spring 2025

math 411 · topics in advanced analysis

Fourier series and Hilbert spaces, with side trips to Weyl equidistribution and values of the Riemann zeta function.

syllabus: Syllabus
meetings: MWF 9-9:50 in Lib 389
office hours: Tu 14-15, Th 11-12 in Lib 306
discussion: Zulip workspace
submissions: Gradescope

jump to this week ↓39 meetings

week 1

mon jan 27

Welcome & warmup
linksintake survey❦notes❦demo 1❦demo 2

wed jan 29review

Review of inner product spaces
linksnotes

fri jan 31review

Review of $L^{2}$ and Hilbert spaces
linksnotes

week 2

mon feb 3

linksFourier series notes❦demo 1❦demo 2

wed feb 5

Dirac kernels and the inversion theorem
linkshomework due❦notes❦demo

fri feb 7

Examples

week 3

mon feb 10

Weyl's equidistribution theorem
linksnotes❦demo 1❦demo 2

wed feb 12

Riemann zeta values
linkshomework due❦notes

fri feb 14

Snow day ❄️

week 4

mon feb 17

Operators on Hilbert spaces
linksnotes

wed feb 19

linksHeat equation homework due❦notes❦3b1b

fri feb 21

Solving the heat equation
linksnotes

week 5

mon feb 24

Eigenbasis method and the wave equation
linksnotes❦demo

wed feb 26

Fourier transform
linkshomework due❦notes

fri feb 28

Schwartz space
linksnotes

week 6

mon mar 3

Inversion theorem for Fourier transforms
linksnotes

wed mar 5

Plancherel's theorem and Poisson summation
linkshomework due❦notes

fri mar 7exam

Exam review

week 7

mon mar 10

Harmonic analysis preview
linksmidterm exam distributed❦notes

wed mar 12no class

no class, midterm exam due

fri mar 14

Discuss projects
linksproject assignment❦notes

week 8

mon mar 17

Locally compact Abelian groups
linksnotes

wed mar 19

Pontryagin duals of LCA groups
linkshomework due❦notes

fri mar 21

Pontryagin duality for LCA groups

week 9

mon mar 31

Haar integration
linksproject proposal due❦notes

wed apr 2

Fubini's theorem for Haar integrals
linkshomework due❦notes

fri apr 4

Guest lecture: Josh Zahl
linksnotes

week 10

mon apr 7

Plancherel's theorem for LCA groups
linksnotes

wed apr 9

Fourier analysis on Euclidean space
linksnotes

fri apr 11

Minkowski-Siegel's geometry of numbers via Fourier analysis I
linksnotes

week 11

mon apr 14

Minkowski-Siegel's geometry of numbers via Fourier analysis II
linksnotes

wed apr 16

Primes as sums of squares
linkshomework due❦notes❦extremal body demo❦ $p = a^{2} + b^{2}$ demo

fri apr 18

Poisson summation as a trace formula
linksnotes

week 12

mon apr 21

class optional (work time on drafts), paper draft due

wed apr 23no class

no class (practice presentations)

fri apr 25

student presentations: DD (distributions), SG (Heisenberg uncertainty)

week 13

mon apr 28

student presentations: EW-S (discrete Fourier transform), NS (spectral bias and machine learning)

wed apr 30

student presentations: GW (central limit theorem), NE (Levy's continuity theorem)

fri may 2

student presentations: AK (Fourier transforms for non-Abelian groups)