assembling the pieces via some algebraic rules. The theory of higher
categories is more or less the theory of possible ways to assemble
algebraic data associated to small pieces of manifolds. I'll tell you
about some of the aspects of this problem that I've worked on, and how
I ended up working on them!
- September 21, The blob complex, Algebra/Geometry/Topology Seminar at the University of Melbourne. (abstract, notes, video)
Abstract:
Topological field theories associate vector spaces to n-manifolds (and sometimes numbers to (n+1)-manifolds, although that won't concern us much). The blob complex provides a natural way to see these vector spaces as the 0-th homology of an invariant chain complex. I'll explain some nice properties of this chain complex, and tell you what we might hope to learn about manifolds by looking at the higher homologies of this chain complex.
- August 28, Khovanov homology for 4-manifolds and the blob complex, New Perspectives in Topological Field Theories, Hamburg. (notes, video)
- July 16, Turaev-Viro theory and connections, Subfactors in Maui. (notes)
- July 4, The blob complex, ANU. (notes)
- May 21, The blob complex, Stanford RT+QA/Topology seminar. (notes)
- March 14, Constructing exotic subfactors, AIM workshop on classifying small fusion categories.
- February 17, The $5-\epsilon$ colour conjecture, UC Berkeley subfactor seminar. (notes)
- January 31, A 4-manifold invariant from Khovanov homology, UC Davis geometry/topology seminar. (notes, video)
- January 25, Computing pushforwards of data sets, Amgen bio-informatics group.
- January 9, Can we get to $3+\sqrt{5}$?, Kyoto conference on von Neumann algebras. (rough notes)
- December 6, Knots and quantum computation, Miller Institute lunch talk. (video, slides)
- November 14, Khovanov homology and 4-manifold invariants, Eugene colloquium. (abstract, video, notes)
Abstract:
I'll describe an invariant of smooth 4-manifolds defined in terms of
Khovanov homology. It associates a doubly-graded vector space to each
4-manifold (optionally with a link in its boundary), generalizing the
Khovanov homology of a link in the boundary of the standard 4-ball.
I'll finish by talking about relations to TQFT, and the prospects for
concrete calculations. This is joint work with Chris Douglas and Kevin
Walker.
- November 10, Invariants of 4-manifolds from Khovanov homology, Boston College. (video, slides)
- October 29, Connections and planar algebras, Von Neumann algebras and Conformal Field Theory at Vanderbilt. (abstract, handout)
Abstract:
With Emily Peters, I've been exploring subfactors with index in the
interval $(5, 3+\sqrt{5})$. We've recently obtained a classification
of 1-supertransitive subfactors in this range, and performed an
extensive computer search in higher supertransitivities. I'll describe
the examples of subfactors we've found. We have two main new
techniques. First, even when we only know a fragment of a principal
graph, we can extract certain inequalities by considering the norms of
the entries of a connection. This allows the new classification
result. Second, we extend the theory of bi-unitary connections to the
bi-invertible case, and find we can then work over a fixed number
field. This allows effective use of the "hybrid method" of
constructing subfactors: given a not-necessarily flat bi-invertible
connection, we can efficiently solve the equations for a flat element
in the graph planar algebra. This lets us completely analyse the
examples.
- October 21, The blob complex, Toronto. (abstract, video, Bar-Natan's notes)
Abstract:
In my paper with Kevin Walker, we defined the blob complex, which pairs an $n$-manifold and an $n$-category to produce a chain complex. When $n=1$ and the manifold is the circle, this is the Hochschild complex. Thus the blob complex is a generalization of Hochschild homology to higher dimensional algebraic structures, where we can specify the $n$-dimensional shape "along which" we compute. Alternatively, the $0$-th homology is the usual TQFT invariant. Thus we can think of the extra information in the chain complex as extending TQFT invariants in the same way that Hochschild homology extends the coinvariants of an algebra. I'll define the blob complex, sketch its important properties, and outline its intended applications, calculating invariants of manifolds based on exotic higher categories.
