Talk information will be posted

Möbius transformations, projective geometry,

Either full videos or thorough notes of all talks will be posted. For Möbius transformations, my reference was Jones and Singerman's

**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 $\text{PSL}(2,\mathbf{R})$.

**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"

**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*