Explore advanced concepts in high-dimensional probability theory through this comprehensive lecture series featuring leading researchers in the field. Delve into the geometric approach to invertibility of random matrices across four detailed lectures by Mark Rudelson, examining fundamental techniques and applications. Investigate complex stochastic models and their real-world applications through multiple sessions, while studying the intricate behavior of random matrices including spectral properties, large deviations, and eigenvalue distributions. Learn about spatial interacting models, limit theorems, and their applications to understanding complex systems. Examine specialized topics including Gaussian Brunn-Minkowski theory, line ensembles and resampling methods, spin glass models at zero temperature, and the Edwards-Anderson model. Discover cutting-edge research on random regular graphs, nodal domains, and percolation theory in random environments. Study advanced probability techniques including anti-concentration methods for random polynomials, replica symmetry breaking, and metastability phenomena. Explore applications to quantum gravity through Liouville quantum gravity and geodesic roughness, while investigating homogenization of stochastic gradient descent in high dimensions and algebraic statistics of random integral matrices.

Syllabus

Geometric Approach to Invertibility of Random Matrices (Lecture 1) by Mark Rudelson
Sausage Volume of the Random String and Survival in a Medium of Poisson Traps by Siva Athreya
Outliers in weakly Confined Coulomb-type systems by Alon Nishry
Geometric Approach to Invertibility of Random Matrices (Lecture 2) by Mark Rudelson
Gaussian Brunn-Minkowski Theory by Mokshay Madiman
Geometric Approach to Invertibility of Random Matrices (Lecture 3) by Mark Rudelson
Line Ensembles, Resampling and the Tangent Method by Shirshendu Ganguly
Monotonicity of the Logarithmic Energy for Random Matrices by Benjamin Dadoun
A One-dimensional Spin Model with Kac type Interaction, and a Continuous ...by Erwin Bolthausen
The Homogenization of SGD in High Dimensions by Elliot Paquette
Algebraic Statistics of Random Integral Matrices by Hoi H. Nguyen
Regularized Functional Inequalities and Applications to Markov Chains by Pierre Youssef
Geometric Approach to Invertibility of Random Matrices (Lecture 4) by Mark Rudelson
Complex Stochastic Models and their Applications by Subhroshekhar Ghosh
Limit Theorems for Spatial Interacting Models by Yogeshwaran D
A Limit Law for the Most Favorite point of a simple Random walk on a Regular tree by Oren Louidor
Spin Glass Phase at Zero Temperature in the Edwards--Anderson Model by Sourav Chatterjee
Complex Stochastic Models and their Applications by Subhroshekhar Ghosh
Extremal Landscape for the CbetaE Ensemble by Ofer Zeitouni
Large Deviations for the Largest Eigenvalue of Sub-Gaussian Wigner Matrices by Nicholas Cook
Many Nodal Domains in Random Regular Graphs by Nikhil Srivastava
Complex Stochastic Models and their Applications by Subhroshekhar Ghosh
Spectral Large Deviations for Sparse Random Matrices by Kyeongsik Nam
Large Deviations of the Largest Eigenvalue of Supercritical Sparse Wigner Matrices by Fanny Augeri
Complex Stochastic Models and their Applications by Subhroshekhar Ghosh
Replica Symmetry Breaking, Shattering, and Metastability by Aukosh Jagannath
Propagation of Lagrangian States Under random Potentials by Martin Vogel
Continuum Percolation in Random Environments by Benedikt Jahnel
Anti-concentration and application to random polynomials by Oanh Nguyen
Roughness of Geodesics in Liouville Quantum Gravity by Subhajit Goswami