IAS | PCMI Park City Mathematics Institute Courses

A¹-Algebraic Topology and Introduction to Unstable Motivic Homotopy Theory - Part 1

Delve into unstable motivic homotopy theory, exploring Morel-Voevodsky spaces, homotopy sheaves, and motivic spheres through expert guidance in algebraic topology and geometry fundamentals.

• YouTube
• 1 hour 34 minutes
• On-Demand
• Free Video

Torsors Over Affine Curves - Part 2

Delve into advanced algebraic geometry concepts focusing on G-torsors and G-bundles, exploring their behavior over affine curves and Dedekind rings through étale cohomology and patching techniques.

• YouTube
• 1 hour 1 minute
• On-Demand
• Free Video

The Motivic Brouwer Degree

Delve into advanced concepts of motivic homotopy theory and the Brouwer degree through an expert lecture exploring fundamental mathematical principles and their applications in algebraic geometry.

• YouTube
• 1 hour 2 minutes
• On-Demand
• Free Video

A1-Homotopy Theory and the Weil Conjectures - Part 1

Delve into advanced algebraic geometry concepts through A1-homotopy theory, exploring cellular homology of Morel and Sawant while examining analogues of the Weil conjectures in modern mathematical frameworks.

• YouTube
• 57 minutes
• On-Demand
• Free Video

Torsors Over Affine Curves - Part 1

Delve into the theory of G-torsors in algebraic geometry, focusing on affine smooth connected curves over algebraically closed fields and exploring vector bundles, quadratic bundles, and semisimple k-group properties.

• YouTube
• 1 hour 1 minute
• On-Demand
• Free Video

From Matrices to Motivic Homotopy Theory

Explore the foundations of motivic homotopy theory, from matrix fundamentals to advanced concepts, with applications in algebra and algebraic geometry for graduate-level mathematicians.

• YouTube
• 1 hour 4 minutes
• On-Demand
• Free Video

Algebraic Vector Bundles Over Smooth Affine Varieties

Delve into advanced mathematical concepts of algebraic vector bundles over smooth affine varieties, exploring fundamental theories and applications in motivic homotopy theory and algebraic geometry.

• YouTube
• 1 hour 1 minute
• On-Demand
• Free Video

Artistic Mathematics - Truth and Beauty in Mathematical Visualization

Explore mathematical visualization through 3D printing, virtual reality, and spherical video, discovering how to create compelling and beautiful representations of complex mathematical concepts.

• YouTube
• 52 minutes
• On-Demand
• Free Video

A-Upper Motives in Motivic Homotopy Theory

Delve into advanced algebraic geometry concepts focusing on motivic structures of projective homogeneous varieties and reductive groups, featuring research by Karpenko, de Clercq, and Quéguiner-Mathieu.

• YouTube
• 59 minutes
• On-Demand
• Free Video

Chow Groups and Algebraic Cycles in Complex Algebraic Geometry - Part 4

Delve into algebraic geometry's Chow groups, exploring how complex algebraic subvarieties contribute to cohomology theory and connect to major mathematical conjectures like the Hodge conjecture.

• YouTube
• 57 minutes
• On-Demand
• Free Video

Massey Products in Galois Cohomology - Part 3

Delve into advanced Galois cohomology theory, focusing on Massey products, the Norm-Residue Theorem, and recent developments in the Massey Vanishing Conjecture for mod 2 fourfold products over arbitrary fields.

• YouTube
• 1 hour 1 minute
• On-Demand
• Free Video

Local Systems in Arithmetic Geometry - Lecture 3

Delve into fundamental group theory and local systems in arithmetic geometry, exploring motivic conjectures and their implications for complex quasi-projective varieties and geometric representations.

• YouTube
• 45 minutes
• On-Demand
• Free Video

Chow Groups and Algebraic Cycles in Complex Algebraic Geometry - Part 3

Delve into algebraic geometry's Chow groups, exploring how complex algebraic subvarieties relate to cohomology, the Hodge conjecture, and fundamental mathematical problems in number theory and geometry.

• YouTube
• 1 hour 1 minute
• On-Demand
• Free Video

Massey Products in Galois Cohomology - Lecture 2

Delve into advanced Galois cohomology theory, exploring Massey products, the Norm-Residue Theorem, and recent developments in the Massey Vanishing Conjecture through expert-led mathematical discussions.

• YouTube
• 55 minutes
• On-Demand
• Free Video

Arithmetic Properties of Local Systems - Lecture 2

Delve into fundamental group theory and local systems in arithmetic geometry, exploring motivic conjectures and their implications for complex quasi-projective varieties and linear representations.

• YouTube
• 50 minutes
• On-Demand
• Free Video