Topics in Geometry: curves, planes, forms, exceptions and their applications.
An idea of the course:
This course is at the cross-roads of algebra, analysis, arithmetic, geometry, topology and trigonometry. It is designed not as an introduction to any particular subject, but rather as a motivation for further studies. Most of the objects will be revisited or recycled, seen from different perspectives. No knowledge beyond the undergraduate math program is assumed, and I will be happy to review the necessary pre-requisites and to adjust the syllabus to the needs of the audience. Similarly to Conway's book [2] we aim to help the student establish a sensual contact with quadratic (and even cubic!) forms and geometric notions. Similarly to Clemens [1] we will try to keep every lecture coherent. Asterisk (*) means that the proofs are out of scope of this course.

List of lectures (f - only formulation):

  1. [3] (f) Quaternions, modular functions (theta, eta) and their applications in arithmetics.
  2. Euler's 4-square identity, Lagrange's 4-square theorem, Fermat 2-square theorem and Gauss integers. Valuations.
  3. Oriented area and determinant. Area of triangle in coordinates. Sperner lemma. Monsky theorem.
  4. Valuation ring, valuation ideal, units, fraction field. Valuation implies Metric implies Topology. Cauchy sequences and completion. p-adic numbers (construction and homeomorphism with a Cantor set). Teichmuller representatives. Frobenius map.
  5. [Proof of Ostrowki theorem.] Convergence of exponent and logarithm in (non-)-archimidean topologies and over p-adics. Laurent and Puiseux series.
  6. Functional and differential equations for Exp, Log. Convergence. De Moivre and Euler formulae, trigonomentric formulae, trigonometric polynomials of Tchebyshev and Korkin-Zolorarev.
  7. Newton polygon and its additivity. Tropicalization of a polynomial over non-archimedean field. Newton's method and Hensel lemma. Solving algebraic equations in Puiseux series.
  8. Coverings, liftings. Fiber products, pullbacks and fibers. Monodromy. Critical points and values. Belyi polynomials and plane trees.
  9. Riemann-Hurwitz formula. Mason-Stothers theorem. Belyi rational functions. Cyclic type of local monodromy in terms of multiplicities.
  10. Applications of Mason-Stothers. A proof of genus zero case of the "difficult part" of Belyi theorem. Monodromy of Belyi polynomial in terms of the tree.
  11. Belyi functions and 8-color stratifications of the sphere. Normalization. Dirichlet domains, Voronoi diagrams. Quotients and invariant functions.
  12. Orbits. Invariants. Quasi-Invariants and (non-)liftings.
  13. Group extensions and compositions of functions. Invariants of Kleinian groups of types An, Dn, E7. Dihedral and octahedral functions as compositions. Commutative square of cyclic, Joukovsky and Tchebyshev. Barycentric triangulations and flags. Big sums of inverses of integers.
  14. [Jensen formula, Mahler measure, Lehmer conjecture.] Gluings of polygons, Harer-Zagier formula (f). Equivalence between graphs embedded in oriented surfaces, ribbon graphs, abstract graphs equipped with cyclic orders of semi-edges in every vertex, triples of permutations, homomorphisms from a free group F2 to permutation group.
  15. Polar duality on a sphere and real projective plane. Concurence of altitudes in a spherical triangle via Jacobi identity. Area of a spherical triangle as angle defect. Regular polygons on sphere and hyperbolic plane. Geometrization of topological tesselations. Dual regular tesselation by geodesics. Normalization and a metric proof of existence of Belyi functions with given graph.
  16. [Balanced necklaces (f).] Ice melting. Pick's theorem from ice melting and sum of angles. Lattice polygons and reflexive broken lines, formulation of 12-theorem.
  17. Broken watches problem. Discussion of the homework: two solutions for balanced necklace problem, a quadrangle without equiareal triangulations.
  18. Arithmetic and topological ways to solve ad-bc=1. Intersection of curves on the torus. First homology of a map between tori as linearization and as the best linear approximation for universal lifting.
