Introduction to Geometry and Topology in the Plane

Have you ever wondered how mathematicians study shapes, curves, and spaces without relying on rigid measurements? Geometry and topology in the plane offer a fascinating entry point into modern mathematics, focusing on properties that remain unchanged under continuous deformations. This course provides a gentle, structured pathway into these profound mathematical fields using concepts that are easy to conceptualize and map out mentally. You will transition from basic geometric intuition to formal mathematical reasoning, building a solid foundation in how spaces are structured and analyzed. Through clear written explanations, you will explore the fascinating properties of two-dimensional surfaces and curves. What you'll learn: - Understand foundational concepts of topology, including open sets, continuity, and homeomorphisms in the plane - Explore non-Euclidean spaces by studying the basic principles and models of hyperbolic geometry - Analyze the behavior of loops, curves, and winding numbers in two-dimensional space - Study geometric dynamical systems through the mathematics of mathematical billiards - Practice formulating basic mathematical proofs and reasoning about abstract spatial properties - Connect classical geometric theorems to modern topological applications This course begins with essential definitions and core terminology, ensuring you understand the language of topology before moving on to specific geometric models and dynamical systems. You will progress systematically through self-contained written modules that balance intuitive concepts with rigorous mathematical thinking. This course is designed entirely for beginners, requiring only a basic background in high school algebra and geometry, with no prior exposure to advanced university-level mathematics. Start your journey into the beautiful landscape of modern geometry and topology today.