AGA problem answers
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Macaulay2 exercises
Exercises by day
- Mon, June 18:
- Convex Algebraic Geometry: Newton Polytopes and Sums of Squares, Motzkin Form, Sums of Squares on a Lemniscate, Convex Hulls of Planar Curves, Conical Hull of a Compact Convex Set, Nonnegative and SOS cones
- Tropical Exercises: Minkowski sums, Non-constant coefficients.
- Computational algebraic geometry: Every system = a system of quadrics, Gamma exceptional set, …
- Tue, June 19:
- Computational algebraic geometry: Study coordinates, Grassmannian of lines
- Convexity: Polar convex body, …
- Groebner bases: Ascending Chain Condition, Elimination Example, Lex Groebner basis and weight orders, …
- Wed, June 20:
- Lifts of convex sets: Lemniscate of Gerono
- Thu, June 21:
- Lifts of Convex Sets: Cross Polytopes, Rank of a Slack Matrix, Vanishing Ideals of Finitely Many Points, Birkhoff Polytope and Permutahedron
- Groebner Fans and Tropical Varieties: Principal Ideals, Term Orders and Weights, Tropical Varieties and Gfan
- Semidefinite and Polynomial Optimization: Explicit Example of Semidefinite Duality, Univariate Nonnegative Polynomial
- Fri, June 22:
- Computational algebraic geometry: Descartes's Circle Theorem via elimination, Intersection of Ideals and Radical Membership
- Semidefinite and Polynomial Optimization: Stability number of a graph, Positive maps and decomposable maps
- Mon, June 25:
- Mixed Volume and Bernstein's Theorem: Newton polygons with no common slopes, Mixed volume computation, MV(P,P,...,P), Mixed volume is symmetric, Mixed Volume is multi-linear, Mixed volume and containment, Bezout Bound, Mixed volume formula in dimension 2, Criterion for non-zero mixed volume
- Convex Algebraic Geometry: Dual Faces, Maximal Faces of the Nonnegative Cone, Exposed Extreme Rays of the Nonnegative Cone
- Primary Decomposition: Intersection of primary ideals
- Tues, June 26:
- Primary Decompositions: Splitting polynomial and saturations, Saturation by Elimination
- Computational Algebraic Geometry: Binomial start system, Deflation.
- Binomial Ideals: Cellular binomial ideals, Using binomial ideals for combinatorics
- Thu, June 28:
- Computational Algebraic Geometry: Numerical primary decomposition, Macaulay dual space.
- Tropical: Projection of tropical hypersurface, Minkowski sum and mixed faces
- Fri, June 29:
- Convex Algebraic Geometry: Inner product on the vector space of polynomials, Extreme rays of a spectrahedron, Extreme rays of the cone of sums of squares
- Primary Decomposition: Radicals of zero-dimensional ideals
- Tropical: Dimension Formula, Finite Combinatorial Types, Dim 3 Resultant Polytopes
- Mon, July 02:
- Biochemical Reaction Networks: 1-site phosphorylation,
- (Convex) Algebraic Geometry: Olympic Rings, Product of the unit disk, Associahedron, Cycle space of K5 Note: exercise 1 (planar curves) is on Greg's first exercise sheet, see Mon, June 18
- Tues, July 03:
- Tropical Geometry: Valuations, Residue field, p-adic Gröbner fans