Mar 27: Nikolay Romanovskii, Novosibirsk State University

Hilbert's Nullstellensatz in algebraic geometry over rigid solvable groups
Abstract: The classical Hilbert's Nullstellensatz says: if $K$ is algebraically closed field and there is a system of polynomial equations over $K$, $\{f_i (x_1, \ldots, x_n) = 0 \ | \ i \in I \}$, then an equation $ f (x_1, \ldots, x_n) = 0 $ is a logical consequence of this system (satisfies all the solutions of the system in $K^n$) if and only if some nonzero power of $f$ belongs to the ideal $ (f_i \ | \ i \in I) $ of the ring $ K [x_1, \ldots, x_n]. $ One can say say that we give an algebraic method for constructing all logical  consequences of the given system of equations: $f$ is obtained from $f_i \ (i \in I)$ using the operations of addition, subtraction, multiplication by elements of $K [x_1, \ldots, x_n]$, and extraction of roots.
Our approach to Hilbert's theorem in algebraic geometry over groups is as follows.
1. We should consider some good class of equationally Noetherian groups, let it be a hypothetical class $\mathcal{K}$.
2. In this class, we need to define and allocate an algebraically closed objects and to prove that any group of $\mathcal {K}$ is embedded into some algebraically closed group. Hilbert's theorem should be formulated and proved for algebraically closed in $\mathcal{K}$ groups.
3. Further, let $G$ be an algebraically closed group in $\mathcal{K}$. We think about equations over $G$ as about expressions $v=1$, where $v$ is an element of the coordinate group of the affine space $G^n$.
4. Since an arbitrary closed subset of $G^n$ is defined in general not by a system of equations, but by a positive quantifier free formula (Boolean combination without negations of a finite set of equations) we should consider as basic blocks not equations, but positive formulas.
5. We should specify and fix some set of algebraic rules of deduction on the set of positive formulas over $G$.
6. If the above conditions Hilbert's theorem will consist in a statement that all logical consequences of given positive formula over $G$ are exactly the algebraic consequences.
We realized this approach in algebraic geometry over rigid solvable groups.

Sep 4, Alexander Treyer (Omsk state University), 16:00, Room 5417

posted by NY Group Theory

 "Canonical and existentially closed groups for universal classes of abelian groups"
Abstract. The talk is based on joint work with A. Mishenko and V.Remeslennikov and devoted to universal classes of abelian groups. In our work we classify universal classes of abelian groups in terms of f.g. groups closed under discriminating operator. Also we introduce the principal universal classes of abelian groups and canonical groups for them. For arbitrary universal class K we describe the class of existentially closed groups relatively universal theory of class K and show that this class is axiomatizable. 

May 8: Andy Putman (Rice U.) Room 5417, 16:00-17:00

posted May 4, 2015, 4:09 AM by NY Group Theory

May 8: Andy Putman (Rice U.) TITLE: The stable cohomology of congruence subgroups
I will explain how to use representation-theoretic tools to understand patterns in the cohomology of congruence subgroups of SL(n,Z) and related groups.  This is joint work with Steven Sam.

May 1, Lee Mosher (Rutgers), Room 5417, 4:00-5:00

posted Apr 27, 2015, 7:18 AM by NY Group Theory

Title: Hyperbolic actions and second bounded cohomology for subgroups of
Out(F_n) (joint with Michael Handel)

Abstract: After surveying the co-evolution of the theories of hyperbolic actions and of second bounded cohomology of groups, we will report on recent progress on this topic for subgroups of Out(F_n).

April 29, 1-2 pm, Hunter College, E920, E. Plotkin (Bar Ilan U)

posted Apr 22, 2015, 6:20 AM by NY Group Theory

Kac-Moody Algebras

The talk is a short survey of recent developments in the area of word maps evaluated on groups and 
algebras. It is aimed to pose questions relevant to Kac-Moody theory.

