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Abel prize awarded to Furstenberg and Margulis

Just a short post to note that this year’s Abel prize has been awarded jointly to Hillel Furstenberg and Grigory Margulis for “for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics”.  I was not involved in the decision making process of the Abel committee this year, but […]

Distinguished Lecture Series III: Gregory Margulis, “Homogeneous dynamics and number theory III.”

In his final lecture, Prof. Margulis talked about some of the ideas around the theory of unipotent flows on homogeneous spaces, culminating in the orbit closure, equidsitribution, and measure classification theorems of Ratner in the subject.  Margulis also discussed the application to metric theory of Diophantine approximation which was not covered in the preceding lecture.

Distinguished Lecture Series II: Gregory Margulis, “Homogeneous dynamics and number theory II.”

Today, Prof. Margulis continued his lecture series, focusing on two specific examples of homogeneous dynamics applications to number theory, namely counting lattice points on algebraic varieties, and quantitative versions of the Oppenheim conjecture.  (Due to lack of time, the third application mentioned in the previous lecture, namely metric theory of Diophantine approximation, was not covered.)

Distinguished Lecture Series I: Gregory Margulis, “Homogeneous dynamics and number theory I.”

The final distinguished lecture series for the academic year here at UCLA is being given this week by Gregory Margulis, who is giving three lectures on “homogeneous dynamics and number theory”.  In his first lecture, Prof. Margulis surveyed some classical problems in number theory that turn out, rather surprisingly, to have more or less equivalent […]

Open thread for mathematicians on the immigration executive order

The self-chosen remit of my blog is “Updates on my research and expository papers, discussion of open problems, and other maths-related topics”.  Of the 774 posts on this blog, I estimate that about 99% of the posts indeed relate to mathematics, mathematicians, or the administration of this mathematical blog, and only about 1% are not […]

Avila, Bhargava, Hairer, Mirzakhani

The 2014 Fields medallists have just been announced as (in alphabetical order of surname) Artur Avila, Manjul Bhargava, Martin Hairer, and Maryam Mirzakhani (see also these nice video profiles for the winners, which is a new initiative of the IMU and the Simons foundation). This time four years ago, I wrote a blog post discussing […]

254B, Notes 3: Quasirandom groups, expansion, and Selberg’s 3/16 theorem

In the previous set of notes we saw how a representation-theoretic property of groups, namely Kazhdan’s property (T), could be used to demonstrate expansion in Cayley graphs. In this set of notes we discuss a different representation-theoretic property of groups, namely quasirandomness, which is also useful for demonstrating expansion in Cayley graphs, though in a […]

254B, Notes 2: Cayley graphs and Kazhdan’s property (T)

In the previous set of notes we introduced the notion of expansion in arbitrary -regular graphs. For the rest of the course, we will now focus attention primarily to a special type of -regular graph, namely a Cayley graph. Definition 1 (Cayley graph) Let be a group, and let be a finite subset of . […]

Course announcement: 254B, expansion in groups of Lie type

In the Winter quarter (starting on January 9), I will be teaching a graduate course on expansion in groups of Lie type.  This course will focus on constructions of expanding Cayley graphs on finite groups of Lie type (such as the special linear groups , or their simple quotients , but also including more exotic […]

254A, Notes 9: Applications of the structural theory of approximate groups

In the last set of notes, we obtained the following structural theorem concerning approximate groups: Theorem 1 Let be a finite -approximate group. Then there exists a coset nilprogression of rank and step contained in , such that is covered by left-translates of (and hence also by right-translates of ). Remark 1 Under some mild […]