Lecture contents anticipated and then after delivery

1. September 9

   Space of continuous functions, dual space, positivity.

2. September 14

   Outer measures and measures.

3. September 16

   Caratheodory's theorem.

4. September 21

   Measurable functions and the integral - including Lebesgue's theorem of dominated convergence.

5. September 23

   Riesz representation, $L^p$ spaces and completeness, $L^2(X,\mu)$ and Hilbert space.

6. September 28

   Riesz representation for Hilbert space. Differentiability and Schwartz space of test functions.

7. September 30

   Properties of $\mathcal{S}(\mathbb{R}^n).$ Tempered distributions. Differentiation and differential operators. Fourier transform.

8. October 5

   Bump functions. Characterization of $\delta.$ Fourier inversion. Plancherel formula.

9. October 7

   Convolution and density. Fourier transform on $L^2(\mathbb{R}^n).$

10. October 12

    Sobolev spaces and Sobolev embedding. Duality between Sobolev spaces.

11. October 14

    Schwartz representation theorem. Fundamental solution of $\partial_x+i\partial_y.$ Support of a distribution; distributions of compact support (start).

12. October 19

    Compact supports. Convolution of distributions

       supp$$(u*v)\subset$$supp$$(u)+$$supp$$(v)$$
    if one, at least, has compact support. Fundamental solutions.

13. October 21

    Singular support, hypoellipticity, ellipticity - parametrices for elliptic operators.

14. October 26

    Fundamental solution of the heat operator, hypoellipticity, initial value problem. Homogeneity. The distributions $x_{\pm}^z,$ $z\in\mathbb{C}\setminus(-\mathbb{N}).$

15. October 28

    Distributions supported at$0.$ Homogeneous distributions of order $z\notin-\mathbb{N}$ on the line. Hadamard regularization. Cone supports.

16. November 2

    Singular support and products. Conic support and convolution.

17. November 4

    Wavefront set refines singular support. Scattering wavefront set. Product and wavefront set. The wave equation.

18. November 9

    Fundamental solution of the wave equation. Solution to the Cauchy problem.

19. November 16

    Operators and Schwartz' kernel theorem.

20. November 18

21. November 23

22. November 30

23. December 2

24. December 7

    Lidskii's theorem. $\varDelta$ on the torus. Self-adjointness of $\varDelta+V.$ Spectral decomposition. Wave equation on the torus. Wave equation on torus with potential. $\varDelta+V$ on $\mathbb{R}^n$ with $V\in\mathcal{C}_{\text{c}}(\mathbb{R}^n).$

25. December 9

    Questions:- Trace as integral of the kernel over the diagonal. Microlocal analysis.