Date  Topics 
M 1/6  The Euclidean heat kernel. Parabolic maximum principles. Tychonov's uniqueness theorem 
W 1/8  Heat equations with variable coefficients. Heat kernels. The LaplaceBeltrami operator 
M 1/13  The MinakshisundaramPleijel recursion formula and heat parametrices 
W 1/15  Duhamel's formula. Construction of the heat kernel 
M 1/20  Heat trace asymptotics. Weyl's law. The zeta function and the determinant of the Laplacian 
W 1/22  The Hodge theorem 
M 1/27  The CauchyKovalevskaya theorem. The JanetCartan isometric embedding theorem 
W 1/29  Proof of the CauchyKovalevskaya theorem 
M 2/3  Characteristic surfaces 
W 2/5  Holmgren's theorem 
M 2/10  Locally convex spaces 
W 2/12  Fréchet spaces. Embeddings 
M 2/17  Test functions and distributions 
W 2/19  Subspaces of distributions. Basic operations 
M 2/24  Compactly supported distributions. Local structure 
W 2/26  More on compactly supported distributions 
M 3/10  Complex valued distributions. Fundamental solutions 
W 3/12  Convolutions 
M 3/17  Schwartz's theorem on hypoellipticity 
W 3/19  The Fourier transform. PaleyWiener theorem 
M 3/24  Tempered distributions. Liouville's theorem 
W 3/26  Laurent expansion. MalgrangeEhrenpreis theorem 
M 3/31  Hypoelliptic polynomials. The SeidenbergTarski theorem 
W 4/2  Hörmander's theorem 
M 4/7  No class 
W 4/9  Petrowsky's theorem. Fundamental solution of the Cauchy problem 
Date  Topics  Speaker 
Th 1/23  HahnBanach theorem 

Th 1/30  Basic Riemannian geometry 

Th 2/6  Baire's theorem and its consequences 

Th 2/13  Fredholm operators 

Th 2/20  Weyl's law 

Th 2/27  Weyl's law 

Th 3/13  No seminar 

Th 3/20  Fredholm property of elliptic operators 

Th 3/27  Optimal transport problem 

Th 4/3  The heat kernel for forms 

Th 4/10  Spectral theorem 

Instructor: Dr. Gantumur Tsogtgerel
Prerequisite: MATH 580 (PDE1), MATH 355 (Honours Analysis 4) or equivalent
Note: If you plan to take this course without taking MATH 580, please consult with the instructor.
Topics:
After covering some topics related to the heat equation and the CauchyKovalevskaya theorem,
the main focus of the course is going to be on the theory of distributions, general constant coefficient operators, semilinear evolution equations, and strongly elliptic systems.
The approach will be from the viewpoint of a general theory, rather than as apparently ad hoc treatments of specific examples.
As such, distributions, the Fourier transform, and functional analytic methods will be heavily used.
More precisely, the planned topics are
Calendar description: Systems of conservation laws and Riemann invariants. CauchyKowalevskaya theorem, powers series solutions. Distributions and transforms. Weak solutions; introduction to Sobolev spaces with applications. Elliptic equations, Fredholm theory and spectra of elliptic operators. Second order parabolic and hyperbolic equations. Further advanced topics may be included.
Books:
There is no required textbook.
The followings are recommended.
Homework: Assigned and graded roughly every other week.
Weakly seminars: We will organize weekly seminars on standard results from analysis and geometry, and other stuff related to the course.
Course project: The course project consists of the student reading a paper or monograph on an advanced topic, typing up notes, and giving a lecture.
Grading: The final grade will be the weighted average of homework 20%, the takehome midterm exam 30%, and the course project 50%.