Math 207A: How much covolume does tell us about a lattice?

Fall 2015
Lectures:

MWF

10:00  10:50

APM 5829

Office Hour:

Send me an email.



Send me an email, we can meet and discuss math (possibly in a coffee shop).

Course description:
The following is a tentative list of topics to be covered:
 Hyperbolic plane: GaussBonnet in this case, fundamental domain of SL(2,Z), lattice of minimum covolume in SL(2,R).
 Geometry of numbers: space of lattices in R^{n}, Mahler's compactness criteria, Minkowski's reduction theory.
 KazhdanMargulis theorem, existence of a lattice of minimum
covolume in semisimple Lie groups without compact factors, Wang's
theorem on the finiteness of the number of lattices with covolume
at most x, up to conjugation.
 Covolume of SL(n,Z): Siegel's approach.
 Strong approximation, adeles and covolume of SL(n,Q) in SL(n,A).
 Siegel's mass formula and Tamagawa number of an orthogonal group.
 EskinRudnickSarnak's approach: counting integer points and Tamagawa number of orthogonal groups.
 Weil's conjecture on Tamagawa number.
 Prasad's volume formula.
 Related open problems and projects.

Prerequisite:
will be kept to a minimum.

Resources:
I will not follow a particular book, but I will post the related books and articles in the course's webpage.
Here are a few related references:
 M. S. Raghunathan, Discrete Subgroups of Lie Groups, Springer, New York, 1972
 D. W. Morris, Introduction to arithmetic groups, go here for an electronic version.
 S. Katok, Fuchsian groups.
 A. Weil, Adeles and algebraic groups.
 C. Maclachan, A. Reid, The arithmetic of hyperbolic 3manifolds.
 C. L. Siegel, A mean value theorem in geometry of numbers,
Annals of math. (2nd ser.) 46, no. 2, (1945), 340347.
 T. Tamagawa, Adeles, Proc. Symp. Pure Math., 9 113121, Amer. Math. Soc, Providence, 1966.
 G. Prasad, Volume of Sarithmetic quotients of semisimple groups,
Publ. math. IHES 69 (1989) 91114.

Notes related to lectures and supplementary materials:

In the first lecture, I said what a lattice in a topological group is,
explained that any lattice in R is cyclic, and gave an overview of the course. Then I defined the hyperbolic metric, and explored the basic properties of
hyperbolic plane.

In the second lecture, I showed that Mobius transformations are hyperbolic isometries and understood hyperbolic geodesics.

In the third lecture, we saw the classification of hyperbolic isometries, and Schottky's pingpong argument.

In the forth and the fifth lecture, we computed the area of a hyperbolic triangle, reviewed covering spaces and
the group of deck transformations, and mentioned the connection between the Teichmuller space and the character variety. You can see more precise statements
in the note.

In the sixth and seventh lecture, we introduced the
symmetric space of SL(n,R).

In the eighth lecture,
we introduced Dirichlet domain, and found a fundamental domain of PSL(2,Z).

In the ninth and tenth lecture, we talked about
Haar measure, found explicit Haar measures of certain groups, found various volume
forms of SL(2,R).
Here is the exercise on the decomposition of the Haar measure.

In the eleventh lecture, we talked about reduction theory.

In the twelfth and thirteenth lectures, we continued the study of reduction theory.

In the fortheenth and fifteenthlectures, we proved Mahler's compactness criterion, SL(n,Z) is a lattice in SL(n,R), etc.

In the sixteenth, seventeenth, and eighteenth lectures, we defined Siegel transform, proved the average of the Siegel transform is equal to
the Lebesgue integral of the function over R^{n}, computed the covolume of SL_{n}(Z) inductively, and proved Hlawka's theorem
(Siegel's approach) which asserts that for any bounded region A in R^{n} with area less that ζ(n) one can find a unimodular lattice
Δ in R^{n} that meets A only possibily at the origin.

In the ninteenth and twentieth lectures, we found an SL(n)invariant gauge form and compute the covolume of SL_{n}(Z) with respect to the
induced Haar measure. Then we defined padic numbers, and proved a generalization of Hensel's lemma.

In the lectures
2124, we covered basics of algebraic number thoery: ring of
integers in a number field is a Dedekind domain, it is a
finitely generated free abelain group, defined the discriminant of
a number field, proved the class number is finite, defined valuations and
their completion, defined ring of adeles, proved strong approximation (in
the additive case), showed k is a lattice in the ring of adeles of k and
computed its covolume.
