Spring 2025
Natasa Sesum
Subtitle:
Mean Curvature Flow
Course Description:
We will introduce the mean curvature flow, give characterization of the first singular time, and then focus on singularity analysis of the flow. We will prove the monotonicity formula of Huisken, which has been extremely helpful in understanding the singularities of the mean curvature flow.
We will also study ancient solutions to the MCF that appear as singularity models of the flow, and their classification in some cases. We will also discuss weak solutions to the MCF, such as the level set flow and the Brakke flow, in particular the compactness theorem for integral Brakke flows, and the regularity theorems.
Text:
Regularity theory for the MCF by Klaus Ecker and various research papers.
Prerequisites:
Basic knowledge of PDE
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Fall 2023
Guangbo Xu
Subtitle:
Atiyah-Singer Index Theorem
Course Description:
Atiyah-Singer index theorem is a landmark of mathematics of the 20th century. It is regarded as a bridge between two broad branches of mathematics: topology and analysis. It also summarizes several important classical theorems, such as the Riemann-Roch theorem in algebraic geometry and the Gauss-Bonnet-Chern theorem in differential geometry. The development of the index theory was also intertwined with the development of physics, especially quantum field theory. The Atiyah-
Singer index theorem is a manifestation of the unity of mathematics and the close companionship between math and physics.
This course will be a mixture of topology, geometry and analysis. I will start with basic functional analytic properties of Fredholm operators such as the homotopy invariance of the Fredholm index. Then I will move to the discussion of topological K-theory, including the Bott periodicity theorem and the definition of topological index of elliptic operators. Instead of the original K-theory proof of the index theorem by Atiyah and Singer, I will present the analytic proof using heat kernels for the most elegant case of Dirac operators.
Text:
The main references are 1. Booss-Bleecker, Topology and Analysis, 2. Berline-Getzler-Vergne, Heat kernels and Dirac operators
Prerequisites:
1. absolute necessary ones: basic differential geometry on smooth manifolds, basic algebraic topology (homoloy groups), basic functional analysis (Hilbert spaces) 2. preferrable ones: PDE (elliptic equation, Sobolev spaces), more advanced algebraic topology (homotopy groups, characteristic classes of vector bundles)
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Spring 2023
Natasa Sesum
Subtitle:
Mean curvature flow
Course Description:
We will introduce the mean curvature flow and some of its basic properties, such as characterization of the first singular time, Huisken's monotonicity formula, special solutions to the flow that only move by homotheties and are called the self-similar solutions, the notion of weak solutions to the flow. We will also discuss singularity analysis of the flow, give a possible description of singularities in some special situations, such as in the mean convex curvature setting. We will also talk about the importance of ancient solutions in singularity analysis of the flow and mention how the classification of those can help us prove some nice theorem in the field, such as the Mean convex neighborhood conjecture for neck singularities or uniqueness of weak solutions through spherical or neck singularities.
Text:
Regularity of mean curvature flow by Klaus Ecker
Prerequisites:
Basic PDE course
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Spring 2022
Mariusz Mirek
Subtitle:
Fourier methods in ergodic theory and combinatorics
Course Description:
We begin by illustrating a difference between norm and pointwise convergence by proving respectively von Neuman’s mean ergodic theorem and Birkhoff’s pointwise ergodic theorem. Our ultimate goal will be to build the necessary Fourier analytic and number theoretic tools needed to discuss Bourgain’s pointwise ergodic theorem along polynomials orbits. The second part of the course will cover some aspects of Gowers’ norms — the basics of the so-called higher order Fourier analysis — in the context of Gowers’ proof of Szemerédi’s theorem as well as Peluse and Prendiville theory.
Text:
No textbook
Prerequisites:
Measure Theory
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Fall 2020
Avy Soffer
Subtitle:
Functional Analysis and the Related Quantum Theory.
Course Description:
This is an introductory course to modern functional analysis and its use in the study of PDE of Mathematical-Physics. Beginning with a review of Hilbert spaces and linear operators, we then proceed to the basics of spectral theory of self adjoint operators in Hilbert space, and applications to Fundamental mathematical aspects of Quantum mechanics, and more general Dispersive wave equations.
The necessary Analysis will be developed in class.
This course is also available by arrangement.
Text:
Functional Analysis, by M. Reed and B. Simon
Recommended Text: Functional Analysis, by Yoshida, Kosaku
Operator Theory, part 4, by Barry Simon
Prerequisites:
Real Analysis and Linear Algebra are prerequisites.
Schedule of Sections:
16:640:510 Schedule of Classes