Past

UCLA Math 32B: Calculus of Several Variables - Winter 2021

The course 32B treats topics related to integration in several variables, culminating in the theorems of Green, Gauss and Stokes. Each of these theorems asserts that an integral over some domain is equal to an integral over the boundary of the domain. In the case of Green's theorem the domain is an area in the plane, in the case of Gauss's theorem the domain is a volume in three-dimensional space, and in the case of Stokes' theorem the domain is a surface in three-dimensional space. These theorems are generalizations of the fundamental theorem of calculus, which corresponds to the case where the domain is an interval on the real line. The theorems play an important role in electrostatics, fluid mechanics, and other areas in engineering and physics where conservative vector fields play a role. TA for Sections 1A, 1B. (Taught by [March Boedihardjo](https://www.math.ucla.edu/~march/)). Link to the [CCLE site](https://ccle.ucla.edu/course/view/21W-MATH32B-1).

UCLA Math 33B: Differential equations

First-order, linear differential equations; second-order, linear differential equations with constant coefficients; power series solutions; linear systems. TA for Sections 1C, 1D. (Taught by [Chengxi Wang](http://www.math.ucla.edu/~chwang)). Link to the [CCLE site](https://ccle.ucla.edu/course/view/21S-MATH33B-1?section=15).

Uniform boundedness in operators parametrized by polynomial curves

Multiple results in harmonic analysis involving integrals of functions over curves (such as restriction theorems, convolution estimates, maximal function estimates or decoupling estimates) depend strongly on the non-vanishing of the torsion of the associated curve.

Uniform boundedness in operators parametrized by polynomial curves

Multiple results in harmonic analysis involving integrals of functions over curves (such as restriction theorems, convolution estimates, maximal function estimates or decoupling estimates) depend strongly on the non-vanishing of the torsion of the associated curve.

Uniform boundedness in operators parametrized by polynomial curves

Uniform boundedness in operators parametrized by polynomial curves

Uniform boundedness in operators parametrized by polynomial curves

Uniform boundedness in operators parametrized by polynomial curves

Decoupling and applications: from PDEs to Number Theory.

Decoupling estimates were introduced by Wolff [1] in order to improve local smoothing estimates for the wave equation. Since then, they have found multiple applications in analysis: from PDEs and restriction theory, to additive number theory, where Bourgain, Demeter and Guth[2] used decoupling-type estimates to prove the main conjecture of the Vinogradov mean value theorem for d>3.

Decoupling and applications: from PDEs to Number Theory.

Decoupling estimates were introduced by Wolff [1] in order to improve local smoothing estimates for the wave equation. Since then, they have found multiple applications in analysis: from PDEs and restriction theory, to additive number theory, where Bourgain, Demeter and Guth[2] used decoupling-type estimates to prove the main conjecture of the Vinogradov mean value theorem for d>3.