14.1 Tropical geometry and its applications
5 min read•july 30, 2024
Tropical geometry is a fascinating field that bridges algebraic geometry and combinatorics. It replaces classical arithmetic with tropical operations, transforming complex algebraic objects into simpler piecewise linear structures. This shift allows for easier computation and analysis of geometric problems.
By studying tropical varieties, mathematicians gain insights into classical algebraic geometry and related fields. Tropical methods have found applications in optimization, phylogenetics, game theory, and mathematical physics, offering new perspectives on longstanding problems and opening up exciting avenues for research.
Tropical Geometry Defined
Overview and Relation to Classical Algebraic Geometry
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- Tropical geometry studies geometric objects defined by polynomial equations, replacing classical arithmetic operations with tropical operations (minimum and addition)
- Solution sets of polynomial equations in tropical geometry are piecewise linear objects called tropical varieties
- Tropical varieties can be viewed as combinatorial shadows of classical algebraic varieties
- Provides a way to study algebraic geometry over non-Archimedean fields (field of Puiseux series) by considering valuations of coefficients
- Many classical algebraic geometry concepts have tropical analogues that are often easier to compute and understand
- Bézout's theorem and intersection theory have tropical analogues
- Has connections to other areas of mathematics (combinatorics, polyhedral geometry, mathematical physics)
Key Features and Advantages
- Tropical geometry offers a new perspective on algebraic geometry by replacing classical arithmetic with tropical operations
- This change in operations leads to piecewise linear objects that are easier to study and compute
- Provides a bridge between algebraic geometry and combinatorics
- Tropical varieties have a strong combinatorial flavor and can be studied using combinatorial techniques
- Allows for the study of algebraic geometry over non-Archimedean fields
- Non-Archimedean fields (Puiseux series) have a natural valuation that can be used to define tropical varieties
- Simplifies many classical algebraic geometry concepts and makes them more accessible
- Tropical analogues of Bézout's theorem and intersection theory are often easier to understand and compute
Objects and Operations in Tropical Geometry
Tropical Semiring and Polynomials
- The tropical semiring (ℝ ∪ {∞}, ⊕, ⊗) is the basic algebraic structure in tropical geometry
- ⊕ denotes the minimum operation and ⊗ denotes the addition operation
- Tropical polynomials are polynomials in the tropical semiring
- Coefficients are real numbers or infinity
- Operations are tropical addition and multiplication
- Example of a :
- In classical notation:
Tropical Varieties and Morphisms
- The tropical hypersurface defined by a tropical polynomial is the set of points where the minimum of the monomials is attained at least twice
- Forms a piecewise linear object
- Tropical varieties are the solution sets of systems of tropical polynomial equations
- Can be represented as the intersection of tropical hypersurfaces
- Example of a tropical line in ℝ^2: defined by the equation
- In classical notation:
- Tropical morphisms are piecewise linear maps between tropical varieties that preserve the tropical structure
- Analogous to morphisms in classical algebraic geometry
Applications of Tropical Geometry
Optimization and Linear Programming
- Tropical geometry has applications in optimization, particularly in linear programming and the simplex method
- The tropical semiring captures the essential properties of the min-plus algebra used in these areas
- The tropical version of the simplex method can be used to solve linear programming problems
- Provides a new perspective on the geometric aspects of linear programming
- Example: The tropical version of the assignment problem can be solved using tropical matrix multiplication
Phylogenetics and Evolutionary Biology
- In phylogenetics, tropical geometry is used to study the space of phylogenetic trees
- Phylogenetic trees represent evolutionary relationships among species
- Tropical varieties can be used to model the space of all possible phylogenetic trees satisfying certain constraints
- Allows for the study of the combinatorial and geometric properties of phylogenetic tree spaces
- Example: The Bergman fan of a matroid can be used to represent the space of phylogenetic trees with a given set of splits
Game Theory and Economics
- Tropical geometry has connections to game theory and economics
- The tropical semiring can be used to model certain types of games and markets (bargaining problem, housing market)
- Tropical methods can be used to study the equilibria and dynamics of these systems
- Provides a new perspective on the geometric aspects of game theory and economics
