University of California, San Diego
January 19-20, 2013

Organizers: Dan Rogalski, Alireza Salehi Golsefidy, Efim Zelmanov.
Funded by NSF and Alfred Sloan Foundation.


       Misha Kapovich (UC, Davis)
       Bert Kostant (MIT)
       Alex Lubotzky (Hebrew University)
       Amir Mohammadi (University of Texas at Austin)
       Vera Serganova (UC, Berkeley)
       Terry Tao (UCLA)
       Nolan Wallach (UC, San Diego)


      Saturday, January 19                                 Sunday,    January 20                           
  9:30-10:30 Light breakfast   9:00-10:00 Light breakfast
10:30-11:30 Nolan Wallach Ricci flow on Wallach flag varieties 10:00-11:00 Misha Kapovich Noncoherence of arithmetic groups  
11:30-13:00 Lunch/Conversation 11:00-13:00 Lunch/Conversation
13:00-14:00 Bert Kostant On the algebraic set of singular elements in a complex simple Lie algebra 13:00-14:00 Amir Mohammadi Unipotent flows and infinite measures
14:30-15:30 Vera Serganova On the Kostant theorem for Lie superalgebras 14:00-14:30 Tea time
15:30-16:30 Tea time 14:30-15:30 Alex Lubotzky Arithmetic groups, Ramanujan graphs and error correcting codes
16:30-17:30 Terry Tao Hilbert's fifth problem and approximate groups
19:00- Conference dinner Sammy's pizza  (Del Mar highlands town center)

Abstracts (and related PDF files)

Misha Kapovich
      Title:  Noncoherence of arithmetic groups (Here is the related PDF file.)
      Abstract:   It follows from the work of Borel and Serre on bordification of locally-symmetric spaces that all arithmetic groups have finite type. This, of course, does not extend to subgroups of arithmetic lattices in semisimple Lie groups, since these lattices always contain free subgroups of infinite rank. A group is called coherent if all its finitely-generated subgroups are also finitely-presented. In the talk I will discuss coherence and non-coherence of arithmetic lattices in semisimple Lie groups.

Bert Kostant
      TitleOn the algebraic set of singular elements in a complex simple Lie algebra.
      Abstract:  Let G be a complex simple Lie group and let g = Lie G. Let S(g) be the G-module of polynomial functions on g and let Sing(g) be the closed algebraic cone of singular elements in g. Let L ⊂ S(g) be the (graded) ideal defining Sing(g) and let 2r be the dimension of a G-orbit of a regular element in g. Then Lk = 0 for any k < r. On the other hand, there exists a remarkable G-module M ⊂ Lr which already defines Sing(g). The main results of this paper are a determination of the structure of M.

Alex Lubotzky
      TitleArithmetic groups, Ramanujan graphs and error correcting codes (Here is the related PDF file.)
      Abstract:   While many of the classical codes are cyclic, a long standing conjecture asserts that there are no `good' cyclic codes. In recent years, interest in symmetric codes has been stimulated by Kaufman, Sudan, Wigderson and others (where symmetric means that the acting group can be any group). Answering their main question (and contrary to common expectation), we show that there DO exist symmetric good codes. In fact, our codes satisfy all the "golden standards" of coding theory. Our construction is based on the Ramanujan graphs constructed by Lubotzky-Samuels-Vishne as a special case of Ramanujan complexes. The crucial point is that these graphs are edge transitive and not just vertex transitive as in previous constructions of Ramanujan graphs. These complexes are obtained as quotients of the Bruhat-Tits building modulo the action of suitable arithmetic groups. We will discuss the potential of these complexes and their cohomology to yield more applications to coding theory. All notions will be explained. Joint work with Tali Kaufman.

Amir Mohammadi
      TitleUnipotent flows and infinite measures
      Abstract:  Unipotent flows on homogeneous spaces obtained as a quotient of a Lie group by a lattice have been an important subject of study in the past four decades or so. Recently some developments have been made towards the study of unipotent flows on homogeneous spaces preserving a geometric but infinite measure; in this talk we will highlight the similarities and main differences between these two settings. This talk is based on a joint work with H. Oh.

Vera Serganova
      TitleOn the Kostant theorem for Lie superalgebras
      Abstract:  The classical result of Kostant that the algebra of endomorphisms of the Whitaker module coincides with the center of the universal enveloping algebra does not hold in the supercase. We formulate a conjecture for simple Lie superalgebras, discuss in detail the case of the Lie superalgebra q(n) and connection with Yangians.

Terry Tao
      TitleHilbert's fifth problem and approximate groups
      Abstract:  Approximate groups are, roughly speaking, finite subsets of groups that are approximately closed under the group operations, such as the discrete interval {-N,...,N} in the integers. Originally studied in arithmetic combinatorics, they also make an appearance in geometric group theory and in the theory of expansion in Cayley graphs.

Hilbert's fifth problem asked for a topological description of Lie groups, and in particular whether any topological group that was a continuous (but not necessarily smooth) manifold was automatically a Lie group. This problem was famously solved in the affirmative by Montgomery-Zippin and Gleason in the 1950s.

These two mathematical topics initially seem unrelated, but there is a remarkable correspondence principle (first implicitly used by Gromov, and later developed by Hrushovski and Breuillard, Green, and myself) that connects the combinatorics of approximate groups to problems in topological group theory such as Hilbert's fifth problem. This correspondence has led to recent advances both in the understanding of approximate groups and in Hilbert's fifth problem, leading in particular to a classification theorem for approximate groups, which in turn has led to refinements of Gromov's theorem on groups of polynomial growth that have applications to the study of the topology of manifolds. We will survey these interconnected topics in this talk.

Nolan Wallach
      Title Ricci flow on Wallach flag varieties (Here is the related PDF file.)
      Abstract:  "Wallach flag varieties" consist of the manifold of flags in 2 dimensional projective space over the complexes, the quaternions and the octonions. We study the effect of Ricci flow on the Sectional and Ricci curvature extending and sharpening work of Bohm and Wilking. This is joint work with Man Wai (Mandy) Cheung.

    All talks will be held in Mathematics Department Room 6402 on the UCSD campus.
    The refreshments will be served in the same building Room 7356 .
    Click on campus map to locate mathematics department (the AP&M building is the red building on the map.)
    Here are directions to the UCSD campus and mathematics department (AP&M building).

    Parking on the UCSD campus is free during the weekend. You may park in any of the parking lots next to the AP&M building.
    Please do not park in the "reserved" or "A parking only" spaces.

    You can find Hotel information here. Residence Inn and Sheraton La Jolla are in walking distance. Also, the shuttle to UCSD from the Del Mar Inn does not run during weekends. So anyone staying there should either have a car, arrange to get picked up by someone, or else it is possible to take the 101 bus. This runs every half hour in the mornings on Sat/Sun, but becomes every hour after about 6pm and ends early (the last bus north from campus is about 9:30pm).

Registration and funding
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