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18.704

In 18.704, students take turns presenting the subject matter week by week. The Spring ’22 topic is harmonic analysis on finite groups and its applications. In the first half of the course, we introduce the discrete Fourier transform and apply it to graph theory, probability, coding theory, and physics. In the second half, we discuss its nonabelian generalization, focusing on matrix groups over finite fields.

pdf Syllabus

pdf Tentative schedule

Time & Place: MWF, 1–2 PM, Room 2-151

Textbook: Terras, Fourier Analysis on Finite Groups and Applications

Supplements:
  • Etingof et al., Introduction to Representation Theory
  • pdf Mackey, “Harmonic Analysis as the Exploitation of Symmetry”
  • Piatetski-Shapiro, Complex Representations of \(GL(2, K)\) for Finite Fields \(K\)
  • pdf Sloane, “An Introduction to Association Schemes and Coding Theory”

Notes

pdf My notes for the first week

pdf Susan Ruff’s slides about known vs. new information in scientific writing

pdf Xiangkai’s notes on Laplacians and their spectra

Final Papers

Krit Boonsiriseth pdf Radar Cross-Ambiguity and the Heisenberg Group
Merrick Cai pdf The Volume of \(\mathrm{SL}(2, \mathbb{Z})\backslash \mathrm{SL}(2, \mathbb{R})/\mathrm{SO}(2, \mathbb{R})\)
Chang-Han Chen pdf Conjugacy Classes of \(\mathrm{GL}(2, \mathbb{F}_q)\)
Matthew Cho pdf The Wavelet Transform
Matthew Cox pdf The Poincaré Upper Half-Plane
Rupert Li pdf Mackey–Wigner’s Little Group Method with an Application to \(\mathrm{Aff}(q)\)
Atharv Oak pdf \(k\)-Bessel Functions as Eigenfunctions of the Laplacian
Jeffery Opoku-Mensah pdf Constructing the Discrete Series Representation of \(\mathrm{GL}(2, \mathbb{F}_q)\)
Tristan Shin pdf Quadratic Reciprocity and Gauss Sums
Carlos Solano pdf Exceptional Isomorphisms of \(\mathrm{SL}\), \(\mathrm{PSL}\) in Rank \(2\)
Xiangkai Sun pdf Criteria for Finite Symmetric Spaces
Derrick Xiong pdf On the Finite Upper Half Plane
Jessica Yuan-Chen Yeh pdf Conjugacy Classes and Irreducible Representations of \(\mathrm{Heis}(q)\)