- Example: The tropical version of the Nash equilibrium can be studied using tropical polyhedra
Mathematical Physics and Integrable Systems
- In mathematical physics, tropical geometry has been applied to the study of integrable systems (KP equation, ultradiscrete Toda lattice)
- Considers the tropical limit of these systems
- Tropical methods can be used to study the combinatorial and geometric aspects of integrable systems
- Provides a new perspective on the structure and dynamics of these systems
- Example: The tropical version of the KP equation can be studied using tropical Grassmannians and the ultradiscrete Toda lattice can be studied using tropical Prym varieties
Tropical Geometry Connections
Combinatorics and Matroids
- Tropical geometry has deep connections to combinatorics
- Many combinatorial objects (matroids, polyhedra) can be studied using tropical methods
- The Bergman fan of a matroid is a that encodes the combinatorial structure of the matroid
- Provides a geometric perspective on matroid theory
- Tropical methods can be used to study the combinatorial properties of polyhedra
- The tropical version of a polytope is a polyhedral complex that retains many of the combinatorial properties of the original polytope
Toric Geometry and Convex Polytopes
- Tropical geometry is related to toric geometry, which studies algebraic varieties that arise from convex polytopes
- The of a toric variety is a tropical variety that retains many of the combinatorial properties of the original toric variety
- Provides a way to study toric varieties using tropical methods
- Tropical methods can be used to study the combinatorial and geometric properties of convex polytopes
- The tropical version of a polytope is a polyhedral complex that encodes the combinatorial structure of the polytope
Non-Archimedean Geometry and p-adic Numbers
- Tropical geometry has applications in non-Archimedean geometry, which studies geometric objects over fields with a non-Archimedean valuation (p-adic numbers)
- Tropical varieties can be used to study the structure of non-Archimedean analytic spaces
- Provides a way to study non-Archimedean geometry using combinatorial and polyhedral methods
- The tropicalization of a p-adic analytic space is a tropical variety that encodes the non-Archimedean structure of the space
- Allows for the study of p-adic geometry using tropical methods
Mirror Symmetry and Enumerative Geometry
- Tropical geometry has connections to mirror symmetry, a conjectural duality between symplectic geometry and complex geometry
- The tropical version of mirror symmetry relates to Landau-Ginzburg models
- Has been used to study the enumerative geometry of rational curves
- Tropical methods can be used to study the combinatorial and geometric aspects of mirror symmetry
- Provides a new perspective on the structure and properties of mirror pairs
- Example: The tropical version of the Gromov-Witten invariants can be studied using tropical curves and Landau-Ginzburg models
Key Terms to Review (16)
Algebraic variety: An algebraic variety is a fundamental concept in algebraic geometry that represents the set of solutions to a system of polynomial equations. These varieties can be either affine or projective, and they can exhibit a wide range of geometric and topological properties. Understanding algebraic varieties is essential for exploring advanced topics such as singularities, computational techniques, and tropical geometry.
Applications in combinatorics: Applications in combinatorics refer to the various ways that combinatorial methods and structures can be utilized to solve problems across different fields, including mathematics, computer science, and engineering. This area involves counting, arranging, and analyzing discrete structures, providing tools to tackle complex problems like optimization, graph theory, and network analysis. The rich framework of combinatorial techniques often connects with other areas like algebraic geometry and optimization problems.
Bernd Sturmfels: Bernd Sturmfels is a prominent mathematician known for his contributions to algebraic geometry, particularly in the areas of polynomial systems and their applications. He has played a vital role in bridging the gap between symbolic and numerical methods, leading to significant advancements in hybrid algorithms and homotopy continuation techniques that solve complex polynomial equations. His work also extends into tropical geometry, influencing how these mathematical concepts are applied in various fields, including optimization and computational biology.
Gianluigi Zappalà: Gianluigi Zappalà is an influential mathematician known for his contributions to tropical geometry, particularly in relation to its applications in algebraic geometry. His work has helped bridge the gap between classical algebraic concepts and tropical methods, making complex ideas more accessible and applicable to various mathematical problems.
