Course Identification

I expect the students to be familiar on a basic level with at least 80% of the following notions:
- Linear algebra.
- Vector space, linear map, subspace, quotient space, dual space, Tensor product.
- Topology.
- Topological space, Locally compact space, metric space, Complete metric space, completion of a metric space.
- Geometry.
- Differentiable manifold, tangent space, tangent bundle.
- Group theory.
- Group, group action, abelian group,
- Functional analysis.
- Hilbert space, Fourier series, measure, Fourier transform

We will study the theory of generalized functions and distributions (which are almost the same thing) on various geometric objects, operations with distributions (like pushforward, pullback and Fourier transform), and invariants of distributions (like the support and the wave front set).

The topic by its nature is analytic, but my point of view on this topic is oriented towards representation theory and algebraic geometry, so the course will have some algebraic and geometric flavor. We will discuss both the Archimedean case (i.e. distributions on real geometric objects) and the non-Archimedean case (i.e. distributions on p-adic geometric objects). We will discuss the similarity and difference of both cases.

During the later stages of the course, we will discuss distributions in the presence of a group action, the notion of an invariant distribution, and different methods to prove vanishing of invariant distributions. Those topics are closely related to representation theory.

In addition to the main topic of the course, we will have "digressions" (i.e. some lectures that are related to the main topic but not part of it) on: Functional analysis, p-adic numbers, Harmonic analysis on locally compact abelian groups, Differentiable manifolds, Nuclear spaces, algebraic and semi-algebraicgeometry, D-modules, the Weil representation and geometric invariant theory. Those digressions will be done on a very basic level, with the aim of making the students familiar with the basic notions in this topics. In case some of these topics will turn out to be too complicated, we will exclude them together with the related parts of the main topic.

We'll try to include in the course discussion some open (or semi-open) questions, which might interest
M.Sc. or Ph.D. students.

Upon successful completion of this course students should be able to:

[1] Demonstrate knowledge of generalized functions and distributions.

[2] Discuss the relation between generalized functions and analysis on manifold, algebraic geometry and (hopefully) representation theory

[GS], Volume 1, Chapters 1-3.1, and Volume 2, Chapters 1-3]
[Hor, chapters 1-8]
[BZ76, Chapter 1, section 1].
[BZ76] J. Bernstein, A.V. Zelevinsky, Representations of the group GL(n,F), where F is a local non-Archimedean field, Uspekhi Mat. Nauk.
10/3, (1976).
[GS] I.M. Gelfand, G. Shilov Generalized functions volumes I,II.
[H?r] L. Hormander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Grundlehren der Mathematischen Wissenschaften 256. Springer-Verlag, Berlin, 1990.