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 18. Rachel Skipper (Binghamton University)

Title: The congruence subgroup problem for a family of branch groups Abstract: A group, G, acting on a regular rooted tree has the congruence subgroup property (CSP) if every subgroup of finite index contains the stabilizer of a level of the tree. When the subgroup structure of G resembles that of the full automorphism group of the tree, additional tools are available for determining if G has the CSP. In this talk, we look at the Hanoi towers group which has fails to have the CSP in a particular way. Then we will generalize this construction to a new family of groups and discuss the CSP for them. |

#### Friday Nov 11 Shamgar Gurevich (Madison and Yale)

Title: Small Representations of finite classical groups.Abstract: Suppose you have a finite group G and you want to study certain related structures (random walks, expander graphs, word maps, etc.). In many cases, this might be done using sums over the characters of G. A serious obstacle in applying these formulas seemed to be lack of knowledge over the low dimensional representations of G. In fact, the “small" representations tend to contribute the largest terms to these sums, so a systematic knowledge of them might lead to proofs of some important conjectures. The “standard" method to construct representations of finite classical group is due to Deligne and Lusztig (1976). However, it seems that their approach has relatively little to say about the small representations. This talk will discuss a joint project with Roger Howe (Yale), where we introduce a language to define, and a new method for systematically construct, the small representations of finite classical groups. I will demonstrate our theory with concrete motivations and numerical data obtained with John Cannon (MAGMA, Sydney) and Steve Goldstein (Scientific computing, Madison). |

#### Nov 4, Denis Ovchinnikov (Stevens Institute)

Title: Random nilpotent groups. Notion of a random (finitely presented) group gives a well-established approach to think about the question "what most groups look like". In standard models (few relator and density models), with probability 1 (or, more precisely, asymptotically almost surely), these groups turn out to be either hyperbolic or trivial. This way, if one desires to study the question "what most groups in class N look like", for some class of non-hyperbolic groups N, classical model cannot be applied directly. I will discuss the question above for the class of (finitely generated) nilpotent groups, provide an outline of known approaches to define a random nilpotent group, and present our results about typical groups in some of these models. The talk is based on joint work with Albert Garreta-Fontelles and Alexei Miasnikov. |

#### Friday, Oct 28, 16:00-17:00, Room 5417 Dima Savchuk (University of South Florida)

Title: Lamplighter groups and (bi)reversible automata from affine transformations of $\mathbb Z_p[[t]]$.Abstract: The ring $\mathbb Z_p[[t]]$ of formal power series over $\Z_p$ can be naturally identified with the boundary of $p$-ary rooted tree $T_p$. For each $a(t),b(t)\in\mathbb Z_p[[t]]$ with $b(t)$ being a unit, we consider the affine transformations of $\mathbb Z_p[[t]]$ defined by $f(t)\mapsto a(t)+f(t)\cdot b(t)$. This transformation defines automorphisms of $T_p$ that can be explicitly described by an automaton that is finite if and only if both $a(t)$ and $b(t)$ are rational power series. We prove that the multiplication by a power series corresponding to a rational function $p(t)/q(t)\in\mathbb Z_p(t)$ is defined by a finite automaton that is reversible if and only if $\deg p\leq \deg q$. In particular, if $\deg p=\deg q$ the corresponding automaton is bireversible. This covers several examples that were studied earlier. We also describe algebraic structure of corresponding self-similar groups generated by such automata and show that they are isomorphic to lamplighter groups of various ranks. This is a joint result with Ievgen Bondarenko. |

#### Friday, Oct 21, Pascal Weil, CNRS and Université de Bordeaux

The study of random algebraic objects sheds a different light on these objects, which complements the algebraic, but also the algorithmic points of view. I will discuss random finitely generated subgroups of free groups from several perspectives: when they are given by a random tuple of generators (of reduced words), and when they are given by a random Stallings graph. The Stallings graph of a subgroup H is a finite labeled graph uniquely associated with H, from which one can efficiently compute invariants of H. It is an interesting combinatorial object in and of itself, whose structure must be understood to enumerate and randomly generate Stallings graphs and subgroups. While both approaches to random subgroups, generators and Stallings graphs, are natural, they yield different distributions, and a different view of what ‘most subgroups’ look like. |

#### Sep 30, Alexei Miasnikov (Stevens Institute), Room 5417, 16:00-17:00

“What the group rings know about the groups?” How much information about a group G is contained in the group ring K(G) for an arbitrary field K? Can one recover the algebraic or geometric structure of G from the ring? Are the algorithmic properties of K(G) similar to that of G? I will discuss all these questions in conjunction with the classical Kaplansky-type problems for some interesting classes of groups, in particular, for limit, hyperbolic, and solvable groups. At the end I will touch on the solution to the generalized 10 |

#### Sep 23, Ben Steinberg, 16:00-17:00, Room 5417

Title: Homological and topological finiteness conditions for monoids Abstract: Homological and topological finiteness properties of groups has long been of interest in connection with topology. Interest in homological finiteness conditions for monoids began with the Anick-Squier-Groves-Kobayashi theorem which says that a monoid with a finite complete rewriting system is of type $FP_{\infty}$. Starting in the early nineties Pride, Otto, Kobayashi and Guba began to investigate homological finiteness properties of monoids in connection with complete rewriting systems (there is also some work of Ivanov and of Sapir). In group theory, one normally studies homological properties via topology by using Eilenberg-MacLane spaces. For monoids, the work has been almost entirely algebraic in nature and for this reason progress on understanding finiteness conditions for such basic operations as free product with amalgamation has been slow. In this talk, we introduce the topological finiteness condition $F_n$ for monoids. It extends the usual notion for groups and seems to be surprisingly robust. We can then extend Ken Brown's topological proof of the Anick-Squier-Groves-Kobayashi theorem to monoids and we have made new progress on understanding finiteness properties of amalgamations, HNN extensions and HNN-like extensions (in the sense of Otto and Pride). In the process we develop some very rudimentary Bass-Serre theory for monoids. This is joint work with Bob Gray. |

#### May 13. E. Zelmanov (U. San Diego), 16:00-17:00, Science Center, room 4102

Groups with Identities. Abstract. We will discuss groups satisfying pro-p and prounipotent identities : examples, theory and possible applications. |