Intersection number: Intersection number is a concept that quantifies the intersection of algebraic varieties, providing a way to measure how many points two varieties meet, taking into account their multiplicities. This number can reveal crucial information about the geometric and topological properties of the varieties involved. It plays a significant role in determining their degrees, analyzing their behavior in projective spaces, and applying ideas in tropical geometry.
Macaulay2: Macaulay2 is a software system designed specifically for research in algebraic geometry and commutative algebra. It provides a powerful environment for performing computations with polynomial rings, ideal theory, and various algebraic structures, making it an essential tool for tackling complex problems in these areas.
Max-plus algebra: Max-plus algebra is a mathematical framework that extends traditional algebra by using the maximum operation in place of addition and addition in place of multiplication. In this system, the set of real numbers is equipped with two operations: 'max' for addition and '+' for multiplication, allowing for a unique approach to problems in optimization, scheduling, and combinatorial structures. This algebra is particularly useful in tropical geometry, where it helps to study geometrical properties of polynomials and provides insights into various applications.
Tropicad: A tropicad is a geometric structure that arises in tropical geometry, which can be seen as a piecewise linear version of classical algebraic geometry. In this context, tropicads serve as tropical analogs to algebraic varieties, enabling the study of polynomial equations through a combinatorial lens. They allow for new insights into the relationships between algebraic curves and their tropical counterparts, particularly in the realm of enumerative geometry and intersection theory.
Tropical Algebraic Geometry: Tropical algebraic geometry is a branch of mathematics that studies algebraic varieties using tropical semiring, where the usual operations of addition and multiplication are replaced with min and plus operations, respectively. This transformation allows complex algebraic problems to be analyzed in a more combinatorial and piecewise linear manner, leading to a new perspective on classical algebraic geometry concepts.
Tropical Curves: Tropical curves are geometric objects that arise in tropical geometry, which is a piecewise-linear version of algebraic geometry. These curves are defined over the tropical semiring, which replaces classical addition with minimum and classical multiplication with addition, transforming polynomial equations into combinatorial structures. Tropical curves can be seen as graphs in the plane where points correspond to the solutions of certain polynomial equations, providing a new perspective on classical algebraic curves.
Tropical intersection theory: Tropical intersection theory is a framework within tropical geometry that studies the intersections of tropical varieties, which are combinatorial analogues of algebraic varieties. This theory provides tools to analyze geometric properties using a piecewise-linear structure, making it easier to understand complex interactions among varieties in the tropical setting.
Tropical polynomial: A tropical polynomial is a mathematical expression in tropical algebra, where the usual operations of addition and multiplication are replaced by taking the maximum and summation of numbers, respectively. This concept is crucial in tropical geometry, as it helps in studying geometric structures by transforming classical polynomial equations into a more combinatorial framework, enabling the exploration of their properties and applications in various fields such as optimization and algebraic geometry.
Tropical variety: A tropical variety is a geometric object in tropical geometry that represents the solution set of a system of polynomial equations when analyzed through the lens of tropical mathematics. In this context, tropical varieties allow for the study of algebraic varieties using piecewise linear structures, which can simplify complex problems in algebraic geometry and connect them with combinatorial and polyhedral methods.
Tropical version of Bézout's theorem: The tropical version of Bézout's theorem states that for two tropical polynomials in a tropical projective space, the number of intersection points, counted with multiplicities, is equal to the product of their degrees. This theorem reflects how classical algebraic geometry concepts are transformed into the tropical setting, emphasizing the importance of valuations and piecewise-linear structures.
Tropicalization: Tropicalization is a mathematical process that transforms algebraic varieties into a piecewise linear setting by replacing the usual operations of addition and multiplication with maximum and addition, respectively. This transformation allows complex geometric problems to be simplified and analyzed using combinatorial techniques, linking classical algebraic geometry with tropical geometry. The concept plays a vital role in understanding toric varieties and finding applications in various fields, including optimization and computational geometry.
Valuated fields: Valuated fields are algebraic structures that extend fields by introducing a valuation, which is a function that measures the 'size' or 'value' of elements in the field. This concept allows for the study of how algebraic properties relate to geometric structures, particularly in tropical geometry, where valuations help transform classical geometric problems into combinatorial ones.