For a compact Lie group action on a smooth manifold, we will introduce a complex of basic relative forms on the inertia space, which was originally constructed by Brylinski. We will explain how basic relative forms can be used to study the Hochschild homology of the convolution algebra. This is work in progress with Markus Pflaum and Hessel Posthuma.
Hochschild Homology of Proper Lie Groupoids Sponsored by the Meyer Fund
Apr. 26, 2018 2pm (MATH 350)
Topology
Jordan Watts
X
Fix an irrational number A, and consider the action of the group of pairs of integers on the real line defined as follows: the pair (m,n) sends a point x to x + m + nA. The orbits of this action are dense, and so the quotient topology on the orbit space is trivial. Any reasonable notion of smooth function on the orbit space is constant. However, the orbit space is a group: the orbits of the action are cosets of a normal subgroup. Can we give the space any type of useful "smooth" group structure?
The answer is "yes": its natural diffeological group structure. It turns out this is not just some pathological example. Known in the literature as the irrational torus, as well as the infra-circle, this diffeological group is diffeomorphic to the quotient of the torus by the irrational Kronecker flow, it has a Lie algebra equal to the real line, and given two irrational numbers A and B, the resulting irrational tori are diffeomorphic if and only if there is a fractional linear transformation with integer coefficients relating A and B, and so it is of interest in many fields of mathematics. Moreover, it shows up in geometric quantisation and the integration of certain Lie algebroids as the structure group of certain principal bundles, the main topic of this talk.
We will perform Milnor's construction in the realm of diffeology to obtain a diffeological classifying space for a diffeological group G, such as the irrational torus. After mentioning a few hoped-for properties, we then construct a connection 1-form on the G-bundle EG --> BG, which will naturally pull back to a connection 1-form on sufficiently nice principal G-bundles. We then look at what this can tell us about irrational torus bundles.
Classifying Spaces of Diffeological Groups Sponsored by the Meyer Fund
Apr. 26, 2018 3pm (MATH 350)
Probability
Leonard Huang (CU Boulder)
X
This talk is inspired by the following problem, which has appeared in real-analysis exams at several universities around the world:
Let be a sequence in . If for some , then prove that also. One may now ask: Is this result still true if we replace by some other topological vector space? In this talk, we will prove that the result is true for a wide class of topological vector spaces that includes all locally-convex topological vector spaces, as well as some non-locally-convex ones, such as the -spaces for . We will also construct, using probability theory, an example of a topological vector space for which the result is false.
Generalization of a Real-Analysis Exam Problem to a Class of Topological Vector Spaces
Hochschild Homology of Proper Lie Groupoids