Hilbert's Nullstellensatz in algebraic geometry over rigid solvable groupsAbstract: 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. |

### Mar 27: Nikolay Romanovskii, Novosibirsk State University

#### Nov 13, Ruth Charney (Brandeis U.), Room 5417, 16:00-17:00

Random Graph Products of Cyclic Groups For a finite graph $\Gamma$, the right-angled Artin group $A_\Gamma$ is the group generated by the vertices of $\Gamma$ with relations given by commutators of adjacent vertices. More generally, if we also allow (all or some) of the generators to have finite order, we get a collection of groups $G_\Gamma$ known as graph products of cyclic groups. In random group theory, one asks what the probability is that a randomly chosen group will satisfy a given property. While the notion of a ``random group" usually involves fixing a generating set and randomly choosing relators, in the case of graph products, the natural model to consider is a group $G_\Gamma$ associated to a random graph in the sense of Erdos-Renyi. In this talk I will review the basics of random graph theory and survey applications to graph products of cyclic groups and their automorphism groups, including my joint work with M. Farber, as well as subsequent work by various other authors. |

#### Nov 6 Aner Shalev (Hebrew U.), Joint meeting with GRECS seminar, GC 5417, 4pm-5pm

Words and generarion in linear groups Abstract: We will discuss two recent results showing that finitely generated linear groups with certain properties are virtually solvable. The first one (joint with Kantor and Lubotzky) concerns invariable generation by a finite subset. The second one (joint with Larsen) concerns probabilistic identities. We will deduce the following probabilistic Tits alternative: a finitely generated linear group is either virtually solvable or almost all n-tuples in its profinite completion generate a free subgroup of rank n. Related open problems will also be discussed. |

#### Oct 30 Alex Gamburd (Grad Center), 16:00-17:00, Room 5417

Markoff Triples and Strong Approximation. Abstract: Markoff triples are integer solutions to Markoff equation $x^2+y^2+z^2=3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond. We will review some of these (in particular connection to Product Replacement Graphs) and will then discuss recent joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo $p$ under the action of the group generated by Vieta involutions. We show in particular that for almost all prime the induced graph is connected. Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite. |

#### Oct 23, J. Behrstock (Lehman College, CUNY), 16:00-17:00, Room 5417

Title: Random graphs and applications to Coxeter groups. Abstract: Erdos and Renyi introduced a model for studying random graphs of a given "density" and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected. Motivated by ideas in geometric group theory we will explain some new threshold theorems we have discovered for random graphs. We will then, explain applications of these results to the geometry of Coxeter groups. Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen, Susse, and Falgas-Ravry. |

#### Oct 16, I. Lysenok (Stevens Inst.), 16:00 Room 5417

Towards optimization of the Novikov-Adian exponent Abstract: The Novikov-Adian theorem states that a non-cyclic Burnside group B(m,n) of odd exponent n greater or equal 665 is infinite. I will discuss ideas behind different known proofs of infiniteness of groups B(m,n) as well as possibility of a substantional reduction of the lower bound 665. |

#### Oct 9 Lisa Carbone (Rutgers University), 4 p.m., Room 5417

Title: Finite presentations of hyperbolic Kac-Moody groups Abstract: Tits defined Kac--Moody groups over commutative rings, providing infinite dimensional analogues of the Chevalley-Demazure group schemes. Tits' presentation can be simplified considerably when the Dynkin diagram is hyperbolic and simply laced. Over finitely generated rings R, we give finitely many generators and defining relations parametrized over R and we describe a further simplification for R=Z. We highlight the role of the group E10(R), conjectured to play a role in the unification of superstring theories. |

#### Oct. 2 R. Gilman (Stevens Institute), 16:00, Room 5417

Title: How to Sample Hard Instances of the Word problem. Abstract: It is well known that there are finitely presented groups with unsolvable word problem. Any (partial) algorithm for the word problem of such a group must fail on infinitely many instances. Nevertheless it can be hard to find these instances. In this talk we exhibit a finitely presented group whose hard instances can be sampled in linear time. |

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

"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

May 8: Andy Putman (Rice U.) TITLE: The stable cohomology of congruence subgroupsABSTRACT: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. |