Geometry
Tropical Geometry
Цели урока
- Define the tropical semiring (R union {-inf}, max, +) and compute in it
- Understand tropical polynomials as piecewise-linear functions and their zero sets
- Construct tropical curves as weighted balanced graphs dual to Newton polytope subdivisions
- State Mikhalkin's correspondence theorem and compute tropical curve multiplicities
Предварительные знания
- Mirror symmetry and Calabi-Yau geometry
- Algebraic curves and Gromov-Witten invariants
- Basic combinatorics: convex hulls, polytopes
Can replacing + and * by max and + in polynomial equations yield rigorous geometric information about classical algebraic varieties - and actually count complex curves combinatorially?
- **Network routing:** tropical matrix multiplication (A*B)_{ij} = max_k(A_{ik} + B_{kj}) computes shortest paths - Bellman-Ford and Floyd-Warshall are tropical linear algebra
- **Economics:** optimal assignment problems and auction theory use the tropical (min-plus) semiring; the dual of a transportation problem is a tropical linear program
- **Phylogenetics:** tree metrics on biological sequences form a tropical Grassmannian; tropical geometry classifies evolutionary trees from DNA data
- **Enumerative geometry:** Mikhalkin's theorem makes counting genus-g degree-d curves through points a finite combinatorial computation instead of a transcendental integral
From Maslov to Mikhalkin
1979: Maslov introduced idempotent analysis (min-plus semiring) for asymptotics in classical mechanics. 1998: Kapranov proved that amoebas of algebraic hypersurfaces degenerate to tropical hypersurfaces under the Maslov dequantization. 2002: Mikhalkin formulated tropical curves. 2004: Mikhalkin proved the correspondence theorem, showing tropical curve counts reproduce Gromov-Witten invariants of P^2. The name 'tropical' honors Brazilian mathematician Imre Simon, who pioneered the min-plus semiring in computer science.
Tropical Semiring and Polynomials
The tropical semiring T = (R union {-infinity}, max, +) replaces classical addition with max and classical multiplication with ordinary addition. The additive identity is -infinity (since max(a, -infinity) = a) and the multiplicative identity is 0 (since a + 0 = a). There are no additive inverses, so T is a semiring, not a ring.
The tropical semiring arises as the Maslov dequantization limit of the ordinary semiring (R_{>0}, +, *): replace a with e^{a/h} and let h -> 0+. In the limit, max replaces + and + replaces *. Tropical geometry is thus the 'classical limit' of algebraic geometry over C as the geometry degenerates.
In the tropical semiring (R union {-inf}, max, +), what is the tropical product 3 * 7 and tropical sum 3 + 7?
In the tropical semiring: tropical multiplication is ordinary addition, so 3 * 7 = 3 + 7 = 10. Tropical addition is max, so 3 + 7 = max(3,7) = 7. The idempotent identity max(a,a) = a means tropical addition is not ordinary addition.
Tropical Curves and Newton Polytopes
A tropical curve in R^2 is the 'zero set' of a tropical polynomial: the locus where at least two monomials c_{ij} + ix + jy achieve the maximum simultaneously. This is always a piecewise-linear graph. The Newton polytope of the polynomial governs the combinatorial type of the curve.
A degree-1 tropical curve (tropical line) has Newton polytope Delta_1 = triangle (0,0),(1,0),(0,1). Its dual subdivision has one vertex inside Delta_1, so the tropical line is a trivalent graph with exactly one vertex and three unbounded edges - matching the classical line in P^2 having three intersection points with the torus boundary.
The balancing condition for a tropical curve states that at each vertex, the weighted sum of primitive direction vectors of incident edges is zero. What classical geometric property does this condition encode?
The balancing condition is the tropical analogue of a closed (compact, boundaryless) curve in classical geometry. It is equivalent to the graph being the zero set of a tropical polynomial, and it encodes that all 'flows' on the curve are conserved - analogous to a closed 1-form integrating to zero around any loop.
Mikhalkin Correspondence Theorem
Mikhalkin's correspondence theorem (2004) bridges classical enumerative geometry and tropical combinatorics: the number of genus-g degree-d complex curves in P^2 through 3d+g-1 generic points equals a weighted count of tropical curves through the corresponding tropical points. This converts a transcendental geometric problem into finite combinatorics.
The correspondence theorem has been extended to other toric surfaces and to counts with tangency conditions. In all cases the tropical count is a finite combinatorial problem, making previously intractable enumerative questions computable by hand or by simple algorithms.
Mikhalkin's theorem gives N_1^0 = 1: there is exactly one line through 2 points. From the tropical side, how does this count arise?
A tropical line max(x, y, 0) has one trivalent vertex with direction vectors (-1,0), (0,-1), (1,1); these have det = |det((-1,0),(0,-1))| = 1, so mult = 1. Through any 2 tropically generic points there is exactly one tropical line (the vertex position is uniquely determined). Thus N_1^0 = 1 * 1 = 1.
Connections to Other Topics
Tropical geometry converts hard analytic questions in algebraic geometry into tractable piecewise-linear combinatorics.
- Berkovich analytification — The tropicalization map factors through the Berkovich analytification of a variety over a non-Archimedean field; tropical geometry is a skeleton of the Berkovich space
- Enumerative geometry — Gromov-Witten invariants of toric surfaces are computable via tropical curve counts by Mikhalkin's theorem
- Optimization — Tropical linear programming captures combinatorial optimization problems; the tropical simplex algorithm computes shortest paths
- Mirror symmetry — The Gross-Siebert program constructs mirror pairs using tropical geometry to encode the degeneration data
Итоги
- Tropical semiring: a + b = max(a,b), a * b = a + b, with identity -infinity for addition and 0 for multiplication
- Tropical polynomial f(x,y) = max_{(i,j)} (c_{ij} + ix + jy) is a piecewise-linear function; its 'zero set' is the locus of non-differentiability
- A tropical curve is a weighted balanced graph in R^2 dual to a regular subdivision of the Newton polytope Delta(f)
- Balancing condition: at each vertex, the weighted sum of primitive outgoing direction vectors is zero
- Mikhalkin's theorem: the classical count N_d^g of genus-g degree-d curves through 3d+g-1 points equals the weighted count of tropical curves with mult(Gamma) = product of vertex multiplicities
- Tropical matrix multiplication gives shortest paths; tropical geometry is the algebraic structure underlying dynamic programming