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).
In this talk we discuss sharp ℓ2L2n estimates for Cantor sets. These estimates are related to the work of Biggs bounding the number of solutions to a certain type of Diophantine equations for integers contained in Ellipsephic sets, sets of numbers missing certain digits in base p. We discuss the connection between both problems, and exploit it to find computational methods to find sharp decoupling estimates. Joint work with A. Chang, R. Greenfeld, A. Jamneshan, Z.K. Li and J. Madrid.
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. Over the past years there has been considerable interest in extending these results to a degenerate case where the torsion vanishes at a finite number of points by using the affine arc-length as an alternative integration measure. As a model case, multiple results have been proven in which the coordinate functions of the curve are polynomials. In this case one expects the bounds of the operators to depend only on the degree of the polynomial. In this talk I will introduce and motivate the concept of affine arclength measure, provide new decomposition theorems for polynomial curves over characteristic zero local fields, and provide some applications to uniformity results in harmonic analysis.
The course 32A treats topics related to differential calculus in several variables, including curves in the plane, curves and surfaces in space, various coordinate systems, partial differentiation, tangent planes to surfaces, and directional derivatives. The course culminates with the solution of optimization problems by the method of Lagrange multipliers.
TA for Sections 1A, 1B. (Taught by [Suzuki Fumiaki](https://www.math.ucla.edu/~suzuki/)). Link to the [CCLE site](https://ccle.ucla.edu/course/view/21W-MATH32A-1).
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).
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. Over the past years there has been considerable interest in extending these results to a degenerate case where the torsion vanishes at a finite number of points by using the affine arc-length as an alternative integration measure. As a model case, multiple results have been proven in which the coordinate functions of the curve are polynomials. In this case one expects the bounds of the operators to depend only on the degree of the polynomial. In this talk I will introduce and motivate the concept of affine arclength measure, provide new decomposition theorems for polynomial curves over characteristic zero local fields, and provide some applications to uniformity results in harmonic analysis.
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. In this talk I will explain what decoupling estimates are, I will talk about its applications to the Vinogradov Mean Value theorem and local smoothing, and I will explain the main ingredients that go into (most) decoupling proofs [1] Wolff, T. (2000). Local smoothing type estimates on Lp for large p. Geometric & Functional Analysis GAFA [2] Bourgain, J., Demeter, C., & Guth, L. (2016). Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three. Annals of Mathematics, 633-682.