Lectures on Random Matrices

IAS | PCMI Park City Mathematics Institute via YouTube

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YouTube
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English
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1 day 15 hours 42 minutes
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On-Demand
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Advanced

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Explore random matrix theory through a comprehensive lecture series from the 27th Annual PCMI Summer Session. Delve into the mathematical foundations and applications of random matrices, which sit at the intersection of mathematics, physics, computer science, and statistics. Master key concepts including Wigner random matrices and the Semicircle Law through detailed presentations by Ioana Dumitriu, examining the fundamental properties and behaviors of these important matrix ensembles. Understand universality phenomena for random matrix ensembles of Wigner type with Terry Tao's in-depth analysis of how different random matrix models exhibit similar statistical properties. Investigate the microscopic description of logarithmic and Coulomb gases with Sylvia Serfaty, exploring the connection between particle systems and random matrix eigenvalues. Learn about the relationship between random matrices and free probability theory through Dimitri Shlyakhtenko's lectures, discovering how non-commutative probability provides powerful tools for analyzing random matrix behavior. Study the matrix Dyson equation and its applications in random matrix analysis with László Erdős, focusing on local eigenvalue statistics and universality results. Examine Riemann-Hilbert problems and their role in random matrix theory through Percy Deift's presentations on integrable systems and asymptotic analysis. Explore how random matrices can be used for counting equilibria in complex systems with Yan Fyodorov's lectures on applications to theoretical physics and complex analysis. Investigate the delocalization properties of eigenvectors in random matrices with Mark Rudelson, understanding how eigenvectors spread across the entire space. Discover operator limits of random matrices through Balint Virag's analysis of infinite-dimensional generalizations and their limiting behaviors. Learn about the Kardar-Parisi-Zhang (KPZ) equation and its connections to random matrix theory with Jeremy Quastel's introduction to this important area of mathematical physics. Each topic is presented through multiple detailed sessions, providing thorough coverage of both theoretical foundations and recent research developments in random matrix theory.

