Iwaniec-Kowalski, Davenport, Bump (Automorphic Forms and Representations), Einsiedler-Ward (Homogeneous Dynamics)
WEEK
LECTURE 1 VIDEO
LECTURE 2 VIDEO
COMMENTS/TOPICS
Week 1 01/18, 01/21
Lecture 0: Overview Lecture 1
Leibniz/Huygens sum of reciprocals of triangular numbers, Euler evaluation of zeta(2), Euler product formula, Divergence of sum of reciprocals of primes, Euler's (!!!) Functional Equation for zeta, trivial zeros, Poisson summation as a Trace Formula
Read: Euler's 1749 paper Gauss's hand computations
Lecture 3
Gauss's Conjecture, Eratosthenes, Mellin Transform of Theta Function, Gamma Function, Analytic Continuation and Functional Equation of Zeta, Zeros and Polar Behavior, von Mangoldt and Chebyshev functions, Riemann's Explicit Formula as Poisson Summation in Primes
Week 3 02/01, 02/04
Lecture 4
Prime Number Theorem, Hadamard, de la Vallee Poussin, Hoffstein-Lockhart, Goldfeld-Hoffstein-Lieman, zero-free region, Mellin convolution, smoothing and unsmoothing, Prime Number Theorem Riemann's 1859 Memoir Riemann's handwritten memoir (And you thought my handwriting was bad!)
Lecture 5
Primes in Arithmetic Progressions, Via Euclid/Fermat, Dirichlet characters, Dirichlet's Theorem, Reduced to Nonvanishing of Real L-function at s=1
Dirichlet's 1837 Paper
Lecture 7
Explicit computation of L(1,chi), Binary Quadratic Forms up to (Proper) Equivalence, Fundamental Domain for SL(2,Z), Class Group, Finiteness of Class Number
Week 5 02/15, 02/18
Lecture 9a technical issues
Lecture 8
Group Actions, Iwasawa Decomposition, Siegel Domain, GL(n,Z), Finiteness of Class Number Lean Formalization
Lecture 9
Manifolds, Riemannian geometry/structure, Geodesics, Tangent Space,
Unit Tangent Bundle,
Hyperbolic Structure on H,
Isometric Action of PSL(2,R), Geodesic Flow as Right-Regular Diagonal Flow,
Hyperbolic Distance Function
Week 7 03/01, 03/04
Lecture 10
Laplacian, Lie algebra, Action by Differential Operators, Universal Enveloping Algebra, Casimir Operator, Laplacian commutes with (left) group action, nonholomorphic Eisenstein series, Dirichlet convolution, Mobius inversion, Epstein zeta function
Lecture 11
Eisenstein series, Class Group, Class Number, Representation by Quadratic Form, Dedekind Zeta Function, Ring of Integers, (Prime) Ideals, Split/Inert Primes, Fourier Expansion of Eisenstein Series, K-Bessel Function
Week 8 03/8, 03/11
Lecture 12
Eisenstein series, Fourier Expansion, Meromorphic Continuation, Polar Behavior, Finish Proof of Dirichlet's Class Number Formula, and Theorem on Primes in Progressions. Mellin Convolutions, K-Bessel Function, Asymptotics, Randomness of Modular Inversion, Discrepancy, Kloosterman Sums
Lecture 13
Kloosterman's Bound for Kloosterman Sums Modular Inversion is Pseudorandom Modular Multiplication is Pseudorandom Continued Fractions
Week 9 03/15, 03/18
SPRING BREAK
Week 10 03/22, 03/25
Lecture 14
Modular Multiplication is Random, Continued Fractions, Uniformly Badly Approximable Numbers, Cantor Sets, Hausdorff Dimension, Zaremba's Conjecture, Hensley's Conjecture, Density One Theorem
A to Z survey
Lecture 15
