🌴Tropical Geometry Unit 7 – Tropical Grassmannians and Stiefel Spaces

Key Concepts and Definitions

• Tropical geometry studies geometric objects defined by polynomial equations, where the arithmetic operations are replaced by the tropical operations of minimum and addition
• Tropical Grassmannians are tropical analogs of classical Grassmannians, which parametrize linear subspaces of a vector space
  • Tropical Grassmannians are defined using tropical polynomials and the tropical semiring $(\mathbb{R} \cup \{\infty\}, \min, +)$
• Stiefel spaces in tropical geometry are tropical analogs of classical Stiefel manifolds, which represent orthonormal frames in a vector space
• Tropical hyperplanes are the building blocks of tropical geometry, defined as the set of points where a tropical linear form attains its minimum
• The tropical convex hull of a set of points is the smallest tropically convex set containing those points, analogous to the classical convex hull
• Tropical matrices are matrices with entries in the tropical semiring, used to represent linear transformations in tropical linear algebra
• The tropical rank of a matrix is the largest size of a square submatrix with a finite tropical determinant, analogous to the classical rank

Historical Context and Development

• Tropical geometry emerged in the early 2000s as a combination of ideas from algebraic geometry, combinatorics, and mathematical physics
• The term "tropical" was coined by French computer scientist Jean-Eric Pin in the 1980s, referring to the use of the min-plus semiring in optimization problems
• Early work on tropical geometry was motivated by applications in enumerative geometry, particularly the study of Gromov-Witten invariants and Welschinger invariants
  • These invariants count curves satisfying certain conditions in algebraic varieties, and tropical geometry provides a combinatorial approach to their computation
• The development of tropical geometry was influenced by connections to toric geometry, which studies algebraic varieties equipped with an action of an algebraic torus
• Tropical geometry has also been applied to problems in mathematical physics, such as the study of amoebas and the Ronkin function in the context of complex analysis and statistical mechanics
• The field has grown rapidly since its inception, with contributions from mathematicians, computer scientists, and physicists around the world
• Recent developments include the study of tropical moduli spaces, tropical intersection theory, and the application of tropical techniques to problems in combinatorial optimization and data science

Mathematical Foundations

• Tropical geometry is built on the tropical semiring $(\mathbb{R} \cup \{\infty\}, \min, +)$, where addition is replaced by the minimum operation and multiplication is replaced by standard addition
  • This semiring is idempotent, meaning that $a + a = a$ for all elements $a$, which leads to many unique properties compared to classical algebra
• Tropical polynomials are polynomials with coefficients in the tropical semiring, and their solutions define tropical hypersurfaces
  • The tropical zero set of a polynomial is the set of points where the minimum of the polynomial's terms is attained at least twice
• Tropical varieties are defined as the intersection of tropical hypersurfaces, analogous to classical algebraic varieties
• The Fundamental Theorem of Tropical Geometry states that every tropical variety is the tropicalization of a classical algebraic variety, providing a link between the two theories
• Tropical linear algebra studies linear equations and their solutions in the tropical semiring, leading to concepts such as tropical eigenvalues and eigenvectors
• Tropical convexity is the study of convex sets and functions in the tropical setting, with applications to optimization and discrete geometry
• Tropical intersection theory investigates the intersections of tropical varieties, with analogues of classical results such as Bézout's theorem and the Bernstein-Kushnirenko theorem

Tropical Grassmannians: Structure and Properties

• Tropical Grassmannians $\mathcal{G}(k,n)$ parametrize tropical linear spaces of dimension $k$ in a tropical projective space of dimension $n-1$
  • They are tropical analogs of classical Grassmannians, which represent $k$-dimensional subspaces of an $n$-dimensional vector space
• Tropical Grassmannians can be realized as the tropicalization of classical Grassmannians, by replacing the defining equations with their tropical counterparts
• The Plücker embedding of the classical Grassmannian $\mathcal{G}(k,n)$ into a projective space $\mathbb{P}^{\binom{n}{k}-1}$ has a tropical analog, which embeds the tropical Grassmannian into a tropical projective space
• Tropical Grassmannians have a rich combinatorial structure, with a stratification into cells indexed by matroid subdivisions of the hypersimplex $\Delta(k,n)$
  • Each cell corresponds to a unique arrangement of tropical hyperplanes defining the linear spaces parametrized by the cell
• The tropical Grassmannian $\mathcal{G}(2,n)$ is closely related to the space of phylogenetic trees with $n$ labeled leaves, which arise in evolutionary biology
• Tropical Grassmannians play a role in the study of tropical flag varieties, which parametrize chains of tropical linear spaces and are related to representation theory and combinatorics
• The intersection theory of tropical Grassmannians is an active area of research, with connections to enumerative geometry and the study of moduli spaces

Stiefel Spaces in Tropical Geometry

• Stiefel spaces in tropical geometry, denoted $\mathcal{V}(k,n)$, are tropical analogs of classical Stiefel manifolds, which represent ordered sets of $k$ orthonormal vectors in an $n$-dimensional vector space
  • In the tropical setting, orthogonality is replaced by the condition that the vectors have tropically independent coordinates
• Tropical Stiefel spaces can be realized as the tropicalization of classical Stiefel manifolds, by replacing the orthogonality conditions with their tropical counterparts
• The tropical Stiefel space $\mathcal{V}(k,n)$ is closely related to the tropical Grassmannian $\mathcal{G}(k,n)$, with a natural surjective map from the Stiefel space to the Grassmannian
  • This map forgets the order of the vectors in each $k$-tuple, analogous to the classical relationship between Stiefel manifolds and Grassmannians
• Tropical Stiefel spaces have a cell decomposition similar to that of tropical Grassmannians, with cells indexed by ordered matroid subdivisions of the hypersimplex $\Delta(k,n)$
• The study of tropical Stiefel spaces is motivated by applications in optimization and data science, particularly in the context of low-rank matrix approximation and subspace clustering
• Tropical flag varieties can be constructed as iterated bundles of tropical Stiefel spaces, providing a connection between the two objects
• The topology and combinatorics of tropical Stiefel spaces are active areas of research, with potential applications to problems in algebraic geometry and representation theory

