Ruijie Yang
March 12, 2021

In complex algebraic geometry, the decomposition theorem asserts that semisimple geometric objects remain semisimple after taking direct images under proper algebraic maps. This was conjectured by Kashiwara and is proved by Mochizuki and Sabbah in a series of long papers via harmonic analysis and D-modules.

In this thesis defense, we would like to explain a simpler proof in the case of semisimple local systems, using a more geometric approach adapting de Cataldo- Migliorini. On the one hand, we complement Simpson’s theory of weights for local systems by proving a global invariant cycle theorem in the setting of local systems. On the other hand, we define a notion of polarization via Hermitian forms on pure twistor structures. This is partially based on joint work with Chuanhao Wei.

March 8-12, 2021

The goal of this Workshop is to explore connections between various aspects of Renormalization in Dynamics (unimodal and circle, holomorphic and cocyclic, Henon, KAM, and stochastic renormalizations) and Physics (QFT and statistical mechanics, fluid dynamics, and KPZ), which could help to reveal a unifying theme for all these phenomena.

This workshop is part of the Program: Renormalization and universality in Conformal Geometry, Dynamics, Random Processes, and Field Theory: February 3 – June 5, 2020. There will also be Renormalization retrospective: Feigenbaum Memorial Conference held right before (May 28-29, 2020).

March 4-7, 2021

This Conference will pay tribute to the great discovery made by Feigenbaum in the mid 1970s and its ramifications (mostly in math) in the past 45 years. It will also serve as an introduction to the SCGP Workshop Many faces of renormalization held during the following week (June 1–5). Both events are part of the Program: Renormalization and universality in Conformal Geometry, Dynamics, Random Processes, and Field Theory: February 22 – March 19, 2021

Claude LeBrun is one of the foremost geometers in the world today. His original and profoundly influential research over the past forty years has consistently changed the landscape of the field. He has used ideas from physics, such as Penrose’s twistor theory, the Gibbons-Hawking Ansatz, and Seiberg-Witten invariants, to establish results that no one expected, and at the same time to disprove long-standing conjectures. His deep mathematical knowledge and talent has enabled him to make transformative contributions to a wide range of subjects. He has been at Stony Brook since 1983, and is a leading member of the differential geometry group in our department.

Spring 2020 Workshop

Stony Brook University
October 23-25, 2020

The August Comprehensive Exams will be offered both in-person and online. If you wish to take the exam online, please contact Christine Gathman.

The Stony Brook Trustees Faculty Awards recognize early-career faculty whose research, creative activities, and scholarly achievements predict an exceptional trajectory.

Project: "Analytic Number Theory and Analysis of Discrete Structures"

Robert Hough is an Assistant Professor of Mathematics at Stony Brook University. His research focuses on analytic number theory and discrete probability. started as a resource for Boston-area number theorists but has grown rapidly into a catalog of virtual seminars around the world. It was developed by Collaboration research scientists Edgar Costa and David Roe, with the guidance of PIs Andrew Sutherland and Bjorn Poonen, all based at MIT. It relies on database infrastructure created for the current version of the L-functions and Modular Forms Database.

Robert Hough is an Assistant Professor of Mathematics at SUNY Stony Brook. His research focuses on analytic number theory and discrete probability. His best known work solved one of Paul Erdos' favorite problems, the "minimum modulus problem for covering systems" and has appeared in the Annals of Mathematics. Dr. Hough won the Math Association of America's David P. Robbins Prize for this work. His current research concerns questions related to the enumeration of low degree number fields, extending some works cited in Manjul Bhargava's Fields medal, and studies the asymptotic mixing of large statistical physics models such as the abelian sandpile model and Kac model. A discussion of Dr. Hough's work on the 15-puzzle recently appeared in Quanta.

Stanislav Smirnov (University of Geneva, Skoltech)

Percolation is a mathematical model for the filtering of a liquid through a porous material or the spread of a forest fire or an epidemic: the edges of some graph are declared open or closed depending on independent coin tosses, and then connected open clusters are studied. While simple to define, this model exhibits very complicated behavior, with non-trivial scaling exponents and dimensions.Centering on the 2D setting, we will discuss simple proofs of some important theorems, connection of percolation to other models,as well as remaining open questions.  …