Zaremba's Conjecture, Hensley's Conjecture, Bourgain-Kontorovich Theorem, Uniformly Badly Approximable Semigroup, Thin Sets/Groups/Semigroups, Zariski Closure, Local Obstructions, Admissible Numbers, Strong Approximation, Failure of Strong Approximation for Reductive Groups, Modular Semigroups are Groups, Alphabets having Local Obstructions
Week 11 03/29, 04/01
Lecture 16
Circle Method, Dirichlet Approximation Theorem (Pigeonhole), Major and Minor Arcs, L2 Cancellation To Density 1, Dyadic Decomposition of the Circle
Lecture 17
Minor Arcs Analysis Bilinear Forms Cancellation in Exponential Sums Kloosterman Refinement
Week 12 04/05, 04/8
Lecture 18
1st and 2nd Kloosterman Refinement
Techniques for Cancellation of Exponential Sums Multilinear Analysis Minor Arcs
Lecture 19
Finishing Minor Arcs Analysis for Zaremba Conjecture: Cauchy-Schwarz, Poisson Summation, Distance to Nearest Integer, LCM, GCD, Exponential Sums
Week 13 04/12, 04/15
Lecture 20
McMullen's Arithmetic Chaos Conjecture, Real Quadratic Fields, Fundamental Units, Local-Global Conjecture for Traces, Pell equation
Lecture 21
Geodesic Flow on Unit Tangent Bundle of Modular Surface, Right Diagonal Action, Billiards vs Geodesics, Classification of Hyperbolic/Parabolic/Elliptic Motions, (Primitive) Closed Geodesics, (Primitive) (Conjugacy Classes of) Hyperbolic Elements, Cutting Sequence of the Geodesic Flow, Farey Tessellation, Continued Fractions
More discussion on these topics
Week 14 04/19, 04/22
Lecture 22
Correspondence between Closed Geodesics, Hyperbolic Conjugacy Classes, and Classes of Indefinite Binary Quadratic Forms; Finiteness of the Class Number in Indefinite Case, Deuring-Heilbronn/Landau-Siegel/Goldfeld+Gross-Zagier, Is Class Number One infinitely often, Equivalently, Can we make Large Pell fundamental solutions, Dirichlet Class Number Formula, Duke's Equidistribution Theorem
Lecture 23
Duke's Theorem, Einsiedler-Lindenstrauss-Michel-Venkatesh problem, Low-Lying Fundamental Closed Geodesics, Sieve Black Box, Expansion (Expander Graphs + Thermodynamic Formalism + Ruelle transfer operators), Square-free Sieve, Beyond Expansion Program, Replace Anabelian Harmonic Analysis with Abelian
Paper with Bourgain
Week 15 04/26, 04/29
Lecture 24
Twin Primes, Brun's Theorem, Eratosthenes, Legendre, Selberg Upper Bound Sieve, Diagonalizing Quadratic Form, Minimizing Form with Linear Constraint, Mobius Inversion Selberg Sieve Notes
Lecture 25
Dirichlet's theorem in Real Quadratic Fields, Integrating Eisenstein series along closed geodesics corresponding to inequivalent forms...
Week 16 (Makeups) 05/03, 05/06 in 425 Hill
Lecture 26
Modular Forms, Fourier Expansions, Holomorphic at Infinity, Holomorphic Eisenstein series, Ramanujan discriminant, Ramanujan tau function, Hecke L-function, Analytic Continuation, Functional Equation, Trivial bound on Fourier coefficients, Hecke operators, commute and are self-adjoint with respect to Petersson inner product, Normalized Hecke forms, Eigenvalues *are* Coefficients!, Euler product of degree 2, Deligne Riemann hypothesis for varieties, Ramanujan conjecture
Lecture 27
Venkatesh's Subconvex Bound for Fourier Coefficients of Modular Forms, Effective Equidistribution of Low-Lying Closed Horocycles, Effective Mixing of Horocycle Flow, Amplification, van der Corput trick, Cusp Forms are Orthogonal to Eisenstein Series, Furstenberg x2x3 philosophy
