Algebraic Geometry Reading Group (2012-2013)

Session 1: Cauchy's Formula

Goal: Let $U$ be an open set in $\mathbb{C}$ and $\gamma:[a,b] \to U$ be a $\mathcal{C}^1$-differentiable loop that is homotopic to a point. If $f$ is a homolophic function on $U$, then the relation $f(z_0)\cdot W(\gamma,z_0)= \frac{1}{2i\pi}\int_{\gamma} \frac{f(z)}{(z-z_0)}\, dz$ holds for every $z_0 \in U-\gamma([a,b])$.
Content: Holomorphic functions; Complex-differentiability; Cauchy-Riemann equations; Cauchy's integral theorem; Cauchy's integral formula; winding numbers $W(\gamma,z_0)$.
Speaker: (23/04/13: done) Bryce

Session 2: Analytic continuation

Goal: Let $U$ be an open set in $\mathbb{C}$, $z_0$ be a point in $U$ and $f$ be a holomorphic function on $U-\{z_0\}$ bounded near $z_0$. Then, there exists a holomorphic function $g$ defined on $U$ such that $f = g$ on $U-\{z_0\}$.
Content: Analytic continuation; Liouville's Theorem; Homological Cauchy's integral formula.
Speaker: (30/04/13: done) Matthew

Session 3: Argument principle

Goal: Let $\gamma:[a,b] \to U$ be a piecewise $\mathcal{C}^1$-differentiable loop dividing the complex plan in two connected components $K_0$ and $K_1$ where $z \in K_i$ iff $W(\gamma,z) = i$. If $f$ is holomorphic on a neighbourhood $U$ of $\gamma([a,b]) \cup K_1$ and does not vanish on $\gamma([a,b])$, then if $x_1,x_2,\dots,x_k$ are the zeros of $f$ (counted with multiplicity), then the relation $\sum_{k \in I}g(x_k) = \frac{1}{2i\pi}\int_{\gamma} \frac{g(z)f'(z)}{f(z)}\, dz$ holds for every holomorphic function $g$.
Content: Isolated zeros of holomorphic functions; argument principle (see Applications and consequences on Wikipedia for the generalized version);
Speaker: (7/05/13: done) Rémy

Session 4: Newton's identities

Goal: Expressing elementary symmetric polynomials in terms of power sums.
Content: See Derivation of the identities on Wikipedia for the beginning of the proof.
Speaker: (7/05/13: done) Rémy

Session 5: Weierstrass preparation theorem

Session 6: Complex analysis in several variables

Goal: Generalization of the Weierstrass preparation theorem and Liouville theorem to several variables.
Content: Holomorphic functions in several variables; Generalization of Taylor expansions to several variables ; Liouville Theorem; Weierstrass polynomials; Weierstrass preparation theorem; see Chapter 1, 7 and 8 of [Sch].
Speaker: (21/05/13: done) Matthew

Session 7: Local rings of holomorphic functions

Goal: The local ring of holomorphic functions at 0 is a UFD.
Content: Definition of a local ring and its properties; Definition of Weierstrass polynomials, Definition of the local rings of holomorphic functions at a point P; Definition of a resultant in a polynomial ring; see Chapter 8 of [Sch].
Speaker: (28/05/13: done) Rémy

Session 8: Chow's Theorem: part I

Session 9: Chow's Theorem: part II

Session 10: Chow's Theorem: part III

Goal: Proving the Chow's Theorem in any dimension.
Content: sections 2.13, 2.14, 2.15, 2.16 in [Kol]; see Remmert-Stein Theorem
Speaker: (18/06/13: done) Matthew

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