mlbaker-run seminar on complex function theory

Waterloo, Fall 2014

Michael Baker

Meetings: Mondays at 5:30pm in MC 5158B; Fridays at 3:30pm in MC 5046.

Talk information will be posted on the Pure Mathematics department page as well as exclusively on this page since I'm clearly too lazy to submit abstracts on time.

Möbius transformations, projective geometry, Riemann surfaces, analytic continuation, elliptic functions, modular forms. More information to come soon.

Either full videos or thorough notes of all talks will be posted. For Möbius transformations, my reference was Jones and Singerman's Complex Functions: An Algebraic and Geometric Viewpoint. For modular forms, my reference was Diamond and Shurman's A First Course in Modular Forms.

September

• Fri Sep 12: Michael Baker, "Introduction"

Discussion of possible topics. Stereographic projection and the Riemann sphere $\Sigma$ (also called the extended complex plane $\mathbf{C} \cup \{ \infty \}$, or complex projective line $\mathbf{CP}^1$). Behaviour of functions at $\infty$. Compactness and sequential compactness. Informal discussion of Riemann surfaces. Standard examples of Riemann surfaces: the complex plane $\mathbf{C}$, the Riemann sphere $\Sigma$, complex tori/elliptic curves $\mathbf{C}/\Lambda$ for $\Lambda$ a lattice in $\mathbf{C}$.

• Mon Sep 15: Michael Baker, "Meromorphic functions on the Riemann sphere"

Recap of behaviour of functions at $\infty$. Extension of usual theorems about analytic/meromorphic functions to the setting of the Riemann sphere. Multiplicity and principal parts. Uniqueness theorems for meromorphic functions. Rational functions. A function is rational if and only if it is meromorphic.

• Fri Sep 19: Michael Baker, "Introduction to Möbius transformations"

Automorphisms of $\Sigma$. The projective general linear group $\text{PGL}(2,\mathbf{C}) = \text{PSL}(2,\mathbf{C})$ and the isomorphism $\text{PGL}(2,\mathbf{C}) \cong \text{Aut}(\Sigma)$ (correspondence between matrices and Möbius transformations). Identification of the Riemann sphere $\Sigma$ with the complex projective line $\mathbf{CP}^1 = \text{P}(\mathbf{C}^2)$. Generators for $\text{PGL}(2,\mathbf{C})$.

• Mon Sep 22: Michael Baker, "Transitivity properties of Möbius transformations"

Circles in $\Sigma$ and how they are preserved by $\text{PGL}(2,\mathbf{C})$. 3-transitivity of $\text{PGL}(2,\mathbf{C})$. The anharmonic group (transformations preserving $\{ 0,1,\infty \}$). The cross ratio and how it classifies exactly which 4-tuples of distinct points are in the same orbit.

• Fri Sep 26: Michael Baker, "More on transitivity"

The effect of permutations on the cross ratio and how this leads to the "exceptional" order 4 normal subgroup of $S_4$.

• Mon Sep 29: Michael Baker, "Inversion and stabilisers"

Inversion through circles and lines. The stabilisers of a circle and a disc in $\Sigma$: $\text{PGL}(2,\mathbf{R})$ and $\text{PSL}(2,\mathbf{R})$.

October

• Fri Oct 3: Michael Baker, "Conjugacy and classification of Möbius transformations" (notes incomplete)

Conjugacy classes in $\text{PGL}(2,\mathbf{C})$. Fixed points of Möbius transformations. The multiplier $\{ \lambda, 1/\lambda \}$ and the squared trace. Limiting behaviour of $T^n$ as $n \to \infty$. Parabolic, elliptic, hyperbolic and loxodromic transformations.

• Mon Oct 6: Michael Baker, "Conformality"

Conformal maps between surfaces. Conformality of stereographic projections $S^2 \to \Sigma$ and $\Sigma \to S^2$. Direct and indirect conformality. Each automorphism (anti-automorphism) $T$ of $\Sigma$ is a directly (indirectly) conformal homeomorphism of $\Sigma$ onto itself, and each directly (indirectly) conformal map from $\Sigma$ to itself is an automorphism (anti-automorphism) of $\Sigma$ (proofs of these two facts omitted). Rotations of $\Sigma$ and the group $\text{PSU}(2,\mathbf{C}) = \text{Rot}(\Sigma) \leq \text{Aut}(\Sigma)$. The isomorphism $\text{PSU}(2,\mathbf{C}) \cong \text{SO}(3,\mathbf{R})$. The algebra of quaternions and its unit sphere $S^3 \cong \text{SU}(2,\mathbf{C})$. Some remarks on Hopf fibrations.