Syllabus

Terry Tao (1.1) Universality for random matrix ensembles of Wigner type, part 1.1
Terry Tao (1.2) Universality for random matrix ensembles of Wigner type, part 1.2
Ioana Dumitriu (1) Wigner random matrices and the Semicircle Law, part 1/PCMI
Yan Fyodorov (1.1) Counting equilibria in complex systems via random matrices, part 1.1
Yan Fyodorov (1.2) Counting equilibria in complex systems via random matrices, part 1.2
Sylvia Serfaty (1.1) Microscopic description of Log and Coulomb gases, part 1.1
Sylvia Serfaty (1.2) Microscopic description of Log and Coulomb gases, part 1.2
Dimitri Shlyakhtenko (1.1) Random matrices and free probability, part 1.1
Dimitri Shlyakhtenko (1.2) Random matrices and free probability, part 1.2
Sylvia Serfaty (2.1) Microscopic description of Log and Coulomb gases, part 2.1
Sylvia Serfaty (2.2) Microscopic description of Log and Coulomb gases, part 2.2
Ioana Dumitriu (2.1) Wigner random matrices and the Semicircle Law, part 2.1
Ioana Dumitriu (2.2) Wigner random matrices and the Semicircle Law, part 2.2
Dimitri Shlyakhtenko (2.1) Random matrices and free probability, part 2.1
Dimitri Shlyakhtenko (2.2) Random matrices and free probability, part 2.2
Dimitri Shlyakhtenko (3.1) Random matrices and free probability, part 3.1
Dimitri Shlyakhtenko (3.2) Random matrices and free probability, part 3.2
Sylvia Serfaty (3.1) Microscopic description of Log and Coulomb gases, part 3.1
Sylvia Serfaty (3.2) Microscopic description of Log and Coulomb gases, part 3.2
Sylvia Serfaty (4.1) Microscopic description of Log and Coulomb gases, part 4.1
Sylvia Serfaty (4.2) Microscopic description of Log and Coulomb gases, part 4.2
Terry Tao (3.1) Universality for random matrix ensembles of Wigner type, part 3.1
Terry Tao (3.2) Universality for random matrix ensembles of Wigner type, part 3.2
Yan Fyodorov (2.1) Counting equilibria in complex systems via random matrices, part 2.1
Yan Fyodorov (2.2) Counting equilibria in complex systems via random matrices, part 2.2
Ioana Dumitriu (3.1) Wigner random matrices and the Semicircle Law, part 3.1
Ioana Dumitriu (3.2) Wigner random matrices and the Semicircle Law, part 3.2
Ioana Dumitriu (4.1) Wigner random matrices and the Semicircle Law, part 4.1
Ioana Dumitriu (4.2) Wigner random matrices and the Semicircle Law, part 4.1
Yan Fyodorov (3.1) Counting equilibria in complex systems via random matrices, part 3.1
Yan Fyodorov (3.2) Counting equilibria in complex systems via random matrices, part 3.2
Yan Fyodorov (3.3) Counting equilibria in complex systems via random matrices, part 3.3
Dimitri Shlyakhtenko (4.1) Random matrices and free probability, part 4.1
Dimitri Shlyakhtenko (4.2) Random matrices and free probability, part 4.2
Yan Fyodorov (4.1) Counting equilibria in complex systems via random matrices, part 4.1
Laszlo Erdos (1.1) The matrix Dyson equation for random matrices, part 1.1
Laszlo Erdos (1.2) The matrix Dyson equation for random matrices, part 1.2
Laszlo Erdos (1.3) The matrix Dyson equation for random matrices, part 1.3
Terry Tao (2.1) Universality for random matrix ensembles of Wigner type, part 2.1
Percy Deift (1.1) Riemann-Hilbert problems, part 1.1
Percy Deift (1.2) Riemann-Hilbert problems, part 1.2
Laszlo Erdos (2.1) The matrix Dyson equation for random matrices, part 2.1
Laszlo Erdos (2.2) The matrix Dyson equation for random matrices, part 2.2
Laszlo Erdos (3.1) The matrix Dyson equation for random matrices, part 3.1
Laszlo Erdos (3.2) The matrix Dyson equation for random matrices, part 3.2
Terry Tao (4.2) Universality for random matrix ensembles of Wigner type, part 4.2
Terry Tao (4.1) Universality for random matrix ensembles of Wigner type, part 4.1
Percy Deift (2.1) Riemann-Hilbert problems, part 2.1
Percy Deift (2.2) Riemann-Hilbert problems, part 2.2
Percy Deift (3.1) Riemann-Hilbert problems, part 3.1
Percy Deift (3.2) Riemann-Hilbert problems, part 3.2
Laszlo Erdos (4.1) The matrix Dyson equation for random matrices, part 4.1
Laszlo Erdos (4.2) The matrix Dyson equation for random matrices, part 4.2
Percy Deift (4.1) Riemann-Hilbert problems, part 4.1
Percy Deift (4.2) Riemann-Hilbert problems, part 4.2
Mark Rudelson (1.1) Delocalization of the eigenvectors of random matrices, part 1.1
Mark Rudelson (1.2) Delocalization of the eigenvectors of random matrices, part 1.2
Jeremy Quastel (1.2) Introduction to KPZ (Kardar-Parisi-Zhang), part 1.2
Jeremy Quastel (1.1) Introduction to KPZ (Kardar-Parisi-Zhang), part 1.1
Balint Virag (1.1) Operator limits of random matrices, part 1.1
Balint Virag (1.2) Operator limits of random matrices, part 1.2
Balint Virag (2.1) Operator limits of random matrices, part 2.1
Balint Virag (2.2) Operator limits of random matrices, part 2.2
Mark Rudelson (2.1) Delocalization of the eigenvectors of random matrices, part 2.1
Mark Rudelson (2.2) Delocalization of the eigenvectors of random matrices, part 2.2
Jeremy Quastel (2.1) Introduction to KPZ (Kardar-Parisi-Zhang), part 2.1
Jeremy Quastel (2.2) Introduction to KPZ (Kardar-Parisi-Zhang), part 2.2
Jeremy Quastel (3.1) Introduction to KPZ (Kardar-Parisi-Zhang), part 3.1
Jeremy Quastel (3.2) Introduction to KPZ (Kardar-Parisi-Zhang), part 3.2
Balint Virag (3.1) Operator limits of random matrices, part 3.1
Balint Virag (3.2) Operator limits of random matrices, part 3.2
Jeremy Quastel (4.1) Introduction to KPZ (Kardar-Parisi-Zhang), part 4.1
Jeremy Quastel (4.2) Introduction to KPZ (Kardar-Parisi-Zhang), part 4.2
Mark Rudelson (3.1) Delocalization of the eigenvectors of random matrices, part 3.1
Mark Rudelson (3.2) Delocalization of the eigenvectors of random matrices, part 3.2
Balint Virag (4.1) Operator limits of random matrices, part 4.1
Balint Virag (4.2) Operator limits of random matrices, part 4.2
Mark Rudelson (4.1) Delocalization of the eigenvectors of random matrices, part 4.1