The Degree-Three Bounded Cohomology of Complex Lie Groups of Classical Type

We establish Monod's isomorphism conjecture in degree-three bounded cohomology for every complex simple Lie group of classical type. Our main ingredient is a bounded-cohomological stability theorem with an optimal range in degree three that we bootstrap from previous stability results by the author and Hartnick. The bootstrapping procedure relies on the occurrence in our setting of a variant of the recently observed phenomenon of secondary stability in the sense of Galatius--Kupers--Randal-Williams.Comment: Added norm computations and relevant references, and modified introduction and general structure for better readability. 33 pages. Comments welcom

"Quantization is a mystery"

Expository notes which combine a historical survey of the development of quantum physics with a review of selected mathematical topics in quantization theory (addressed to students that are not complete novices in quantum mechanics). After recalling in the introduction the early stages of the quantum revolution, and recapitulating in Sect. 2.1 some basic notions of symplectic geometry, we survey in Sect. 2.2 the so called prequantization thus preparing the ground for an outline of geometric quantization (Sect. 2.3). In Sect. 3 we apply the general theory to the study of basic examples of quantization of Kaehler manifolds. In Sect. 4 we review the Weyl and Wigner maps and the work of Groenewold and Moyal that laid the foundations of quantum mechanics in phase space, ending with a brief survey of the modern development of deformation quantization. Sect. 5 provides a review of second quantization and its mathematical interpretation. We point out that the treatment of (nonrelativistic) bound states requires going beyond the neat mathematical formalization of the concept of second quantization. An appendix is devoted to Pascual Jordan, the least known among the creators of quantum mechanics and the chief architect of the "theory of quantized matter waves".Comment: lecture notes, 51 page

Some arithmetic Ramsey problems and inverse theorems

In this dissertation we study arithmetic Ramsey type problems and inverse problems, in various settings. This work consists of two parts. In Part I, we study arithmetic Ramsey type problems over abelian groups. This part consists of three chapters. In Chapter 2, using hypergraph containers, we study the rainbow Erdos-Rothschild problem for sum-free sets. In Chapters 3 and 4, we study the avoidance density for (k,l)-sum-free sets. The upper bound constructions are given in Chapter 3, answering a question asked by Bajnok. We also improved the lower bound for infinitely many (k,l) in both chapters, and a special case of the sum-free conjecture is verified in Chapter 4. In Part II, we study inverse problems over nonabelian topological groups. Preliminaries to topological groups are given in Chapter 5. In Chapter 6, we first obtain classifications of connected groups and sets which satisfy the equality in Kemperman's inequality, answering a question asked by Kemperman in 1964. When the ambient group is compact, we also get a near equality version of the above result with a sharp exponent bound, which confirms conjectures by Griesmer and by Tao. A measure expansion gap result for simple Lie groups is also presented. In Chapter 7, we study the small measure expansion problem in noncompact locally compact groups. The question that whether there is a Brunn-Minkowski inequality was asked by Henstock and Macbeath in 1953. We obtain such an inequality and prove it is sharp for a large class of groups (including real linear algebraic groups, Nash groups, semisimple Lie groups with finite center, solvable Lie groups, etc), answering questions by Hrushovski and by Tao

Topological Groups: Yesterday, Today, Tomorrow

In 1900, David Hilbert asked whether each locally euclidean topological group admits a Lie group structure. This was the fifth of his famous 23 questions which foreshadowed much of the mathematical creativity of the twentieth century. It required half a century of effort by several generations of eminent mathematicians until it was settled in the affirmative. These efforts resulted over time in the Peter-Weyl Theorem, the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups, and finally the solution of Hilbert 5 and the structure theory of locally compact groups, through the combined work of Andrew Gleason, Kenkichi Iwasawa, Deane Montgomery, and Leon Zippin. For a presentation of Hilbert 5 see the 2014 book “Hilbert’s Fifth Problem and Related Topics” by the winner of a 2006 Fields Medal and 2014 Breakthrough Prize in Mathematics, Terence Tao. It is not possible to describe briefly the richness of the topological group theory and the many directions taken since Hilbert 5. The 900 page reference book in 2013 “The Structure of Compact Groups” by Karl H. Hofmann and Sidney A. Morris, deals with one aspect of compact group theory. There are several books on profinite groups including those written by John S. Wilson (1998) and by Luis Ribes and ‎Pavel Zalesskii (2012). The 2007 book “The Lie Theory of Connected Pro-Lie Groups” by Karl Hofmann and Sidney A. Morris, demonstrates how powerful Lie Theory is in exposing the structure of infinite-dimensional Lie groups. The study of free topological groups initiated by A.A. Markov, M.I. Graev and S. Kakutani, has resulted in a wealth of interesting results, in particular those of A.V. Arkhangelʹskiĭ and many of his former students who developed this topic and its relations with topology. The book “Topological Groups and Related Structures” by Alexander Arkhangelʹskii and Mikhail Tkachenko has a diverse content including much material on free topological groups. Compactness conditions in topological groups, especially pseudocompactness as exemplified in the many papers of W.W. Comfort, has been another direction which has proved very fruitful to the present day