The GRASP seminar aims to be accessible to graduate students, and focuses on geometry and representation theory and their connections to mathematical physics. The organizers are Theo Johnson-Freyd and Harold Williams. The seminar meets Fridays, 11:10-12:00, in 939 Evans. We stole the name and clever acronym from the eponymous seminar at UT Austin.
If you would like to give a talk or to join the mailing list, send an e-mail to either Theo or Harold.
December 3. Last day of classes. Dave Penneys: The representation theory of a subfactor. (abstract)
Abstract:
I will describe the representation theory of a finite index subfactor, which is a certain unitary 2-category with nice duals. Our 2-category is so nice that we may write planar diagrams to represent 2-morphisms; hence it has the structure of a planar algebra. These 2-categories give a categorified Morita equivalence of fusion categories (in the event that the subfactor is finite depth).
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November 26. No Seminar (Thanksgiving).
November 19. Morgan Brown: Generalizations of Ample Line Bundles. (abstract)
Abstract:
The notion of an ample line bundle is central to algebraic geometry, as it unifies cohomological, geometric, and numerical properties of a variety. I will present some results of Totaro and others which attempt to generalize the notion of amplitude as well as some open problems.
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November 12. Harold Williams: Hochschild Cohomology, Drinfeld Centers, and the Little 2-Disks Operad. (abstract)
Abstract:
A classical fact from topology is that for any space $X$, the set of homotopy classes of maps $S^2 \to X$ naturally forms an abelian group. If instead of modding out by homotopy we consider the space of such maps, we encounter a more subtle algebraic structure, which is described by the little 2-disks operad. In this talk we'll discuss this structure and in particular how it appears implicitly in two apparently untopological constructions, the Hochschild cohomology of an associative algebra and the Drinfeld center of a tensor category.
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November 5. Jakob Blaavand (Institut for Matematiske Fag, Aarhus Universitet): 3-manifold invariants derived from link invariants. (abstract)
Abstract:
In this talk I will give a method of constructing invariants of 3-manifolds from link invariants. The idea is to turn a very effective link invariant, such as the Jones polynomial, into a 3-manifold invariant. I will give an example of this construction using the Jones polynomial. These invariants are closely related to the quantum invariants derived from the BHVM skein theoretical TQFT. If time permits I will talk about how this construction generalizes the Jones polynomial to links in 3-manifolds.
The talk will be aimed at a broad audience, and requires no prior knowledge of knot theory.
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October 29. Søren Fuglede Jørgensen (Institut for Matematiske Fag, Aarhus Universitet): Quantum representations of mapping class groups and TQFTs for newcomers. (abstract)
Abstract:
Roughly, the mapping class group of a given surface can be seen as the symmetry group of the surface in question. In this talk, I will give several examples of mapping classes and see how so-called quantum representations of mapping class groups might enlighten us on the structure of the groups. Furthermore, I will discuss how they relate to and arise from topological quantum field theories. The talk will be expository and aimed at a wide audience — knowing what a surface is and having some clue about what homology is about is probably advantageous. Notes for the talk are available at http://home.imf.au.dk/pred/ (under research).
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October 22. Roland van der Veen (University of Amsterdam): Introduction to spin networks. (abstract)
Abstract:
Spin networks are labeled graphs originally used in classical angular momentum computations. I'll discuss how they have evolved to be a respectable part of representation theory, knot theory and three-dimensional quantum gravity. Along the way we'll see some of the key problems in the field and partial progress towards solving them.
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October 15. No seminar.
October 8. Anton Geraschenko: Toric Artin stacks. (abstract)
Abstract:
Toric varieties provide a great testing ground for ideas about varieties. By understanding how to relate the geometry of a toric variety to the combinatorics of its fan, you can often come up with many interesting examples of a phenomenon. The goal of this talk will be to introduce you to toric Artin stacks and use them to illustrate some of the behaviors of Artin stacks and good moduli spaces. Some experience with toric varieties a plus, but hopefully not absolutely necessary. You don't need to know what an Artin stack is.
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October 1. William Slofstra: Tutorial on Kac-Moody algebras and groups. (abstract)
Abstract:
A slow-paced introduction to loop groups, starting from the perspective of Kac-Moody algebras. This talk should cover some material I didn't have time to get to in my talk last term; in particular, how to work with loop groups using methods from algebraic geometry.
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Sept 24. Dan Berwick Evans: Supersymmetric Field Theories and Generalized Cohomology. (abstract)
Abstract:
The Stolz-Teichner program has illustrated and conjectured deep connections between supersymmetric field theories and generalized cohomology. Starting with a description of the big picture, I will go on to give a feeling for the definitions of these field theories and how they are related to algebraic topology. Time permitting, I will sketch some computations of invariants arising from supersymmetric sigma models with one and two supersymmetries, giving a glimpse at how classic theorems like Gauss-Bonnet can be obtained by field theoretic arguments. No prior knowledge of functorial field theories or super geometry will be assumed.
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Sept 17. Kevin Lin: Tutorial on Hochschild (co)homology. (abstract)
Abstract:
Basic definitions and properties. Applications, time permitting: Deformation theory, Hochschild-Kostant-Rosenberg theorem, topological field theory, Kontsevich formality ...
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Sept 10. Theo Johnson-Freyd: Formal calculus, with applications to quantum mechanics. (abstract)
Abstract:
"Formal" or "Feynman diagrammatic" calculus is nothing more nor less than the differential and integral calculus of formal power series. The latter name is because Feynman's diagrams provide a convenient notation for manipulating formal power series and for understanding their combinatorics. In this talk, I will outline the formal calculus, and then use it to write out the "path integral" description of the asymptotics of the time-evolution operator in quantum mechanics. The diagrammatics make it much easier to prove that the "path integral" is well-defined and satisfies the necessary requirements.
