Tropical Geometry

see also:

• 2018 - Tropical Geometry of Deep Neural Networks\

Tropical geometry is a relatively new and rapidly developing area of mathematics that lies at the intersection of algebraic geometry, combinatorics, and computational geometry. It involves the study of the tropical semiring, which consists of the real numbers augmented with negative infinity, equipped with the operations of tropical addition (taking the maximum of two numbers) and tropical multiplication (usual addition). This seemingly simple change in operations leads to a rich mathematical structure with deep connections to classical algebraic geometry, but with the advantage of often being combinatorially simpler and amenable to explicit calculations.

Key Concepts and Definitions

• Tropical Semiring: The set $\mathbb{T}=\mathbb{R}\cup \{-\infty \}$ with the operations of tropical addition $\oplus$ defined by $a\oplus b=\max (a,b)$ and tropical multiplication $\otimes$ defined by $a\otimes b=a+b$.

• Tropical Polynomials: Analogous to classical polynomials, but defined using tropical operations. A tropical polynomial in one variable might look like $f(x)=\max (3+x,2,4+2x)$, which, in classical terms, corresponds to taking the piecewise linear function that is the maximum of the given linear functions.

• Tropical Varieties: The set of zeroes of a tropical polynomial or a system of tropical polynomials. Unlike classical Algebraic Varieties, tropical varieties tend to be piecewise linear objects, often with a polyhedral or graph-like structure.

Tropical Geometry and Classical Algebraic Geometry

One of the most fascinating aspects of tropical geometry is its relationship with classical algebraic geometry. Tropical geometry can often be seen as a “combinatorial shadow” of classical algebraic geometry, where complex geometric objects are transformed into more combinatorially manageable, piecewise-linear shapes. This correspondence allows for insights into the structure of classical algebraic varieties, including their intersections, degrees, and Betti numbers, through the study of tropical varieties.

Applications and Implications

Tropical geometry has found applications across a range of mathematical disciplines and beyond:

• Enumerative Algebraic Geometry: Tropical geometry provides tools for counting the number of curves of a given degree passing through a specified set of points, a problem that is generally much harder in the classical setting.

• Mirror Symmetry: Connections between tropical geometry and mirror symmetry (a duality between pairs of Calabi-Yau Manifolds) have been explored, leading to new insights into string theory and theoretical physics.

• Computational Geometry and Optimization: The combinatorial nature of tropical geometry, together with its geometric interpretations, has implications for algorithms in computational geometry and optimization problems.

• Network Theory and Phylogenetics: The metric properties of tropical varieties have applications in network theory and phylogenetics, where distances between points (or species) can be analyzed within the tropical framework.

Mathematical Challenges and Developments

The study of tropical geometry poses unique mathematical challenges, including:

• Developing Tropical Intersection Theory: Understanding how tropical varieties intersect and how these intersections relate to classical intersections is an active area of research.

• Tropical Moduli Spaces: Constructing and studying moduli spaces of tropical curves or varieties, analogous to classical moduli spaces, to understand the parameter spaces of such objects.

• Combinatorial and Polyhedral Structures: Exploring the combinatorial and polyhedral structures that arise in tropical geometry, and relating them to classical geometric and algebraic properties.

Tropical geometry, with its blend of algebraic, geometric, and combinatorial techniques, continues to unfold new mathematical landscapes, offering both profound theoretical insights and practical computational tools. Its development is a testament to the richness of mathematical structures that can emerge from simple algebraic and geometric principles.