Feb 9. Bob Gilman (Stevens) Title: Algorithmic Genericity
Abstract: The idea of genericity in geometric group theory was suggested by Gromov and Ol’shanskii in the late 1980's and has attracted many researchers since then. However the several models of genericity have produced occasional contradictory estimates of generic properties of random instances. This talk is aimed at avoiding such contradictions by viewing group theoretic genericity through the lens of Kolmogorov complexity. (No knowledge of Kolmogorov complexity is assumed.)
Feb 16. Wouter Van Limbeek (University of Michigan) Title: Finitely generated non-co-Hopf groups
Abstract: We will consider finitely generated groups with a self-similarity property, namely that there is a subgroup isomorphic to the original group (so-called non-co-Hopf groups). By iterating the process, one obtains a nested chain of subgroups isomorphic to the original. What groups admit such behavior? There appears to be a dichotomy based on the index of the subgroup: If it is infinite, there are many examples (e.g. free groups), but in the finite-index case, all known examples come from nilpotent groups. Are these really all? In this finite-index setting, we will show that if there is a chain consisting of normal subgroups, all examples indeed come from abelian groups.
Feb 23. Nicholas Touikan (Stevens) Title: Collapsing cube complexes (joint with Mark Hagen)
Abstract: I will show a new operation that can be done on cube complexes: panel collapse. Using this technique we immediately get a new proof of the full Stallings's theorem for groups with many ends. I time permitting, I will present other applications: reducing Cashen-Macura complexes, Nielsen realization for free groups, and a special case of Kropholler's conjecture. I will also discuss directions of research this new tool could open up.
March 2. Alex Lubotzky (Hebrew University and Yale) (joint Seminar with GRECS, Hunter College)
Title: First order rigidity of high-rank arithmetic groups
Abstract: The family of high rank arithmetic groups is a class of groups playing an important role in various areas of mathematics. It includes SL(n,Z), for n>2 , SL(n, Z[1/p] ) for n>1, their finite index subgroups and many more.
A number of remarkable results about them have been proven including; Mostow rigidity, Margulis Super rigidity and the Quasi-isometric rigidity.
We will talk about a new type of rigidity : "first order rigidity". Namely if G is such a non-uniform characteristic zero arithmetic group and H a finitely generated group which is elementary equivalent to G then H is isomorphic to G.
This stands in contrast with free groups, surface groups and hyperbolic groups ( many of which are low-rank arithmetic groups) which have many non isomorphic elementary equivalent finitely generated groups.
Joint work with Nir Avni and Chen Meiri.
March 9. Jason Behrstock (CUNY) Quasiflats in hierarchically hyperbolic groups
Abstract: Hierarchically hyperbolic spaces provide a uniform framework for working with many important examples, including mapping class groups, right angled Artin groups, most 3-manifold groups, Teichmuller space, and others. In this talk I'll provide an introduction to studying groups and spaces from this point of view. This discussion will center around recent work in which we classify quasiflats in these spaces, thereby resolving a number of well-known questions and conjectures. This is joint work with Mark Hagen and Alessandro Sisto.
March 16. Rostislav Grigorchuk (Texas A&M) Self-similar groups and aperiodic order.
Abstract: I will give two examples showing how aperiodic order can appear in the study of self-similar groups. The first example will be related to the spectral theory of groups and graphs with links to the theory of random Schrödinger operators. The second example will be related to the theory of automatic and unitriangular representations of self-similar groups.
March 23. Alla Detinko (St. Andrews) Algorithms for infinite linear groups
Abstract: In this talk we survey a novel domain of computational group theory: computing with linear groups over infinite fields. We provide an introduction to the area, available methods, and algorithms obtained so far. This includes deciding finiteness of finitely generated linear groups, and algorithms for infinite solvable matrix groups. Special consideration will be given to the most recent developments in computing with Zariski dense and arithmetic subgroups of semisimple algebraic groups, and some applications.