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Sept 3. Organizational Meeting.
April 30. George Melvin: An Overview of the Kazhdan-Lusztig Conjectures. (abstract).
Abstract:
The Kazhdan-Lusztig conjectures postulate an explicit description of the transition matrix between canonical bases of the Grothendieck group of L-modules in terms of certain polynomials, for a complex semisimple Lie algebra L. Amazingly, these polynomials can be defined in terms of intersection cohomology sheaves on the flag variety associated to L. Using Beilinson-Bernstein localisation and the Riemann-Hilbert correspondence we establish a connection between the world of Lie algebras and the world of perverse sheaves, thereby confirming the conjectures.
This will be an elementary hand-waving exercise (no prior knowledge of perverse sheaves required!) so should be accessible to anyone who appreciates the power of an 'equivalence of categories'.
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April 23. Lilit Martirosyan: Lie Superalgebras and Quivers. (abstract).
Abstract:
I will introduce some basic notions of superalgebras and explain the classification of indecomposable finite-dimentional representations of special Lie Superalgebras using quivers.
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April 16. Jon Aytac: What the bleep is a particle. (abstract).
Abstract:
Particles are rumored to have something to do with representation theory. We will try to clarify this connection.
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April 9. Harold Williams: Dynkin Diagrams and Representations of Quivers. (abstract).
Abstract:
This talk will be an introduction to the representation theory of quivers (a.k.a. finite oriented graphs). Specifically I will discuss the basic question of when the representation category of a quiver has only finitely many indecomposable objects. The remarkable answer, due to Gabriel, is that this happens precisely when the underlying unoriented graph is a Dynkin diagram. Time permitting, I will try to sketch how this foreshadows Lusztig's later work on canonical bases of quantum groups via quiver varieties.
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April 2. Theo Johnson-Freyd: Introduction to Vassiliev invariants. (abstract).
Abstract:
"Vassiliev" or "finite-type" knot invariants include (up to a change of coordinates) most of the popular knot invariants (HOMFLYPT, ...). But they are also closely related to Lie algebraic questions. I will give an introduction to this story.
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March 26. No seminar (Spring Break).
March 19. Anton Geraschenko: Representations of unipotent algebraic groups. (abstract).
Abstract:
Representations of reductive groups are discretely parameterized, but unipotent groups can have non-trivial families of representations, so it is reasonable to ask if there is some geometric space which parameterizes such representations. If you restrict to certain kinds of representations and/or relax your notion of what a ``geometric space'' is, the answer is yes. I'll summarize the few things I know about the general case and then give you a tour of some interesting features that appear in a few small examples.
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March 12. No seminar.
March 5. William Slofstra: How to work with loop groups. (abstract).
Abstract:
A (hopefully) slow-paced introduction to loop groups, their homogeneous spaces, and related concepts like Kac-Moody groups. The focus will be on the different flavours (smooth, analytic, polynomial, formal) and technical approaches (smooth manifolds vs. ideas from algebraic geometry) encountered in working with loop groups.
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Feb 26. Kevin Lin: Landau-Ginzburg B-models and mirror symmetry. (abstract).
Abstract:
Let $X$ be a (not necessarily compact) complex manifold or complex algebraic variety, and let $W : X \to \mathbb{C}$ be a holomorphic function. In the physics literature, the data $(X,W)$ has a fancy name: it is a \emph{Landau-Ginzburg model}. The function $W$ has a fancy name, too: it is called the \emph{superpotential}.
We will look at some invariants of $(X,W)$, in particular its category of matrix factorizations (a notion first introduced and studied by Eisenbud). I will then discuss the appearance of these invariants in string theory and mirror symmetry, which posits that these \emph{algebraic} invariants of $(X,W)$ should, amazingly, match up with certain \emph{symplectic} invariants of a "mirror" \emph{symplectic} manifold $Y$.
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Feb 19. Morgan Brown: Representations and Cox rings of Del Pezzo surfaces. (abstract).
Abstract:
I will give a short introduction to both Cox Rings of algebraic varieties and Del Pezzo Surfaces. The intersection lattice of a Del Pezzo surface contains a canonical root system, and is acted on by the corresponding Weyl group. Batyrev conjectured a geometric explanation for this connection between geometry and representation theory, which was later proved by Popov, Derenthal, and Serganova-Skorobogatov. I will also explain why every smooth cubic surface in $\mathbb{P}^3$ contains exactly 27 lines.
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Feb 12. Matt Tucker-Simmons: What is a spectral triple? (abstract).
Abstract:
Spectral triples are the basic objects in Alain Connes' version of noncommutative geometry. They are meant to be thought of as noncommutative versions of Riemannian manifolds. I will define spectral triples, give at least one example, and discuss how representation theory can be used to analyze spectral triples which arise as homogeneous spaces of (possibly quantum) groups.
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Feb 5. Dave Penneys: What is a subfactor, and why should I care? (abstract).
Abstract:
By deep theorems of Jones and Popa, (suitably nice) subfactors are equivalent to (suitably nice) planar algebras which are, in turn, equivalent to (suitably nice) unitary 2-categories with 2 objects (or "bi-oidal" categories). I will give a brief definition of each of the above, after which I will briefly try to persuade you that you should care.
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Jan 21. Organizational Meeting.