Author Archives: Joe Moeller

Notes for lecture 6 Sheafification Last time: for a space X and a ring k, a presheaf is a functor $latex Op(X)^{op} \to k-mod$, and a sheaf is a presheaf F such that for any open U in X and any open cover $latex (U_\alpha)_{\alpha \in I}$ of U, the sequence is exact. If weContinueContinue reading “Algebraic Analysis notes Lecture 7 (25 Jan 2019)”

Notes for Lecture 5 Last time, for a given topological space X and a ring k, we define a presheaf to be a functor $latex F \colon Op(X)^{op} \to k-mod$. Definition: If F is a presheaf on X, and x is a point in X, then the stalk of F at x, is the k-moduleContinueContinue reading “Algebraic Analysis notes Lecture 6 (18 Jan 2019)”

Notes for Lecture 4 Last time: Hard Lefschetz gives an orthogonal decomposition $latex H^k(X, \mathbb C) = \bigoplus_{2r \leq k}^\bot \eta^r H_{prim}^{k-2r} (X; \mathbb C)$ with respect to a Hermitian form on $latex H^k (X, \mathbb C)$ defined using the Poincare pairing. On the other hand, Hodge theory gives a decomposition $latex H^k (X; \mathbbContinueContinue reading “Algebraic Analysis notes Lecture 5 (16 Jan 2019)”

Notes for Lecture 3 Last time, we proved the Lefschetz hyperplane theorem. Lefschetz Hyperplane Theorem: $latex Let X^n \subset \mathbb P^N$ be a projective variety and $latex H \subset \mathbb P^N$ a hyperplane such that $latex U= X \setminus Y$ (where $latex Y = X \cap H$) is smooth. Then $latex H^k (X, Y; \mathbbContinueContinue reading “Algebraic Analysis notes Lecture 4 (14 Jan 2019)”

Notes for Lecture 2 Theorem: If $latex X \subset \mathbb C^N$ is a smooth affine complex algebraic variety of complex dimension n, then X has the homotopy type of a CW complex of real dimension n. Note that this theorem is saying that certain spaces of real dimension 2n are homotopic to spaces with halfContinueContinue reading “Algebraic Analysis notes Lecture 3 (11 Jan 2019)”

Lecture 1 notes here. Recall that in algebraic topology, we construct homotopy invariants, e.g. homology. Homology measures “global” topology, but its not very sensitive to local structure. However, Poincare noticed that if X is a closed oriented manifold of real dimension n, then $latex dim H_k(X, \mathbb R) = dim H_{n-k} (X, \mathbb R)$. PoincareContinueContinue reading “Algebraic Analysis notes Lecture 2 (9 Jan 2019)”

This quarter I’m taking a course called “Algebraic Analysis”. It’s being taught by Carl Mautner, who does rep theory and algebraic geometry here at UCR. I’ve had a bunch of classes with him before, and they were always fun. These are the notes from the first class. It was sort of running through the motivationContinueContinue reading “Algebraic Analysis notes Lecture 1 (7 Jan 2019)”