📐 Algebraic Geometry

Course Code: 29015-AUX
Duration: 8 Weeks | 2 Sessions per Week (Theory + Applications)

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📚 Course Introduction:

Algebraic Geometry is a branch of mathematics that connects algebra and geometry by studying geometric shapes through algebraic equations. It allows us to understand the structure of mathematical equations through algebraic varieties and curves, and is widely used in theoretical physics, computing, and data science.

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🎯 Course Objectives:

• Understand algebraic equations that represent geometric shapes.

• Study algebraic varieties such as affine and projective spaces.

• Explore algebraic curves and their geometric properties.

• Practical applications in theoretical physics and computer graphics.

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🎯 Target Audience:

• Advanced mathematics students or researchers in algebraic geometry and theoretical physics.

• Programmers specialized in computational mathematics and computer graphics.

• Prerequisite: Prior knowledge of algebra and analytical geometry.

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🛠️ Materials and Resources:

• Textbooks:

  • “Algebraic Geometry” by Robin Hartshorne.

  • “Introduction to Algebraic Geometry” by Serge Lang.

  • “Basic Algebraic Geometry” by Igor V. Dolgachev.

• Software Tools: Mathematica, SageMath, or Maple for graphing and solving algebraic equations.

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🧑‍🏫 Instruction Method:

• 2 Sessions per Week:

  • Theory: Each lecture introduces the core algebraic concepts, equations, and real-world examples.

  • Applications: Practical sessions include solving exercises and applying concepts in physics and computer graphics.

• Project: At the end of the course, students will develop a final project related to algebraic geometry applications in theoretical physics or computer graphics.

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🧑‍🏫 What You Will Learn:

• How to study geometric shapes using multivariable algebraic equations.

• Understand algebraic varieties: affine and projective spaces and how to work with them.

• Study the properties of algebraic curves and represent them using algebraic equations.

• Practical applications of algebraic geometry in physics, such as particle physics, computer graphics, and engineering design.

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🗂️ Detailed Course Outline:

📅 Week 1: Introduction to Algebraic Geometry

• Introduction to algebraic geometry and its history.

• Polynomial equations and their significance in geometry.

📅 Week 2: Affine and Projective Spaces

• Study of affine spaces.

• Understanding projective spaces and how to transition between them.

📅 Week 3: Algebraic Curves

• Definition of algebraic curves.

• Study of algebraic equations for bi-variate curves.

📅 Week 4: Singularities and Singular Points

• Algebraic properties of singular shapes.

• Handling singular points and analyzing them.

📅 Week 5: Morphisms and Maps between Algebraic Varieties

• Definition of algebraic maps between varieties.

• Study of morphisms and their importance in algebraic geometry.

📅 Week 6: Applications in Physics and Particle Theory

• Application of algebraic geometry in theoretical physics.

• Using algebraic geometry in particle physics.

📅 Week 7: Computational Algebraic Geometry

• Using programming for algebraic solutions.

• Computational applications in algebraic geometry.

📅 Week 8: Final Project and Applications in Computer Graphics

• How to apply algebraic geometry in computer graphics.

• Final Project: Using algebraic geometry for a practical application (optional between particle physics or computer graphics).

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🏆 Final Outcome:

By the end of this course, you will have:

• A deep understanding of algebraic geometric equations.

• The ability to analyze algebraic varieties and curves.

• Practical knowledge in applying algebraic geometry in theoretical physics and computing.

• The skills to use algebraic geometry for engineering design and computer graphics.

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