Further Pathologies in Algebraic Geometry

Introductory Geometry: Algebraic Geometry

Algebraic Geometry

Physical objects and constraints may be modeled by polynomial equations and inequalities. For this reason algebraic geometry, the study of solutions to systems of polynomial equations, is a tool for scientists and engineers. Moreover, relations between concepts arising in science and engineering are often described by polynomials. Whatever their source, once polynomials enter the picture, notio...

Algebraic Geometry

Algebraic Geometry

Examples • If S = ∅, then Z(S) = An. • If S = {1}, then Z(S) = ∅. • If S = {x1 − a1, . . . , xn − an}, then Z(S) = {(a1, . . . , an)}. Remark. All rings in this course are commutative and have 1. Remark. If A is a ring, then any subset S ⊆ A generates a minimal ideal 〈S〉 ⊆ A. In fact, we have 〈S〉 = {∑ aj xj : aj ∈ A, xj ∈ S}. Lemma. Z(S) = Z(〈S〉) for all S ⊆ k[x1 , . . . , xn]. Proof. Since 〈S〉...

Developments in Algebraic Geometry

8:00–10:00 Continental breakfast, coffee, registration 10:00–11:00 Vyacheslav Shokurov, Johns Hopkins Univ. Mumford’s legacy in birational geometry. 11:00–11:20 break 11:20–12:20 Ulrike Tillmann, Oxford Univ. Mumford’s conjecture: A topological outlook. 12:20–2:10 Lunch 2:10–3:10 Valery Alexeev, Univ. Georgia Compactification of the Torelli map: from Mumford to tropical geometry. 3:10-3:40 Coff...