Scott Morrison

my publications on the arXiv
my public calendar
photos from my trips

... more at my wiki »

2012

January 24, A 4-manifold invariant from Khovanov homology, UC Davis geometry/topology seminar. (notes, video)
January 9, Can we get to $3+\sqrt{5}$?, Kyoto conference on von Neumann algebras. (rough notes)

2011

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)
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)

2010

November 10, Fusion categories and subfactors, Caltech Colloquium. (slides, slides)
October 29, Classifying subfactors up to index 5, Kyoto Operator Algebra seminar. (slides)
October 8, Subfactors at index 5 and beyond, UCLA/DARPA subfactors meeting. (slides)
September 2, Classification of subfactors up to index 5. Quantum groups, Clermont-Ferrand. (slides)
August 11, Classification of subfactors up to index 5. Operator algebras satellite conference, Chennai (slides)
June 23, The blob complex. Low dimensional topology and categorification, Stony Brook. (slides)
June 4, Classifying fusion categories. Miller Institute Symposium. (poster)
May 11, Classification of subfactors up to index 5., Non-commutative geometry and operator algebras, Vanderbilt. (slides)
April 10, Coincidences of small modular tensor categories related to the D_2n planar algebra Quantum Invariants of 3-manifolds and Modular Categories session of the St Paul AMS meeting. (abstract)
Abstract: I'll quickly describe the D_2n planar algebra, and using this discover some coincidences between small modular tensor categories. One example is that (G_2)_{1/3} is isomorphic to the even part of D_{14}, proving that it is unitary. These coincidences give new identities between quantum knot polynomials.
January 26, Khovanov homology II, functoriality, deformations and the s-invariant, MSRI introductory workshop for the link homology semester. (lecture notes, streaming video, video)
January 25, Khovanov homology I, MSRI introductory workshop for the link homology semester. (lecture notes, streaming video, video)
January 18, Blob homology, TQFT and link homology in Hahei.
January 17, Khovanov homology for 4-manifolds, TQFT and link homology in Hahei.

2009

November 8, Blob homology, part I, special session on Homotopy Theory and Higher Algebraic Structures at AMS Riverside meeting. (slides, video)
- see also slides and video from part II, by Kevin Walker.
November 7, Coincidences of tensor categories, special session on Algebraic Structures in Knot Theory at AMS Riverside meeting. (slides)
October 30, Graph planar algebra embedding theorems, subfactor seminar at UC Berkeley.
October 18, Fusion categories and small index subfactors, part II, special session on Fusion Categories at AMS Waco meeting. (slides)
- see also slides from part I, by Noah Snyder.
October 9, Blob homology, Los Angeles Joint Topology Seminar.
September 3, Fusion categories, UC Berkeley Colloquium. (blackboards)
June 3, Classifying subfactor planar algebras, UC Riverside colloquium.
June 3, Blob homology, UC Riverside.
March 22, Extended Haagerup exists!, Tensor Categories in Bloomington, Indiana. (slides, video and blog post)
- see also March 20, Coincidences of tensor categories, at the same conference, a talk given by Noah Snyder on joint work. (slides and blog post)
January 9, The Cappell-Shaneson spheres and the s-invariant, Knots in Washington. (slides)
January 8, Blob homology, AMS Washington meeting. (slides)
January 6, Khovanov homology of rational tangles, AMS Washington meeting. (slides)

2008

December 4, The Cappell-Shaneson spheres and the s-invariant, UC San Diego. (slides)
October 16, The D_2n planar algebras, University of Tokyo.
October 7, The D_2n planar algebras, University of New South Wales seminar.
September 26, Generators and relations for U_q(sl_n) as a pivotal category, University of Sydney algebra seminar.
September 16, The D_2n planar algebras, PennState noncommutative geometry and analysis seminar.
July 8, Introduction to Khovanov homology and genus bounds for links, Station Q seminar.
May 15, Blob homology, Georgia Topology Conference.
April 18-20, Skein theory for the D_2n subfactors, Planar Algebras and Subfactors meeting at Vanderbilt.
February 1, Link invariants from the D4 subfactor, with Emily Peters and Noah Snyder, UC Berkeley subfactor seminar.

2007

December 12-15, Lasagna composition for Khovanov Homology, in the Quantum Topology Special Session of the joint NZMS/AMS Wellington meeting. (slides)
October 2, Spiders for U_q(sl_n), Station Q Seminar, UC Santa Barbara. (slides)
August 6-10, Lasagna composition for Khovanov Homology, Quantum Topology in Hanoi, Vietnam (slides)
July 5-8, Lasagna composition for Khovanov Homology, Oporto conference on Link Homology, Portugal (slides)
May 23, Functoriality and duality in Khovanov homology, Kyoto. (Nakajima's notes, photos)
May 15, Genus bounds and spectral sequences made easy, Kyoto. (slides)
May 14, Khovanov homology, Kyoto. (slides)
April 24, Khovanov homology is functorial in S^3, Braids In Banff. (slides)
February 21 and 22, Rasmussen's genus bounds from Khovanov homology, UC Berkeley and UC Davis topology seminars.

2006

December 4, A topological categorification of su_3, Colloquium, University of Oregon, Eugene. (slides)
December 1, Khovanov Homology, Basic Notions Seminar, University of Oregon, Eugene. (slides)
November 17, Disoriented and confused: fixing the functoriality of Khovanov homology. Columbia, Gauge Theory Seminar. (slides, handout.)
September 7, Disoriented and confused: fixing the functoriality of Khovanov homology. Uppsala, Categorification in Algebra and Topology. (slides, handout.) My slides got a mention at the n-Category Cafe.
July 3, Disoriented and confused: fixing the functoriality of Khovanov homology. PCMI. (slides)
June 16, A cobordism theory for sl_3 knot homology, Subfactor seminar.
May 6, Disoriented and confused: fixing the functoriality of Khovanov homology. Knots in Washington. (slides, handout, actual handout)
April 28, Disoriented and confused: fixing the functoriality of Khovanov homology. (announcement, slides)
April 11, A cobordism theory for sl_3 knot homology (slides, handout), UCSB topology seminar.
Khovanov Homology, Taipa conference Jan 2006. (slides)

2005

omath, UC Berkeley Subfactor seminar.
The Knot Atlas, at Knots in Washington.
Bar-Natan's local Khovanov homology, UC Berkeley Subfactor seminar.
- Abstracts June 30, July 7 September 9
- You can download my slides from the first two talks and my slides from the 'delooping' talk.
Generators and relations for $Rep(U_q(\mathfrak{sl}_n))$ as a pivotal category.
- Abstracts February 11, February 18.
- I gave a short 25 minute version of this talk at the Quantum Topology AMS meeting at Snowbird, in June. You can download overheads as an ~8mb pdf.

2004

Pivotal categories, planar algebras and subfactors
- Abstracts October 15, October 29, November 5.

2002

October 25, The Temperley-Lieb algebra and $\mathfrak{sl}_2$ representation theory, UC Berkeley subfactor seminar. (abstract)
October 11, Jones-Wenzl projections in the Temperley-Lieb algebra, UC Berkeley subfactor seminar. (abstract)
May 8, Juggling for Geeks, Many Cheerful Facts graduate seminar. (abstract)

2001

June 8, Classifying Spinor Structures, honours talk at the University of New South Wales. (abstract)

Talks

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2001