Institute for Advanced Study Courses

Tropical Subrepresentations and Matroids

Explore the concept of tropical subrepresentations in linear representations, their connection to matroids, and elementary results from Jaiung Jun's joint research work.

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

Flow Categories as Exit Path Categories

Delve into the mathematical exploration of flow categories in symplectic geometry, examining their relationship to exit-path categories and the implications for Morse theory on smooth manifolds.

• YouTube
• 1 hour 5 minutes
• On-Demand
• Free Video

Why Extension-Based Proofs Fail

Delve into the limitations of extension-based proofs in distributed computing, focusing on valency arguments and why they fail for certain impossibility results like 2-set agreement among 3 processes.

• YouTube
• 1 hour 36 minutes
• On-Demand
• Free Video

A Toric Case of the Thomas-Yau Conjecture

Explore the Thomas-Yau conjecture for Lagrangian sections in Calabi-Yau fibrations, proving equivalence between Hamiltonian isotopy and stability conditions via Fukaya-Seidel categories.

• YouTube
• 1 hour 7 minutes
• On-Demand
• Free Video

Coboundary Expansion Inside Chevalley High-Dimensional Expanders

Delve into the topological properties of coset complex high-dimensional expanders (HDXs) over Chevalley groups, focusing on coboundary expansion and their applications in theoretical computer science.

• YouTube
• 1 hour 15 minutes
• On-Demand
• Free Video

From Few-Body to Stellar Clusters: How Gas Reshapes Dynamical Evolution

Explore how gas-rich environments reshape stellar dynamics, from binary interactions to nuclear star clusters, impacting phenomena like tidal disruption events and common envelope formation in astrophysical systems.

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

A Reduction of the F-Conjecture

Delve into the F-Conjecture's reduction to a simpler problem on M_{1,n}\bar, exploring how this reformulation may aid in finding counterexamples rather than proving cases.

• YouTube
• 1 hour 54 minutes
• On-Demand
• Free Video

Enumerating Smooth-Like Permutations Via the Geometry of Schubert Varieties

Dive into the enumeration of smooth-like permutations through Schubert varieties' geometry, exploring pattern avoidance characterization and fiber-bundle structures in this mathematical seminar by Edward Richmond.

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

Combinatorics in Quantum K-theory Schubert Calculus

Delve into the combinatorial model for quantum K-theory of flag manifolds, exploring Weyl group combinatorics, Chevalley-type multiplication formulas, and solutions to longstanding conjectures in Schubert calculus.

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

Aspects of the Large Scale Geometry of Out(Fn)

Delve into the large scale geometry of Out(Fn), exploring hierarchically hyperbolic groups, the Two Over All Theorem, and applications to outer space and free splitting complex geometry.

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

Modular Curves X1(n) as Moduli of Point Arrangements

Explore the mathematical theory of modular curves X1(n) as moduli of point arrangements, focusing on complex elliptic curves and their properties under Weierstrass embedding into CP2.

• YouTube
• 1 hour 8 minutes
• On-Demand
• Free Video

Linear Flows on Translation Prisms

Explore the mathematical study of linear flows in translation prisms, connecting weak mixing properties to ergodic properties and examining unique ergodicity in related billiard flows.

• YouTube
• 1 hour 12 minutes
• On-Demand
• Free Video

From Bourgain's Projection Theorem to Kakeya and Khintchine on Fractals: An Impressionistic Picture

Explore Bourgain's Projection Theorem and its extensions in geometric measure theory, homogeneous dynamics, and number theory with Pablo Shmerkin's overview of recent groundbreaking developments in these fields.

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

Higher Singularities in Positive Characteristic

Delve into Hodge-theoretic invariants of complex singularities and explore positive characteristic methods like Cartier operators for studying such singularities, based on joint research with Tatsuro Kawakami.

• YouTube
• 1 hour 5 minutes
• On-Demand
• Free Video

Cotangent Schubert Calculus and Cohomology Rings in Flag Varieties

Delve into advanced mathematical concepts of Cotangent Schubert Calculus, exploring cohomology rings, characteristic classes, and their connections to Hecke algebra and Kazhdan-Lusztig theory.

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