In this talk I will discuss some investigations of mine within the general area of comparison geometry. The structure will be as follows:
(i) Basic definitions from differential geometry
(ii) Quick survey of the core concepts within Morse theory
(iii) Some emphasis on the connection between Morse theory and comparison geometry
(iv) Mention of the extension of Morse theory to stratified spaces
(v) Brief mention/motivation of the first research project "discrete comparison geometry"
I will conclude in the last 15-20 minutes with a relatively condensed summary of my proof of an interesting new structure result for the cut locus of a Riemannian manifold, which states that any sufficiently "nice" spine of a manifold (that is not a 2-sphere) can be deformed into a cut locus for the same space. This will be motivated by a quick synopsis of a paper due to Alan Weinstein which was the basis of his PhD thesis.
March 9:
Peter Tingley (Melbourne), Quantum groups, braidings and crystals
I will introduce quantized universal enveloping algebras and their representations. I will then discuss two beautiful facts about this theory. The first is that the category of representations is braided, which is a key ingredient in the celebrated quantum group knot invariants. The second is the existence of crystal bases for the representations. These are extremely nice bases which, among other things, describe much of the structure of the representations in a purely combinatorial way. I will then discuss how these things are related. In particular, there is a structure on the combinatorial category of crystals which is analogous to the braiding, except that the braid group is replaced by the so called cactus group. Like the braid group, the cactus group is the fundamental group of a nice space, so I will end with some topology. This talk is largely expository, and will be presented via examples.
March 16:
Mutsuo Oka (Tokyo University of Science), On mixed singularities and their Milnor fibrations
Mixed functions are analytic functions in variables
$z_1,\dots, z_n$ and their conjugates $\bar z_1,\dots, \bar z_n$.
We introduce the notion of Newton non-degeneracy for mixed functions
and develop a basic tool for the study of mixed hypersurface singularities. We show the
existence of a canonical resolution of the singularity,
and the existence of the Milnor fibration under the strong
non-degeneracy condition.
March 23:
Tarje Bargheer Arklint (Copenhagen), Higher dimensional String topology via operads
In the end of the last century Chas and Sullivan discovered some structure on the homology of the free loop space of a smooth manifold M. This structure is now known as the Chas-Sullivan loop product. In the years following the introduction of the Chas-Sullivan loop-product, collaborative work by a good deal of mathematicians have given meaning to this structure via an action of the so-called cacti-operad. There has been interest in finding an analogue of this operadic action, in the case when the free loop space is replaced by Map(N,M), for N more general than S^1. The talk will introduce the concepts above, and hopefully give a glimpse of my -- work in progress -- ideas for giving generalizations to a -- coloured -- operadic action, where the case of N=S^n shall be given special attention.
March 30:
Jeremie Guilhot (Sydney), Semicontinuity properties of Kazhdan-Lusztig cells in affine Weyl groups of rank 2
In this talk I will introduce the semicontinuity conjecture of Cedric Bonnafe concerning the behaviour of the Kazhdan-Lusztig cells when the parameters are varying. I will then describe the partition into cells of the affine Weyl groups of rank 2 and show that the semicontinuity conjecture hold in these cases.
April 6:
Daniel Chan (UNSW), Non-commutative projective geometry
In 1990, Artin-Tate-Van den bergh studied non-commutative algebras called 3-dimensional Sklyanin algebras using many interesting ideas from algebraic geometry. The body of techniques developed now form a school of non-commutative projective geometry. This talk will be primarily be an introduction to some aspects of non-commutative projective geometry. I will show how various geometric concepts in projective geometry such as dimension and intersection theory, can be defined in the non-commutative setting and indicate how they might be useful in studying non-commutative algebra.
April 20:
Peter Trapa (Utah), Functors for representations of GL(n,F)
Let F denote a local field of characteristic zero (like the real numbers or the p-adic numbers), and let GL(n,F) denote the set of n-by-n invertible matrices over F. As the field F varies, the representation theoretic part of the local Langlands correspondence predicts relationships between the Grothendieck groups of representations of the various groups GL(n,F). Most of the talk will be an introduction to these ideas. If time permits, I will introduce some functors (defined in joint work with Dan Ciubotaru for more general groups) which implement the relationships of Grothendieck groups categorically.
April 27:
Steve Rosenberg (Boston University), Chern-Simons classes on loop spaces and diffeomorphism groups
The loop space LM of a Riemannian manifold M is itself an interesting infinite dimensional manifold. LM has a family of Riemannian metrics indexed by a Sobolev parameter. We can construct characteristic classes for LM by using the Wodzicki residue instead of the usual matrix trace. The Pontrjagin classes of LM vanish, but the secondary or Chern-Simons classes may be nonzero. A similar approach applies to diffeomorphism groups of manifolds.
May 4:
Emily Peters (UC Berkeley), Constructing the extended Haagerup planar algebra
The extended Haagerup subfactor is the last unknown item on Haagerup's 1993 list of possible small-index subfactors. In recent work with Stephen Bigelow, Scott Morrison and Noah Snyder we construct this subfactor by constructing its associated planar algebra. This finishes the classification of subfactors with index up to $3+\sqrt{2}$.
I will start this talk by introducing planar algebras, an extremely useful invariant of subfactors, and then discuss the classification of small-index subfactors/planar algebras. This classification points to the existence of `exotic' planar algebras, which are planar algebras that aren't part of known algebraic families (such as those coming from groups or quantum groups), and I will discuss techniques for constructing these objects.
May 11:
Makoto Ozawa (Komazawa University),
Essential surfaces derived from knot and link diagrams
Many knot classes of alternating, positive, Montesinos, adequate, homogeneous, algebraic, etc. are defined from knot and link diagrams, and essential surfaces in their complements are studied. In this talk, we define $\sigma$-adequate and $\sigma$-homogeneous knots and algebraically alternating knots which are extensions of the above knot classes, and describe essential surfaces in their complements.
May 18:
Suhyoung Choi (Kaist), Real projective structures on 3-orbifolds and projective invariants
Real projective structures are given as projectively flat
structures on manifolds or orbifolds.
Hyperbolic structures form good examples. Deforming a hyperbolic
structure into a family of real projective structures
might be interesting from some perspectives. We will try to find
complete projective invariants to deform
projective 3-orbifolds with triangulations and obtain some
deformations of reflection groups based on
tetrahedra, pyramids, octahedra, and so on. (We will give some
introduction to this area of research in the talk.)
May 25:
Hyam Rubinstein (melbourne), k-width of knots and links.
This is joint work with Joel Hass and Abby Thompson. Classically, knots and links are studied using diagrams, which can be projections onto a plane, or slices of knots and links by a family of parallel planes. In both cases, a single projection or normal direction of the family of planes is chosen. We propose instead to study larger families of planes, choosing all normal directions lying in a fixed plane, or all possible normal directions or finally all planes and round spheres to slice a given knot or link. These family of planes have k parameters where k=1,2,3,4 and give rise to natural invariants. Some connections to the curvature of knots and links will be outlined, following ideas of Milnor.