March 31. No Seminar, Spring break
April 6. No Seminar, Spring break
April 13. Rizos Sklinos (Stevens) Homogeneity in nonabelian free groups
Abstract: A (countable) group is homogeneous if for any two tuples of the group that share the same first-order properties there is an automorphism taking one to the other. After the positive answer to Tarski's question, by Kharlampovich-Myasnikov and Sela, on whether free groups share the same common first-order theory, the model theoretic interest on them renewed. Thus, a natural question is whether nonabelian free groups are homogeneous or not. In this talk I will present a proof answering the latter question positively. On the contrary, I will show that most of the surface groups are not homogeneous. This is joint work with C. Perin.
April 20. No Seminar, AMS meeting in Boston.
April 27. O. Kharlampovich, A. Vdovina Low complexity algorithms in knot theory.
Abstract. Agol, Haas and Thurston showed that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP- complete. This shows that (unless P=NP) the genus problem has high computational complexity even for knots in a 3-manifold. We initiate the study of classes of knots where the genus problem and even the equivalence problem have very low computational com- plexity. We show that the genus problem for alternating knots with n crossings has linear time complexity and is in Logspace(n). Al- most all alternating knots of given genus possess additional combi- natorial structure, we call them standard. We show that the genus problem for these knots belongs to TC0 circuit complexity class. We also show, that the equivalence problem for such knots with n crossings has time complexity n log(n) and is in Logspace(n) and TC0 complexity classes.
May 4 Ben Fine ( Joint work with Anthony Gaglione, Gerhard Rosenberger and Dennis Spellman) Elementary Free Groups and Extensions to Group Rings
Abstract: As part of the proof of the Tarski theorems by Myasnikov and Kharlampovich and independently by Sela a complete characterization of elementary free groups was given. These are groups with exactly the same elementary theory as the non-abelian free groups. Prominent among these are the surface groups of appropriately high genus. This provides a powerful method to prove things about surface groups that are otherwise quite difficult. We call these something for nothing results. We explain and discuss these and then show that elementary free groups satisfy many properties that are not elementary. We then change directions and talk about the unjiversl and elementary theory of group rings. This leads us to orderable groups and the connection to Kaplansky’s conjecture about zero divisors in group rings.
May 11 M. Sohrabi, On the elementary theory of SL_n, $n\geq 3$, over the rings of integers of number fields,
Abstract: In this talk I’ll report on some recent work with Alexei Miasnikov on elementary theory of special linear groups of higher rank over the ring of integers of a number field. In particular I’ll provide a characterization of arbitrary models of the complete first-order theory of such groups with parameters and discuss a possible approach to eliminate the need for parameters. Some other developments will also discussed. This is a work in progress.
July 12, 1:30-2:30 pm, room 6421
(Joint with Combinatorics Seminar http://userhome.brooklyn.cuny.edu/skingan/CombinatoricsSeminar/ )
Apoorva Khare (Indian Institute of Science), Title: PolyMath14: Groups with norms
Consider the following three properties of a general group G:
(1) Algebra: G is abelian and torsion-free.
(2) Analysis: G is a metric space that admits a "norm", namely, a translation-invariant metric d(.,.) satisfying: d(1,g^n) = |n| d(1,g) for all g in G and integers n.
(3) Geometry: G admits a length function with "saturated" subadditivity for equal arguments: l(g^2) = 2 l(g) for all g in G.
While these properties may a priori seem different, in fact they turn out to be equivalent. The nontrivial implication amounts to saying that there does not exist a non-abelian group with a "norm".
We will discuss motivations from analysis, probability, and geometry; then the proof of the above equivalences; and finally, the logistics of how the problem was solved, via a PolyMath project that began on a blogpost of Terence Tao.
(Joint - as D.H.J. PolyMath - with Tobias Fritz, Siddhartha Gadgil, Pace Nielsen, Lior Silberman, and Terence Tao.)