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Bms friday Colloquium

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BMS Friday Colloquium

                     Friday 2 December 2011 at 14:15
                     Tea before the lecture begins at 13:00

                     BMS Loft, Urania, An der Urania 17, 10787 Berlin




                     Igor Rivin
                     (Temple University)

                     Probability, Geometry and Algorithms of Matrix
                     Groups
                     Igor Rivin will start with about the simplest situation imaginable:
                     Consider a collection C of invertible matrices (you can assume the
                     entries are integers), and consider the group G(C) they generate.
                     In this talk he will discuss some naive (and not so naive) questions
                     one could ask about G(C), and the sort of answers one might get, or
                     hope to get. Here is a selection of the questions:

                     1. What is G(C) as an abstract group?
                     2. How does G(C) sit in the ambient matrix group? For example, if
                        all we know is that the matrices are 3 × 3 invertible matrices with
                        integer entries, the natural candidates for the ambient group are
                        SL(3, Z or SL(3, R
                              Z)         R).
                     3. What can we say about the generic element of G(C)? For example,
                        what do its eigenvalues look like? What can we say about its cha-
                        racteristic polynomial?

                     It turns out that all of the questions above are intimately connected
                     to central questions in number theory, group theory, probability
                     theory, combinatorics, and, not least, geometry. A lot of the tech-
                     niques have been developed only in the last few years, and our
                     understanding is still quite limited. Igor Rivin will give a (very sub-
                     jective) snapshot of what he knows, and what sort of techniques are
                     being used.

                     Igor Rivin is Professor of Mathematics at Temple University. He is a
                     world-renowned mathematician, whose contributions include solu-
                     tions of long-outstanding problems in geometry, probability theo-
                     ry, combinatorics, dynamics, and other fields. From September to
                     December 2011, Igor Rivin holds the BMS Professorship.
www.math-berlin.de

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Bms friday Colloquium

  • 1. BMS Friday Colloquium Friday 2 December 2011 at 14:15 Tea before the lecture begins at 13:00 BMS Loft, Urania, An der Urania 17, 10787 Berlin Igor Rivin (Temple University) Probability, Geometry and Algorithms of Matrix Groups Igor Rivin will start with about the simplest situation imaginable: Consider a collection C of invertible matrices (you can assume the entries are integers), and consider the group G(C) they generate. In this talk he will discuss some naive (and not so naive) questions one could ask about G(C), and the sort of answers one might get, or hope to get. Here is a selection of the questions: 1. What is G(C) as an abstract group? 2. How does G(C) sit in the ambient matrix group? For example, if all we know is that the matrices are 3 × 3 invertible matrices with integer entries, the natural candidates for the ambient group are SL(3, Z or SL(3, R Z) R). 3. What can we say about the generic element of G(C)? For example, what do its eigenvalues look like? What can we say about its cha- racteristic polynomial? It turns out that all of the questions above are intimately connected to central questions in number theory, group theory, probability theory, combinatorics, and, not least, geometry. A lot of the tech- niques have been developed only in the last few years, and our understanding is still quite limited. Igor Rivin will give a (very sub- jective) snapshot of what he knows, and what sort of techniques are being used. Igor Rivin is Professor of Mathematics at Temple University. He is a world-renowned mathematician, whose contributions include solu- tions of long-outstanding problems in geometry, probability theo- ry, combinatorics, dynamics, and other fields. From September to December 2011, Igor Rivin holds the BMS Professorship. www.math-berlin.de