10:35am - 11:50am on Mondays and Wednesday in 427 Kaprielian Hall (KAP). Note: the meeting time and location are subject to change depending on room availability.
Toric geometry roughly studies spaces equipped with torus actions, with parallel frameworks in both algebraic and symplectic geometry. The torus symmetry allows us to reduce essentially any geometric problem to combinatorics, and this gives a potent laboratory for computing examples and generating and testing conjectures. Almost toric geometry generalizes toric geometry by studying systems with certain extra singularities which arise frequently in nature and which vastly extend the range of applicability. These spaces are also largely combinatorial in nature, but they require additional input from integral affine geometry. In this course, we will introduce almost toric geometry from the perspectives of both algebraic and symplectic geometry, discuss many examples and tools, and finally explore various recent breakthroughs which rely on insights from almost toric geometry.
I will assume familiarity with smooth manifolds, Lie groups, and basic commutative algebra. Some familiarity with the basics of symplectic geometry and complex algebraic geometry will be helpful but not required.
| $\#$ | Date | Material | References | Problem set |
|---|---|---|---|---|
| 1 | Monday 8/26/24 | Impressionistic introduct to toric and almost toric geometry. Exotic Lagrangian tori in $\mathbb{CP}^2$ and the Markov equation. | Fulton §1.1 (for a first taste of toric algebraic geometry) | |
| 2 | Wednesday 8/28/24 | Crash course in algebraic geometry I: affine varieties, coordinate rings, Hilbert basis theorem, Hilbert Nullstellensatz, Zariski topology, irreducibility, localization, normality. | Cox §0 | |
| Monday 9/2/24: Labor day - no class | ||||
| 3 | Wednesday 9/4/24 | Crash course in algebraic geometry II: projective varieties, homogeneous ideals, projective Nullstellensatz, rational functions, weighted projective spaces, valuations, Weil and Cartier divisors, Chow and Picard groups. | Cox §0 | |
| 4 | Monday 9/9/24 | Lattices, characters and cocharacters, examples of toric varieties, convex polyhedral cones, dual cones. | Cox §1 | |
| 5 | Wednesday 9/11/24 | More on cones and dual cones. Faces, facets, and rays. Characterization of strong convexity. Rational polyhedral cones. Monoids of lattice points and Gordan's lemma. Regular and simplicial cones. | Cox §1 | |
| 6 | Monday 9/16/24 | The affine toric variety associated to a cone. First examples. | Cox §1 | Pset 1 (due Wednesday 9/25/24 at 11:59pm). |
| 7 | Wednesday 9/18/24 | Fans and their associated toric varieties. First examples. | Cox §2 | |
| 8 | Monday 9/23/24 | More examples of fans and their associated toric varieties. Complete, regular, simplicial, and polytopal fans and the corresponding properties for the toric variety. | Cox §2 | |
| 9 | Wednesday 9/25/24 | More on face fans and normal fans. Polar dual polytopes. The bijection between cones and orbits and the distinguished point associated to each cone. | Cox §2 and parts of §4, Brasselet §4. | |
| 10 | Monday 9/30/24 | More on distinguished points and orbits associated to cones. Orbit closures and the associated abstract fan. | Fulton §3.1. | |
| 11 | Wednesday 10/2/24 | The divisor associated to a polytope. Proof that it is Cartier and ample. The associated embedding into projective space. Characters as rational functions and their associated divisors. The Chow group of a toric variety and its relation with ordinary homology. | Cox §3.1,§4.1,§4.2,§4.3, Fulton §1.5. | |
| 12 | Monday 10/7/24 | Actions by finite groups and rings of invariant polynomials. The general two-dimensional affine toric variety as a quotient $\mathbb{C}^2/G$. Quotient map induced by a changed of lattice. Simplicial cones give orbifold singularities. Wieghted projective space as a global question of standard projective space. | Cox §2.6,§2.7, Fulton §2.2. | |
| 13 | Wednesday 10/9/24 | Toric morphisms and their criterion for properness. Refinements of fans. Cocharacters and their limit points as distinguished points. Proof that compact toric varieties have complete fans. Brief crash course in vector bundles. The tautological bundle over $\mathbb{P}^1$ as a toric refinement of $\mathbb{C}^2$. | Fulton §1.1,§2.3,§2.4. | Pset 2 (due Friday 10/18/24 at 11:59pm). |
| 14 | Monday 10/14/24 | More on $\mathcal{O}(-1)$ as a toric variety. Toric blowups. More on toric morphisms. Hirzebruch surfaces. Classification of minimal nonsingular compact toric surfaces. Birational maps and rational varieties. Classification of minimal nonsingular rational surfaces. | Fulton §1.1,§1.2,§2.5. | |
| 15 | Wednesday 10/16/24 | Resolution of singularities for toric surfaces. The basic refinement step, and negative continued fraction expansions. Multiplicities of cones, refinements, and resolution of singularities for higher dimensional toric varieties. | Fulton §2.6. | |
| 16 | Monday 10/21/24 | More on resolution of singularities in higher dimensions. The example of the Atiyah flop. First steps in symplectic geometry: symplectic forms, Darboux's theorem, symplectic and Hamiltonian vector fields, conservation of energy, the Poisson bracket. | Fulton §2.6, da Silva §1.1,§1.2,§1.3. | |
| 17 | Wednesday 10/23/24 | Integrable Hamiltonian systems and their basic properties. Little Arnold-Liouville theorem. | da Silva §1.3,§1.4,§1.5,§1.6, Evans §1.3. | |
| 18 | Monday 10/28/24 | More on integrable systems and various examples. Hamiltonian group actions. The Atiyah-Guillemin-Sternberg convexity theorem. Examples of moment maps and moment polytopes. | da Silva §1.3,§1.4,§1.5,§1.6, Evans §1.3. | |
| 19 | Wednesday 10/30/24 | Delzant polytopes and Delzant's theorem. Symplectic reduction and examples. | da Silva §2.1,§2.2,§2.3. | |
| 20 | Monday 11/4/24 | Construction of a toric symplectic manifold from a Delzant polytope. Morse theory using moment polytopes. | da Silva §2.5,§2.6,§3.1,§3.2,§3.3. | |
| 21 | Wednesday 11/6/24 | Examples of Delzant's theorem giving toric symplectic manifolds from Delzant polytopes. Linear toric actions, weight space decompositions, and their moment maps. The equivariant Darboux theorem and computation of the Morse index for a moment map Morse function. | da Silva §2.5,§2.6,§3.1,§3.2,§3.3. | |
| Monday 11/11/24: Veteran's day - no class | ||||
| 22 | Wednesday 11/13/24 | More on integrable systems. Local sections and period lattices. The existence of Liouville coordinates. Lagrangian sections of the standard projection $T^*\mathbb{T}^n \rightarrow \mathbb{R}^n$. | Evans §1.3,§1.4,§1.5,§1.6. | |
| 23 | Monday 11/18/24 | Existence of action angle coordinates and the big Arnold-Liouville theorem. | Evans §1.5,§1.6. | |
| 24 | Wednesday 11/20/24 | Stratified topological spaces and Lagrangian fibrations (possibly singular). Basic properties of regular Lagrangian fibrations and the integral integral affine structure on the base. Integral affine geometry. Recovering a Lagrangian fibration from its integral affine base. | Evans §2.1,§2.2,§2.3. | |
| 25 | Monday 11/25/24 | |||
| Wednesday 11/27/24: Thanksgiving break - no class | ||||
| 26 | Monday 12/2/24 | student presentations | ||
| 27 | Wednesday 12/4/24 | student presentations | ||
| 28 | Monday 12/9/24 | student presentations |