April 24 NYGT Seminar, room 5417 4:15-5:00 – Eugene Plotkin (Bar Ilan U)

posted Apr 19, 2015, 9:00 AM by NY Group Theory   [ updated Apr 22, 2015, 6:07 AM ]

 “Equations over algebras: The logical geometry”

Our aim is to describe a uniform approach to treat equations over algebras and over models. The main goal is to present a machinery which allows us to extend geometry of algebraic sets over algebras to geometry of definable sets. 

April 23 Fairfield University, April 24 Graduate Center.

posted Apr 19, 2015, 8:56 AM by NY Group Theory

Conference  "Infinite Group Theory: From the Past to the Future" A conference celebrating the great achievements
in infinite group theory over the past two decades and in honor of the 70th birthdays of G. Rosenberger and D. Spellman

April 17 Bruce Kleiner, NYU, "Boundaries of hyperbolic groups"

posted Apr 14, 2015, 12:08 PM by NY Group Theory   [ updated Apr 14, 2015, 12:42 PM ]

Every hyperbolic group has a boundary at infinity, which is a compact
space on which the group acts by homeomorphisms.  The boundary carries additional canonical structure in the form of a family of (visual) metrics; in the case of lattices in Lie groups, this metric structure has played a key role in many results, including Mostow's rigidity theorem.
After reviewing the background, the lecture will discuss some further applications of this structure, and related open problems. 

Mar. 20: Alex Taam, Graduate Center CUNY

posted Mar 15, 2015, 6:59 AM by NY Group Theory

 Effective JSJ decompositions of maximal $/Gamma$-limit quotients
Abstract: Analogous to the decomposition of irreducible orientable closed 3-manifolds along incompressible tori from which the name is borrowed, a JSJ decomposition of a group can provide a canonical description of possible group splittings. I will define a JSJ decomposition for the class of finitely generated fully residually $\Gamma$ groups (i.e. $\Gamma$-limit groups), where $\Gamma$ is a fixed torsion-free hyperbolic group. Furthermore, given a group $G$ which is the coordinate group of a finite system of equations over $\Gamma$, I will show that there is an algorithm to find such decompositions of each maximal (with respect to a natural partial order) $\Gamma$-limit group which is a quotient of $G$. This is joint work with O. Kharlampovich and A. Myasnikov.

Mar.13, 16:00-17:00 A. Nikolaev (Stevens Institute)

posted Mar 7, 2015, 6:27 AM by NY Group Theory   [ updated Mar 7, 2015, 6:27 AM ]

Logspace and compressed word computation in finitely generated nilpotent groups.

Abstract: Algorithmic problems in nilpotent groups have been extensively
studied, but few of the algorithms came with robust estimates of
computational complexity. In this talk we present some of the recent
results on the computation of normal forms, the membership problem,
the conjugacy problem, and computation of presentations for subgroups
in nilpotent groups. We show that these problems are solvable using
only logarithmic space and, simultaneously, in quasilinear time. We
also give polynomial time solutions to compressed-word versions of
these problems, in which each input word is provided as a
straight-line program. Time permitting, we will touch on other
applications of our techniques, such as finite separability questions,
distortion of embeddings, and solutions to other algorithmic problems.
This is a joint work with J.Macdonald, A.Myasnikov, S.Vassileva.

March 6, 16:00-17:00, A. Ushakov (Stevens)

posted Mar 1, 2015, 1:38 PM by NY Group Theory

Title: Magnus embedding and algorithmic properties of groups F/N^(d)

Abstract. Let N be a normal subgroup of a free group F. By N' we denote
the derived subgroup of N, and in general by N^(d) the dth derived subgroup.
In my talk I will discuss relations between the following algorithmic problems
for groups F/N^(d): word problem, power problem (membership into cyclic
subgroups), and conjugacy problem.

Joint work with Funda Gul and Mahmood Sohrabi.

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