Applications and Real-World Examples

• Tropical geometry has found applications in various fields, including computer science, biology, and economics
• In computer science, tropical methods have been used to solve problems in combinatorial optimization, such as the minimum cost flow problem and the shortest path problem
  • The tropical semiring provides a natural framework for modeling these problems, and tropical algorithms often have favorable computational complexity compared to classical methods
• In biology, tropical geometry has been applied to the study of phylogenetic trees, which represent evolutionary relationships between species
  • The tropical Grassmannian $\mathcal{G}(2,n)$ is closely related to the space of phylogenetic trees with $n$ labeled leaves, and tropical methods have been used to analyze the geometry and combinatorics of these spaces
• In economics, tropical techniques have been used to study auction theory and the behavior of markets
  • Tropical hypersurfaces can be used to model the indifference curves of buyers and sellers, and tropical intersection theory provides tools for analyzing the equilibria of market systems
• Tropical geometry has also found applications in physics, particularly in the study of statistical mechanics and quantum field theory
  • The amoeba of a complex algebraic variety, which is a tropical object, has been used to analyze the phase transitions and critical phenomena of physical systems
• In data science, tropical methods have been applied to problems in dimensionality reduction and data visualization
  • Tropical principal component analysis and tropical multidimensional scaling are techniques for representing high-dimensional data in lower-dimensional tropical spaces, preserving important geometric features of the data

Computational Methods and Algorithms

• Computational methods play a crucial role in tropical geometry, as many tropical objects are defined combinatorially and can be efficiently computed using algorithms
• The tropical Gröbner basis algorithm is a fundamental tool in tropical computation, analogous to the classical Gröbner basis algorithm in algebraic geometry
  • Given a set of tropical polynomials, the algorithm computes a tropical basis for the ideal generated by the polynomials, which can be used to study the geometry of the corresponding tropical variety
• The Newton polygon method is a technique for computing the tropicalization of a classical algebraic variety, by studying the exponent vectors of the monomials in the defining equations
  • This method is particularly useful for hypersurfaces and curves, and can be used to compute tropical intersection numbers and other geometric invariants
• Tropical linear programming is a variant of classical linear programming, where the arithmetic operations are replaced by their tropical counterparts
  • Tropical linear programs can be solved using combinatorial algorithms, such as the tropical simplex method, which are often more efficient than classical interior point methods
• Tropical matrix factorization is a technique for approximating a tropical matrix as the product of two lower-rank matrices, analogous to classical matrix factorization methods such as SVD and NMF
  • Tropical matrix factorization has applications in data analysis and machine learning, particularly in the context of recommender systems and topic modeling
• Tropical eigenvalue problems involve computing the eigenvalues and eigenvectors of a tropical matrix, which are defined using the tropical semiring operations
  • Tropical eigenvalue problems arise in the study of tropical dynamical systems and tropical optimization, and can be solved using combinatorial algorithms based on the max-plus algebra
• Tropical convex hull algorithms compute the tropical convex hull of a set of points in tropical projective space, which is the smallest tropically convex set containing the points
  • These algorithms have applications in discrete geometry and optimization, and can be used to study the combinatorial structure of tropical polytopes and other tropical convex sets

Advanced Topics and Current Research

• Tropical intersection theory is a rapidly developing area of tropical geometry, which studies the intersections of tropical varieties and their geometric properties
  • Current research focuses on developing tropical analogs of classical results, such as Bézout's theorem and the Bernstein-Kushnirenko theorem, and applying these results to problems in enumerative geometry and mirror symmetry
• Tropical moduli spaces are tropical analogs of classical moduli spaces, which parametrize geometric objects such as curves, surfaces, and vector bundles
  • The study of tropical moduli spaces is motivated by applications in algebraic geometry and mathematical physics, particularly in the context of Gromov-Witten theory and string theory
• Tropical representation theory investigates the connections between tropical geometry and the representation theory of algebraic groups and algebras
  • Current research focuses on developing tropical analogs of classical results, such as the Littlewood-Richardson rule and the Borel-Weil theorem, and applying these results to problems in combinatorics and mathematical physics
• Tropical Hodge theory is an emerging area of research that aims to develop a tropical analog of classical Hodge theory, which studies the cohomology of algebraic varieties and their relationship to complex geometry
  • Tropical Hodge theory has potential applications in mirror symmetry and the study of tropical Calabi-Yau varieties, which are of interest in mathematical physics and string theory
• Tropical arithmetic geometry is a new area of research that combines ideas from tropical geometry and arithmetic geometry, which studies algebraic varieties over number fields and finite fields
  • Current research focuses on developing tropical analogs of classical results, such as the Weil conjectures and the Mordell-Weil theorem, and applying these results to problems in number theory and cryptography
• Tropical geometric analysis investigates the connections between tropical geometry and analysis, particularly in the context of amoebas and the Ronkin function
  • Current research focuses on developing tropical analogs of classical results in complex analysis and potential theory, and applying these results to problems in mathematical physics and optimization
• Tropical combinatorics is a growing area of research that studies the combinatorial properties of tropical objects, such as tropical polytopes, tropical hyperplane arrangements, and tropical matroids
  • Current research focuses on developing tropical analogs of classical results in combinatorics, such as the Cayley-Menger determinant and the Tutte polynomial, and applying these results to problems in discrete geometry and optimization