🌴Tropical Geometry Unit 9 – Tropical Geometry in Algebraic Applications

Fundamentals of Tropical Geometry

• Tropical geometry studies geometric objects defined by polynomial equations using the tropical semiring $(ℝ ∪ \{-∞\}, max, +)$
• Replaces classical arithmetic operations (addition, multiplication) with tropical operations (maximum, addition)
• Leads to piecewise linear geometric objects called tropical varieties
• Provides a combinatorial approach to studying algebraic varieties
• Connections to other areas of mathematics such as combinatorics, graph theory, and optimization
• Useful in applications such as phylogenetics, scheduling problems, and auction theory
• Offers new perspectives on classical algebraic geometry by focusing on the underlying combinatorial structure

Key Concepts and Definitions

• Tropical semiring: $(ℝ ∪ \{-∞\}, max, +)$, where $ℝ$ is the set of real numbers and $-∞$ is the additive identity
  • Tropical addition: $a ⊕ b = max(a, b)$
  • Tropical multiplication: $a ⊗ b = a + b$
• Tropical polynomial: a polynomial with coefficients in the tropical semiring, e.g., $f(x) = 3x^2 ⊕ 1x ⊕ 5$
• Tropical hypersurface: the set of points where a tropical polynomial attains its maximum at least twice
• Tropical variety: the intersection of tropical hypersurfaces
• Tropical convexity: a notion of convexity based on tropical operations
  • Tropical convex hull: the set of all tropical linear combinations of a given set of points
• Tropical basis: a minimal set of polynomials whose tropical hypersurfaces define a tropical variety

Tropical Algebra and Operations

• Tropical addition $(⊕)$ corresponds to taking the maximum of two elements
  • Example: $3 ⊕ 5 = max(3, 5) = 5$
• Tropical multiplication $(⊗)$ corresponds to the usual addition of real numbers
  • Example: $2 ⊗ 4 = 2 + 4 = 6$
• Tropical division: defined as $a ÷ b = a - b$, where $-$ is the usual subtraction of real numbers
• Tropical exponentiation: defined as repeated tropical multiplication, e.g., $a^{⊗n} = a ⊗ a ⊗ ... ⊗ a$ ($n$ times)
• Tropical matrix operations: defined using tropical addition and multiplication
  • Tropical matrix addition: element-wise maximum
  • Tropical matrix multiplication: similar to classical matrix multiplication, but using tropical operations
• Tropical determinant: the maximum weight of a permutation of the matrix entries, where the weight is the sum of the entries

Tropical Curves and Varieties

• Tropical curve: a tropical hypersurface defined by a tropical polynomial in two variables
  • Piecewise linear graph in the plane
  • Consists of line segments and rays with rational slopes
• Tropical variety: the intersection of tropical hypersurfaces
  • Defined by a system of tropical polynomial equations
  • Has a polyhedral structure and can be represented as a polyhedral complex
• Tropical Bézout's theorem: the number of intersection points of two tropical curves is bounded by the product of their degrees
• Tropical Grassmannian: a tropical variety that parametrizes tropical linear spaces of a given dimension
• Tropical Riemann-Roch theorem: relates the dimension of the space of rational functions on a tropical curve to its genus and degree

Applications in Algebraic Geometry

• Provides a piecewise linear approximation of classical algebraic varieties
• Helps to study the combinatorial structure of algebraic varieties
• Tropical compactification: a technique to compactify algebraic varieties using tropical geometry
  • Useful in studying degenerations and limits of algebraic varieties
• Correspondence between tropical and algebraic varieties
  • Tropical varieties can be seen as "skeletons" of algebraic varieties over valued fields
• Tropical Hodge theory: a tropical analog of classical Hodge theory, relating the topology of tropical varieties to the geometry of algebraic varieties
• Tropical Gromov-Witten theory: a tropical version of Gromov-Witten theory, which studies curve counting in algebraic varieties

Computational Techniques and Tools

• Gröbner bases in tropical geometry: a tropical analog of Gröbner bases for solving systems of polynomial equations
  • Used to compute tropical varieties and their polyhedral structures
• Tropical polytopes: a class of polytopes arising from tropical hypersurfaces
  • Can be computed using algorithms from polyhedral geometry
• Software packages for tropical geometry (Gfan, Polymake, Singular)
  • Implement algorithms for computing tropical varieties, tropical bases, and other tropical geometric objects
• Tropical linear programming: a variant of linear programming using tropical algebra
  • Useful in solving optimization problems arising in tropical geometry
• Tropical decision trees: a machine learning technique based on tropical algebra
  • Used for classification and regression tasks in data analysis

Real-World Applications and Case Studies

• Phylogenetics: using tropical geometry to model evolutionary relationships among species
  • Tropical Grassmannians can represent tree-like structures in phylogenetic analysis
• Scheduling problems: applying tropical algebra to optimize resource allocation and minimize delays
  • Tropical polyhedra can model feasible regions in scheduling problems
• Auction theory: using tropical geometry to analyze bidding strategies and market equilibria
  • Tropical hypersurfaces can represent indifference curves of bidders
• Tropical climatology: modeling climate systems using tropical algebra
  • Tropical polynomials can describe the dynamics of atmospheric variables
• Tropical architecture: designing buildings and structures inspired by tropical geometric forms
  • Incorporates principles of sustainability and adaptation to tropical climates

Advanced Topics and Current Research

• Tropical moduli spaces: spaces that parametrize tropical curves or other tropical geometric objects
  • Tropical moduli spaces of curves, abelian varieties, and stable maps
• Tropical mirror symmetry: a tropical version of the mirror symmetry conjecture in string theory
  • Relates tropical Calabi-Yau varieties to their mirror partners
• Tropical intersection theory: a framework for studying intersections of tropical varieties
  • Develops a tropical analog of classical intersection theory in algebraic geometry
• Tropical Donaldson-Thomas theory: a tropical counterpart to Donaldson-Thomas theory in algebraic geometry
  • Studies curve counting and enumerative problems in tropical geometry
• Tropical Langlands correspondence: a proposed tropical version of the Langlands program in number theory and representation theory
  • Aims to relate tropical geometry to the theory of automorphic forms and Galois representations
• Tropical dynamics: the study of dynamical systems defined over tropical semirings
  • Investigates tropical analogs of classical dynamical systems, such as tropical rational maps and tropical lattice maps