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Toric geometry is a mathematical field that combines algebraic geometry with combinatorial methods to study toric varieties, which include a torus as a dense subset. Fundamental concepts like lattices, cones, and fans are crucial in understanding the structure and properties of these varieties. Toric geometry has applications in economics, physics, computer science, and architecture, and influences research across various mathematical disciplines.

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## Introduction to Toric Geometry

### Toric Varieties

Toric varieties are algebraic varieties containing a torus as a dense subset and are defined by combinatorial objects known as fans

### Lattices

Lattices provide a framework for understanding the integral structure of toric varieties

### Cones

Cones represent the local structure of toric varieties at each point

## Specialized Areas of Toric Geometry

### Fan Toric Geometry

Fan toric geometry focuses on the role of fans in the construction and analysis of toric varieties

### Kahler Toric Manifolds

The study of Kahler toric manifolds applies the principles of Kahler geometry to toric varieties, allowing for the exploration of rich geometric and topological properties

### Cox Toric Geometry

Cox toric geometry delves into the algebraic aspects of toric varieties through the study of the Cox ring

## Applications of Toric Geometry

### Economics

Toric geometry provides models for market equilibria and the behavior of trade networks in economics

### Theoretical Physics

Toric geometry plays a crucial role in the study of string theory, particularly in the context of compactification and mirror symmetry

### Computer Science

Toric geometry aids in the development of algorithms for 3D modeling and visualization in computer science

### Architectural Design

Toric geometry inspires innovative structural forms in architectural design

## Interdisciplinary Impact of Toric Geometry

### Algebraic Geometry

Toric geometry enriches the field of algebraic geometry by providing a combinatorial approach to the study of algebraic varieties

### Number Theory

Toric geometry contributes to the understanding of integer solutions to polynomial equations in number theory

### Topology

Toric geometry aids in the analysis of the topological characteristics of toric varieties in topology

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