- October 19, Small index subfactors, Toronto colloquium. (abstract, slides)
Abstract:
Fusion categories provide a simple model for a collection of
particles which can split and fuse. Despite the simplicity of the
model, it has proved useful in describing exotic materials in
condensed matter physics. On the mathematical side, fusion categories
can be thought of as 'quantum' finite groups. I'll describe recent
joint work on the classification of small fusion categories. The new
examples we've encountered are rather strange objects. Moreover, the
classification is much sparser than we had expected, encouraging the
hope of extending our current knowledge to larger and larger classes
of fusion categories. Along the way we'll use analysis (von Neumann
algebras and subfactors), number theory (the geometry of cyclotomic
integers), graph combinatorics, two-dimensional topology, and some
representation theory!
- October 15, Invariants of 4-manifolds from Khovanov homology, Category-theoretic methods in representation theory, Ottawa. (abstract, slides, blog post)
Abstract:
I'll explain how to define invariants of 4-manifolds (possibly with boundary, possibly with a link in the boundary) from Khovanov homology.
The construction relies on a new property of Khovanov homology, roughly 'S^3 functoriality'. I'll try to indicate where the difficulties may
lie establishing this property for the variations of Khovanov homology based on categorified quantum groups other than SU(2). Finally, I'll
explain how this construction is best viewed as a partial categorification of the 3+1 dimensional TQFT of which the usual Witten-
Reshetikhin-Turaev theories are the boundary, rather than a direct categorification of a WRT theory.
- October 8, Classifying fusion categories and subfactors, SIAM Mini-symposium on Algebraic Aspects of Quantum Computing, Raleigh. (slides)
- July 22, The blob complex, Subfactors in Maui. (notes)
- July 21, The $5-\epsilon$ colour theorem., Subfactors in Maui.
- June 7, 4-manifold invariants from Khovanov homology, Michael Freedman's 60th Birthday Conference, Berkeley. (abstract, slides, video)
Abstract:
I'll describe an invariant of smooth 4-manifolds defined in terms of
Khovanov homology. It associates a doubly-graded vector space to each
4-manifold (optionally with a link in its boundary), generalizing the
Khovanov homology of a link in the boundary of the standard 4-ball.
For now, we can only make the construction in characteristic two. I'll
finish by talking about relations to TQFT, and the prospects for
concrete calculations. This is joint work with Chris Douglas and Kevin
Walker.
- May 24, 4-manifold invariants from Khovanov homology, Stanford topology seminar. (abstract, slides, slides)
Abstract:
I'll show you the topological quantum field theory recipe for defining
invariants of n-manifolds from a ``disklike n-category'', illustrated
by the particular example of Khovanov homology. Unfortunately, there's
still a gap in our knowledge --- we don't yet know that our attempts
to construct a disklike 4-category from Khovanov homology really
satisfies all the requisite axioms. I'll carefully explain this defect
and what remains to be done before we can build invariants of
arbitrary 4-manifolds via this recipe.
- May 20, Classifying small index subfactors, Great Plains Operator Theory Symposium. (slides, slides)
- February 9 and 16, Topological field theory and the blob complex, University of Sydney and the Australian National University. (abstract, notes)
Abstract:
An n-dimensional topological field theory (TFT) associates vector
spaces to n-manifolds, and higher algebraic data to lower dimensional
manifolds. In the optimal situation, a TFT associates an n-category to
the point (the 0-dimensional manifold), and this n-category in fact
determines the entire TFT, via gluing formulas. I'll describe an
extension of this framework, which we call the "blob complex". The
blob complex associates a chain complex to each n-manifold, whose 0-th
homology is the usual TFT vector space. When n=1, TFTs are essentially
determined by associative *-algebras, and the blob complex for the
circle is equivalent to the Hochschild complex of the algebra. Thus we
have a generalization of Hochschild homology to higher dimensions
which at the same time extends the usual TFT invariants. The blob
complex satisfies certain "A-infinity gluing formulas", which suggest
connections to important new invariants of manifolds coming from
Seiberg-Witten theory, Heegard-Floer theory, and Khovanov homology.
This is joint work with Kevin Walker.
- January 25, February 1, Field and local relations, Teichner's hot topics course on the blob complex. (notes)
- January 21, Khovanov homology for 4-manifolds, GRASP seminar UC Berkeley. (notes)
- January 13, Fusion categories and subfactors, UCSD Colloquium. (slides, slides)