אירועים מיוחדים Ical Atom

Jurij Volcic (BGU) will give three lectures on the theory of universal skew fields of fractions due to P.M. Cohn, a beautiful and powerful tool of noncommutative algebra little known outside of a small circle of cognoscenti. Everybody is invited (the lectures should be accessible for graduate students).

Time and place:

Tuesday Feb. 6, Thursday Feb. 15, and Sunday Feb. 18 at 11:30-13:00 in Room -101

Abstract:

As opposed to the commutative setting, localization in noncommutative rings is a much more intricate topic, accompanied with various subtle obstacles and at first sight counter-intuitive phenomena. An exceptional contribution to this field has been made by Paul Moritz Cohn (1924-2006) with the introduction of universal skew fields of fractions in the 70’s, and his subsequent development of this theory. In the self-contained course of three talks I will try to present some of Cohn’s core results with the view towards the universal skew field of fractions of a free algebra.

In the first lecture we will consider the general problem of embedding rings into skew fields, and early difficulties accompanying it. Following Cohn we will then introduce the category of epic skew fields and specializations assigned to a fixed ring. This will allow us to talk about universal skew fields of fractions and to draw parallels with the commutative theory.

In the second lecture we will explain the interplay between specializations, singular kernels, matrix ideals in matrix representation of fractions. Our aim will be to determine necessary and sufficient conditions for a ring to admit a (universal) skew field of fractions.

In the last lecture we will finally look at a concrete family of rings admitting universal skew fields of fractions: free ideal rings (firs), among which are free algebras, and their generalizations. We will trace down their distinguishing property that fulfills the sufficient condition from the previous lecture.

All faculty and students are cordially invited.

Program

18:00 Undergraduate math club

Barak Weiss, Tel Aviv University

The illumination problem

The following elementary problem in geometry is still open: given a polygon $P$ in the plane, say that points $x$ and $y$ in $P$ see each other if there is a billiard path from $x$ to $y$. Is there a polygon in which infinitely many pairs of points do not see each other?

Such problems turn out to be easy to state but very difficult to solve. I will explain this and related questions in greater detail, and describe some recent progress which relies on well-known work of Eskin and Mirzakhani (part of the late Maryam Mirzakhani’s Field medal citation).

19:00 A Jazz show by Tsachik and the perverse sheaves

Ehud Ettun on bass

Haruka Yabuno on Piano

Tsachik Gelander on drums

הנכם מוזמנים לכנס חפירות במתמטיקה השני, אשר יתקיים באוניברסיטת בן-גוריון ביום ה’ ה-13.7.17.

יהיו הרצאות מרתקות במגוון נושאים, תוכלו לפגוש ולהכיר אנשים שמתעניינים במתמטיקה, וכמובן – לחפור!

There will be five main Mini-Courses as follows:

1. Higher rank lattices rigidity and arithmeticity, Tsachik Gelander (Weizmann).
2. Exotic lattices, Pierre-Emmanuel Caprace (U.C. Louvin).
3. Strong approximation in linear groups, Nikolay Nikolov (Oxford).
4. Arithmetic hyperbolic 3-manifolds, Nicolas Bergeron (Jussieu)
5. The congruence subgroup property, Nir Avni (Northwestern).

Speaker: Anton Khoroshkin (HSE, Moscow)

Little discs operad, graph complexes and applications

The little cubes operad is one of the key objects of the deformation theory of
the last decades. I will outline its geometric definition, some combinatorial
models and different known results and applications.

The framed little cubes operad is much less studied and I will report on some
recent results of related deformation theory, explain relationships with
Deligne-Mumford compactifications of stable curves of zero genus and use it to
prove some conjectures stated for braid and cacti groups whose representations
appear naturally as tensor products in tensor and coboundary categories
correspondingly.

Dynamics of Spacetime — Einstein’s equations as a geometric flow.

Speaker: Dr. David Fajman (Vienna)

The interpretation of Einstein’s equations as a geometric flow (the Einstein flow) allows to study the evolution of spacetimes from a dynamical point of view. Two types of initial data are mainly considered: Firstly, asymptotically flat data describing initial states of isolated self-gravitating systems and secondly, data on closed manifolds describing initial states for cosmological spacetimes. Studying the evolution of data under the flow we aim to understand its long-time behavior and the global geometry of its time-development. We are interested in the construction of static solutions (or static up to a time-rescaling) as potential attractors of the flow and their nonlinear stability, completeness and incompleteness properties of spacetimes and singularity formation. We present new methods to construct and study solutions by geometric and analytical tools as well as several results in the directions mentioned above. We consider in particular the case of matter models coupled to the Einstein equations, which turns out to provide several interesting phenomena and new classes of solutions.

Speakers

• Christian MAIRE – University of Franche-Comté (France)
• Eliyahu MATZRI – Bar-Ilan University (Israel)
• Thomas WEIGEL – University of Milano-Bicocca (Italy)

Program

11:30 -11:40

Opening with Prof. M. Cohen

11:40 -12:40

C. Maire - Infinite extensions of number fields, analytic pro-p groups and fixed points

12:45 -13:45

Lunch - offered to all participants

14:00 -15:00

E. Matzri - One sided genus of a central simple algebra

15:00 -15:30

Coffee break

15:30 -16:30

T. Weigel - Koszul pro-p groups

The School will focus on Polish Groups. General theory, examples, representations of discrete groups with Polish targets, and applications to discrete groups, geometry and dynamics. The three main courses will be:

Mini-Courses

1. General theory of Polish groups Jullien Melleray, Christian Rosendal and Todor Tsankov.
2. Algebraic groups over Polish fields. Jean Lécureux and Bertrand Rémy.
3. Infinite dimensional Lie groups and their symmetric spaces Bruno Duchesne.

In honor of Prof. Mikhail Klin, on the occasion of his retirement

Speaker: Antoine Ducros (Paris 6)

Title: Stability of Gauss valuations

Abstract:

A valued field $(k,|.|)$ is said to be stable (this terminology has no link with model-theoretic stability theory) if every finite extension $L$ of $k$ is defectless, i.e., satisfies the equality $\sum e_vf_v=[L:k]$, where $v$ goes through the set of extensions of $|.|$ to $L$, and where $e_v$ and $f_v$ are the ramification and inertia indexes of $v$. The purpose of my talk is to present a new proof (which is part of current joint reflexions with E. Hrushovski and F. Loeser) of the following classical fact (Grauert, Kuhlmann, Temkin,…) : let $(k,|.|)$ be a stable valued field, and let $(r_1,\dots,r_n)$ be elements of an ordered abelian group $G$ containing $|k^*|$. Let $|.|’$ be the $G$-valued valuation on $k(T_1,\dots,T_n)$ that sends $\sum a_I T^I$ to $\max_I |a_I|\cdot r^I$. Then $(k(T_1,\dots,T_n),|.|’)$ is stable too.

Our general strategy is purely geometric, but the proof is based upon model-theoretic tools coming from model theory (which I will first present; no knowledge of model theory will be assumed). In particular, it uses in a crucial way a geometric object defined in model-theoretic terms that Hrushovski and Loeser attach to a given $k$-variety $X$, which is called its stable completion; the only case we will have to consider is that of a curve, in which the stable completion has a very nice model-theoretic property, namely the definability, which makes it very easy to work with.

Joint Operator Algebras and Operator Theory Seminar, bringing together people from Ben-Gurion University, the Technion, Tel Aviv University and the University of Haifa