• Fri Oct 10: no talk due to Thanksgiving

• Mon Oct 13: no talk due to Thanksgiving

• Fri Oct 17: Michael Baker, "Lattices" (some extra notes on fundamental groups and Galois groups)

$\text{PGL}(2,\mathbf{C})$ as the Galois group $\text{Gal}(\mathbf{C}(z)/\mathbf{C})$. Discrete subsets of topological spaces. Lattices (discrete additive subgroups of $\mathbf{C}$). Classification of lattices in $\mathbf{C}$ into 3 types: trivial ($\Lambda \cong \{ 0 \}$), degenerate ($\Lambda \cong \mathbf{Z}$), and full rank ($\Lambda \cong \mathbf{Z} \times \mathbf{Z}$).

• Mon Oct 20: Mateusz Olechnowicz, "Elliptic functions 1"

The period set of a nonconstant meromorphic function is a discrete subgroup of $(\mathbf{C},+)$. Jacobi's lemma. The field $E(\Lambda)$ of meromorphic functions on $\mathbf{C}$ doubly periodic with respect to $\Lambda$. Proof that there are no nonconstant elliptic functions with 0 or 1 poles (so there must be 2 poles). The naive attempt, why it fails, and its fix: the Weierstrass $\wp$ function.

• Fri Oct 24: no talk due to my bad sleep habits

• Mon Oct 27: Mateusz Olechnowicz, "Elliptic functions 2" (notes to come)

Elliptic functions.

• Fri Oct 31: Michael Baker, "Fourier series and modular forms"

November

• Mon Nov 3: Mateusz Olechnowicz, "Elliptic functions 3"

Divisors will be quickly introduced, in order to prove that even elliptic functions are rational functions of $\wp$. We will then discuss the elliptic curve associated to a lattice. Every lattice defines an elliptic curve; the converse is this: if $a,b \in \mathbf{C}$ satisfy $a^2 = 27b^3$ then does there exist a lattice $\Lambda$ for which $g_2(\Lambda)=b$ and $g_3(\Lambda)=a$? The answer is yes, and this is (a special case of) the uniformization theorem. The proof uses the $j$-invariant. You get the valency formula for modular forms in general by integrating a particular form along the boundary of a fundamental domain for $\text{SL}(2,\mathbf{Z})$. Applying that valency formula to the $j$-invariant proves that it is a bijection between the fundamental domain and $\mathbf{C}$, so that every point is hit, so that every elliptic curve comes from a lattice.

• Fri Nov 7: Michael Baker, "Fundamental domain for SL(2,Z)"

• Mon Nov 10: Michael Baker, "Valence formula for modular functions"

• Fri Nov 14: no talk due to lack of preparation

• Mon Nov 17: Michael Baker, "Fourier expansions and the j-invariant"

I discuss the Fourier expansion of the Eisenstein series, cusp forms, the modular discriminant, and the $j$-invariant. Finally, I invoke the valence formula to show that the $j$-invariant gives a bijection between the set of $\text{SL}(2,\mathbf{Z})$-orbits and the complex plane.

• Fri Nov 21: Michael Baker, "Congruence subgroups and the four squares problem"

I use a classical problem from number theory to motivate the generalization of the concept of weak modularity, and to motivate the introduction of so-called principal congruence subgroups $\Gamma(N)$ of $\text{SL}(2,\mathbf{Z})$. I also deduce the Poisson summation formula from the Euler-Maclaurin summation formula, by using the Fourier series of the sawtooth wave.

• Mon Nov 24: Michael Baker, "Modular forms for congruence subgroups" (notes coming soon)

...

• Fri Nov 28: no talk due to final projects

• Mon Dec 1: no talk due to final projects