  19. Identification of Euclidean and Minkowski 3-space with the Lie algebra of motions, after some Lie generalities.
  20. Dirichlet (fundamental) domain for the modular group SL(2,Z).
  21. Area of the fundamental domain. Identification of the hyperbolic plane with a quadric in 2x2-matrices. Transition functions for locally trivial fibrations and vector bundles. Line bundles form a group with respect to tensor product. Definition of modular forms. Interpretation of some modular forms as sections of line bundles. Functions on the space of lattices and Eisenstein series.
  22. PSL(2,Z) is generated by two elements, of orders 2 and 3. Moreover, it is a free product (f), only mentioned ping-pong lemma. Binary quadratic forms with fixed negative discriminant: finiteness of isomorphism classes. Modular forms as functions on lattices homogeneous with respect to homotheties. Prove that Eisenstein series are modular and find their constant terms. Sum of valuations/ramification of a modular form is linear with respect to its weight. Formulate construction of Hilbert field for imaginary quadratic irrationality.
  23. Euler's [E41] - infinite products for (sin(s)-y), infinite sum for cotangent function, Leibniz series for pi/4, values of zeta(2), zeta(4), ... Fourier expansions of Eisenstein series. Dimension of spaces of modular forms. Ramanujan's Delta-function as parabolic form of weight 12, j-invariant. Ring of modular forms is freely generated by E4 and E6.
  24. Bernoulli numbers and first two coefficients of Eisenstein series. Transformation law for E2. Quasi-modular forms (f). Transformation law for Dedekind eta-function and Ramanujan's delta-function (via logarithmic derivative). Asymptotics of coefficients of (parabolic) modular forms.
  25. Weierstrass function - definition, Taylor series, uniformization of elliptic curve. Fourier transform of Gaussian. Poisson summation formula. Modularity of Jacobi theta-function. Numbers of representations of an integer as a sum of 2 or 4 squares. Modular forms of higher levels with characters.
  26. Classification of 2x2 matrices into hyperbolic, parabolic and elliptic types; relations between types of conjugacy classes of the group with geometry of the quotient -- elliptic elements and ramification in finite part, parabolic elements and cusps. Reminders: definition of modular forms of higher levels. Gamma(2) is a free group and H/Gamma(2) is a Riemann sphere with 3 cusps and no elliptic points. Modular interpretations of Belyi functions. Platonic solids as Belyi functions on modular curves H/Gamma(N) for N=2,3,4,5. Klein quartic, H/Gamma(7) and PSL(2,7). Eta-function as generalized theta-function; Serre-Stark theorem (f). Theta-functions of lattices. Applications: signature 8 theorem and asymptotics for number of vectors of given length. Theta-functions of Korkin-Zolotarev and Leech lattices. Milnor's pair of 16-dimensionals drums that sound the same, but differ in shape. Venkov's proof of Kneser's theorem on Niemeier lattices via modular forms (f).
  27. the proof of 12-theorem via modular forms (after Bjorn Poonen and Fernando Rodriguez-Villegas).
  28. Weight 2 cusp-forms are holomorphic differentials
  29. Metrics of constant curvature with conical singularities. Consequences of Riemann's uniformization theorem. Moduli spaces of lattices and moduli space of elliptic curves.



Basic bibliography:
  1. C. Herbert Clemens: A Scrapbook of Complex Curve Theory.
  2. John (Horton) Conway. The Sensual (Quadratic) Form.
  3. John Conway, Derek Smith. On Quaternions and Octonions.
  4. Don Zagier. Elliptic Modular Forms and Their Applications.
  5. Sergei Lando, Alexander Zvonkin. Graphs on Surfaces and Their Applications. [also note an appendix by Zagier with crash-course on finite group representations ]
  6. Jean-Pierre Serre. A Course in Arithmetic. (A chapter on modular forms is self-contained.)

Software: PARI.math.u-bordeaux.fr, SageMath.org

Some short inspiring papers: Complementary bibliography: