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. |
Mar 27: Nikolay Romanovskii, Novosibirsk State University
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. |
May 6 M. Bestvina (Utah), 16:00-17:00, Room 5417
Title: On the Farrell-Jones conjecture for mapping class groups Abstract: I will try to describe what the Farrell-Jones conjecture is about, and how one goes about proving it. Then I will try to outline a proof of FJC for mapping class groups, which is work in progress, joint with Arthur Bartels. |
Apr.29 L. Babai (U. of Chicago), Science Center (4102), 16:00-18:00
Title: A little group theory goes a long way: The group theory behind recent progress on the Graph Isomorphism problem Abstract: One of the fundamental computational problems in the complexity class NP on Karp's 1973 list, the Graph Isomorphism problem (GI) asks to decide whether or not two given graphs are isomorphic. While program packages exist that solve this problem remarkably efficiently in practice (McKay, Piperno, and others), for complexity theorists the problem has been notorious for its unresolved asymptotic worst-case complexity: strong theoretical evidence suggests that the problem should not be NP-complete, yet the worst-case complexity has stood at $\exp(O(\sqrt{v\log v}))$ (E. M. Luks, 1983) for decades, where $v$ is the number of vertices. By addressing the bottleneck situation for Luks's algorithm, we recently reduced this ``moderately exponential'' upper bound to quasipolynomial, i.e., $\exp((\log v)^c)$. The problem we actually solve in quasipolynomial time is more general: we solve the String Isomorphism problem (SI), introduced by Luks in his seminal 1980/82 paper in which he brough in-depth applications of group theory to bear on the GI and SI problems. The input to an instance of SI is a permutation group $G$ acting on a set $\Omega$ of $n$ elements, and a pair of strings, $x$ and $y$, over $\Omega$ (functions that map $\Omega$ to a finite alphabet). The question is, does there exist a permutation in $G$ that transforms $x$ into $y$ (``anagrams under group action''). ($G$ is given by a list of generators.) As Luks pointed out, this problem is polynomial-time equivalent to the Coset Intersection problem: given two subcosets of the symmetric group $S_n$, decide whether or not their intersection is empty. The following group theoretic lemma is at the heart of the new algorithm. Let $G$ be a permutation group of degree $n$ and $f$ an epimorphism of $G$ onto $A_k$, the alternating group of degree $k$. We say that a point $p$ in the permutation domain on which $G$ acts is _affected_ by $f$ if the stabilizer $G_p$ is mapped by $f$ to a proper subgroup of $A_k$. Let $U$ be the set of unaffected points and let $H$ be the pointwise stabilizer of $U$ in $G.$ Unaffected Stabilizers Lemma. If $k > \max\{8, 2+log_2 n\}$ then $f$ maps $H$ onto $A_k.$ In the talk we outline the proof of this result and try to convey the basic idea, how, through the Lemma, the affected/unaffected dichotomy plays a central role in the design and analysis of the algorithm. The proof of the lemma is elementary with reference to Schreier's Hypothesis that the outer automorphism group of every finite simple group is solvable. Schreier's Hypothesis follows from the Classification of Finite Simple Groups (CFSG). Under the slightly stronger assumption that $k > (log n)^c$ for some constant $c$, Laszlo Pyber recently announced a CFSG-free proof of the result. The paper is available at arXiv:1512.03547. Helpful reading: E.M. Luks : Isomorphism of graphs of bounded valence can be tested in polynomial time. J. Comp. Sys. Sci., 25:42--65, 1982. |
M. Volkov (Ural Federal University and Hunter College CUNY), April 15, 16:00-17:00
Title: Algebraic properties of monoids of diagrams and 2-cobordisms. Abstract: Partition of diageram monoids first appeared in 1937 in a paper by Brauer in which they serve as vector space bases of certain associative algebras relevant in representation theory of classical groups. Other species of diagram monoids were invented by Temperley and Lieb in the context of statistical mechanics in the 1970s and by Kauffman and Jones in the context of knot theory in the 1980s. Since then diagram monoids have revealed many other connections, e.g., with low-dimensional topology, topological quantum field theory, quantum groups etc. Recently, they have been intensively studied as purely algebraic objects, and these studies have shown that diagram monoids are quite interesting from this viewpoint as well. In the talk, we first present geometric definitions for some classes of infinite diagram monoids and then survey our results on the finite basis problem for their identities. Whilst it is not clear whether or not a study of the identities of infinite diagram monoids may be of any use for any of their non-algebraic applications, such a study has constituted an interesting challenge from the algebraic viewpoint and required to develop new techniques. We also report on a recent application of these new techniques to the finite basis problem for the identities of monoids of 2-cobordisms. |
M. Hagen (U. of Cambridge), Friday April 8, 16:00-17:00, Room 5417
Title: Curve complexes for cube complexes Abstract: I'll discuss a hyperbolic space -- the "contact graph" -- associated to a CAT(0) cube complex. Using the example of a right-angled Artin group, I'll illustrate how the contact graph can help one understand the large-scale geometry of a CAT(0) cube complex in very much the same way that the curve graph of a surface can, by work of Masu-Minsky, be used to understand the geometry of the mapping class group. This is joint work with Jason Behrstock and Alessandro Sisto. |
Apr.1 A. Myropolska ( University of Paris-Sud) Room 5417, 16:00-17:00
Title: Nielsen and Andrews-Curtis equivalence in finitely generated groups Abstract: Various aspects of geometric group theory lead to the study of the natural action of Aut(F_n) on the set Epi(F_n, G) of generating n-tuples of a group G generated by at least n elements. One of the main questions, raised in the context, is the transitivity of this actions. In the talk, we will give an introduction to the subject, then extensively discuss its relation to the Andrews-Curtis conjecture and explain the transitivity results for the class \emph{MN} of finitely generated groups of which every maximal subgroup is normal (this includes nilpotent groups and Grigorchuk-like groups). |
March 18, J. Maher (CUNY), Room 5417, 4:00-5:00 pm
Title: Random walks on weakly hyperbolic groups Abstract: Let G be a group acting by isometries on a Gromov hyperbolicspace, which need not be proper. If G contains two hyperbolic elements with disjoint fixed points, then we show that a random walk on G converges to the boundary almost surely. This gives a unified approach to convergence for the mapping class groups of surfaces, Out(F_n) and acylindrical groups. This is joint work with Giulio Tiozzo. |
March 11, I. Lysenok (Stevens), 16:00-17:00, Room 5417
Quadratic equations in free metabelian groups. |
March 4, Mahmood Sohrabi (Stevens), Room 5417, 4:00-5:00
Title: On groups elementarily equivalent to T(n,R) Abstract: In this talk I will present my joint results with Alexei Miasnikov on the structure of a group elementarily equivalent to the solvable group T(n,R) of all invertible upper triangular nxn matrices over a characteristic zero integral domain where n is greater than or equal to 3. In particular, I will talk about necessary and sufficient conditions for a group being elementarily equivalent to T(n,R) where R is a characteristic zero algebraically closed field, a real closed field, a number field, or the ring of integers of a number field. Time permitting I'll also present a new way to look at Belegradek's work on elementary theory of the group UT(n,R) of upper unitriangular matrices via lattice groups and Lie algebras. |
Feb 26 Khalid Bou-Rabee (CCNY), 4:00-5:00 pm, Room 5417
Title: The topology of local commensurability graphs Abstract: Let $G$ be a group and $p$ a prime number. The $p$-local commensurability graph of $G$ has all finite-index subgroups of $G$ as vertices, and edges are drawn between $A$ and $B$ if $[A : A\cap B][B : A\cap B]$ is a power of $p$. In this talk, I will talk on recent work that draws group-theoretic properties from the diameters of components of these graphs. The connected diameter of a graph is the supremum over all graph diameters of every component of the graph. Every $p$-local commensurability graph of any nilpotent group has connected diameter equal to one, and the reverse implication is true. In contrast, every nonabelian free group has connected diameter equal to infinity for all of their $p$-local commensurability graphs. Moreover, solvable groups, in a sense, fill in the space between free groups and nilpotent groups. This talk covers joint work with Daniel Studenmund and Chen Shi. |