Georgia Institute of Technology College of Sciences

In this talk, we present recent results on the geometry of centrally-symmetric random polytopes generated by N independent copies of a random vector X. We show that under minimal assumptions on X, for N>Cn, and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector - namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body. Moreover, by identifying the floating bodies associated with various random vectors we recover the estimates that have been obtained previously, and thanks to the minimal assumptions on X we derive estimates in cases that had been out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when X is q-stable or when X has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing - noise blind sparse recovery. This is joint work with Olivier Guédon (University of Paris-Est Marne La Vallée), Christian Kümmerle (UNC Charlotte), Shahar Mendelson (Sorbonne University Paris), and Holger Rauhut (LMU Munich).

Bio: Felix Krahmer received his PhD in Mathematics in 2009 from New York University under the supervision of Percy Deift and Sinan Güntürk. He was a Hausdorff postdoctoral fellow in the group of Holger Rauhut at the University of Bonn, Germany from 2009-2012. In 2012 he joined the University of Göttingen as an assistant professor for mathematical data analysis, where he has been awarded an Emmy Noether Junior Research Group. From 2015-2021 he was assistant professor for optimization and data analysis in the department of mathematics at the Technical University of Munich, before he was tenured and promoted to associate professor in 2021. His research interests span various areas at the interface of probability, analysis, machine learning, and signal processing including randomized sensing and reconstruction, fast random embeddings, quantization, and the computational sensing paradigm.

In the early 80's, Freedman discovered that the Whitney trick could be performed in 4-dimensions which quickly led to a complete classification of closed, simply connected topological 4-manifolds. With gauge theory, Donaldson showed that 4-manifolds differ greatly from their higher dimensional counterparts which uncovered the stark differences between topological and smooth results in dimension 4. In this introductory talk, we will give a brief overview this classification and why dimension 4 is so unique. Then, we will describe handlebody decompositions of 4-manifolds and draw several Kirby pictures representing some basic 4-mfds.

We study the global existence of classical solutions to the incompressible viscous MHD system without magnetic diffusion in 2D and 3D. The lack of resistivity or magnetic diffusion poses a major challenge to a global regularity theory even for small smooth initial data. However, the interesting nonlinear structure of the system not only leads to some significant challenges, but some interesting stabilization properties, that leads to the possibility of the theory of global existence of classical and/or strong solutions. This talk is based on joint works with Yi Zhou, Yi Zhu, Shijin Ding, Xiaoying Zeng, and Jingchi Huang.

The fine curve graph of a surface S was introduced by Bowden–Hensel–Webb in 2019 to study the diffeomorphism group of S. We consider a variant of this graph, called the fine 1-curve graph, whose vertices are essential simple closed curves and edges connect curves that intersect in at most one point. Building on the works of Long–Margalit–Pham–Verberne–Yao and Le Roux–Wolff, we show that the automorphism group of the fine 1-curve graph is isomorphic to the homeomorphism group of S. This is joint work with Katherine W. Booth and Daniel Minahan.