UIUC Dept. of Mathematics Seminar Calendar

      July 2012             August 2012           September 2012   
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
  1  2  3  4  5  6  7             1  2  3  4                      1
  8  9 10 11 12 13 14    5  6  7  8  9 10 11    2  3  4  5  6  7  8
 15 16 17 18 19 20 21   12 13 14 15 16 17 18    9 10 11 12 13 14 15
 22 23 24 25 26 27 28   19 20 21 22 23 24 25   16 17 18 19 20 21 22
 29 30 31               26 27 28 29 30 31      23 24 25 26 27 28 29
                                               30

Thursday, January 19, 2012

Group Theory Seminar
1:00 pm in Altgeld Hall 347, Thursday, January 19, 2012

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Submitted by kapovich.

Organizational meeting

Thursday, January 26, 2012

Thursday, February 2, 2012

Group Theory Seminar
1:00 pm in Altgeld Hall 347, Thursday, February 2, 2012

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Submitted by kapovich.

Albert Fisher (University of Sao Paulo)
A flow crossection for Moeckel's theorem on continued fractions
Abstract: We construct a cross-section to the principal congruence modular flow which is represented as a skew product transformation over the natural extension of the Gauss map. This leads to a new proof of Moeckel's theorem on rational approximants. For an irrational number $x$ in the unit interval with continued fraction expansion $[n_0 n_1...]$, let $p_k/q_k= $[n_0 n_1..n_k]$ $ be the rational approximants for $x$. Writing these in lowest terms, they can be of three types: $\frac{O}{E}$, $\frac{E}{O}$, or $\frac{O}{O}$ where $O$ stands for odd and $E$ for even. Moeckel's theorem states that the frequency of each of these exists almost surely. What is unusual in the proof is that this does not follow directly from the ergodic theorem applied to an observable on the Gauss map (the shift on continued fractions): one must first enlarge the space. Moeckel's approach makes use of the geodesic flow on a three-fold cover of the modular surface, together with a geometric argument for counting the time that geodesics spend in cusps. Ergodicity of the flow is automatic (via the Hopf argument) but the counting is somewhat involved. Later Jager and Liardet found a second purely ergodic theoretic proof, constructing a skew product over the Gauss map. There the counting is direct, but the proof of ergodicity is more difficult. Our proof unifies the two earlier arguments, inheriting these strong points of each.

Thursday, February 9, 2012

Group Theory Seminar
1:00 pm in Altgeld Hall 347, Thursday, February 9, 2012

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Submitted by kapovich.

Catherine Pfaff (Rutgers - Newark)
Constructing and Classifying Fully Irreducible Outer Automorphisms of Free Groups
Abstract: The main theorem of my thesis emulates, in the context of $Out(F_r)$ theory, a mapping class group theorem (by H. Masur and J. Smillie) that determines precisely which index lists arise from pseudo-Anosov mapping classes. Since the ideal Whitehead graph gives a finer invariant in the analogous setting of a fully irreducible $\phi \in Out(F_r)$, we instead focus on determining which of the 21 connected 5-vertex graphs are ideal Whitehead graphs of ageometric, fully irreducible $\phi \in Out(F_3)$. Our main theorem accomplishes this. The methods we use for constructing fully irreducible $\phi\in Out(F_r)$, as well as our identification and decomposition techniques, can be used to extend our main theorem, as they are valid in any rank. Our methods of proof rely primarily on Bestvina-Feighn-Handel train track theory and the theory of attracting laminations.

Thursday, February 16, 2012

Thursday, February 23, 2012

Group Theory Seminar
1:00 pm in Altgeld Hall 347, Thursday, February 23, 2012

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Submitted by kapovich.

Nathan Dunfield (UIUC Math)
Integer homology 3-spheres with large injectivity radius
Abstract: Conjecturally, the amount of torsion in the first homology group of a hyperbolic 3-manifold must grow rapidly in any exhaustive tower of covers (see Bergeron-Venkatesh and F. Calegari-Venkatesh). In contrast, the first betti number can stay constant (and zero) in such covers. Here "exhaustive" means that the injectivity radius of the covers goes to infinity. In this talk, I will explain how to construct hyperbolic 3-manifolds with trivial first homology where the injectivity radius is big almost everywhere by using ideas from Kleinian groups. I will then relate this to the recent work of Abert, Bergeron, Biringer, et. al. In particular, these examples show a differing approximation behavior for L^2 torsion as compared to L^2 betti numbers. This is joint work with Jeff Brock.

Thursday, March 1, 2012

Group Theory Seminar
1:00 pm in 347 Altgeld Hall, Thursday, March 1, 2012

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Submitted by dsrobins.

Derek Robinson (Department of Mathematics, University of Illinois at Urbana-Champaign)
Groups with few isomorphism types of derived subgroup.
Abstract: A derived subgroup in a group G is the derived (or commutator) subgroup of some subgroup of G. Recently there has been interest in trying to understand the significance of the set of derived subgroups within the lattice of all subgroups of G. In particular one can ask about the effect on the group structure of imposing restrictions on the set of derived subgroups. In this talk we will describe recent work on groups in which there are at most two isomorphism types of derived subgroup. While this may sound like a very special class of groups, it contains groups of many diverse types. We will describe some of these types of group and show how their construction involves some interesting number theoretic problems.

Thursday, March 8, 2012

Joint Group Theory/Differential Geometry/Ergodic Theory
1:00 pm in 347 Altgeld Hall, Thursday, March 8, 2012

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Submitted by jathreya.

Alex Wright (University of Chicago)
Arithmetic and Non-Arithmetic Teichm�ller Curves
Abstract: Teichm�ller curves are isometrically immersed curves in the moduli space of Riemann surfaces. Their study lies at the intersection of dynamics, Teichm�ller theory, and algebraic geometry. I will begin by summarizing known results on Teichm�ller curves, pointing out some similarities to the study of lattices, for example in PU(n,1). I will then move on to new research involving abelian square-tiled surfaces, Schwarz triangle mappings, and the Veech-Ward-Bouw-Moller Teichm�ller curves.

Tuesday, March 13, 2012

Graph Theory and Combinatorics
3:00 pm in 241 Altgeld Hall, Tuesday, March 13, 2012

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Submitted by west.

Jeffrey Paul Wheeler (University of Pittsburgh)
The Polynomial Method of Alon, Ruzsa, and Nathanson
Abstract: We will explore a particular method of tackling problems in Additive Combinatorics, namely the Polynomial Method of Noga Alon, Imre Ruzsa, and Melvyn Nathanson.� Additive Combinatorics can be described as the study of additive structures of sets. � This area is attractive in that it has numerous connections with other areas of mathematics, including Number Theory, Ergodic Theory, Graph Theory, Finite Geometry, and Group Theory and has drawn the attention of many good mathematicians, including Fields Medalist Terence Tao (2006).

Thursday, March 29, 2012

Thursday, April 5, 2012

Thursday, April 12, 2012

Group Theory Seminar
1:00 pm in Altgeld Hall 347, Thursday, April 12, 2012

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Submitted by kapovich.

Hao Liang (University of Illinois at Chicago)
Centralizers of finite subgroups of the mapping class group and almost fixed points in the curve complex
Abstract: Let S be an orientable surface of finite type, MCG(S) the mapping class group of S, C(S) the curve complex of S and H a finite subgroup of MCG(S). By the hyperbolicity of C(S), there exists points in C(S) whose H-orbit has diameter at most 6\delta; We call such points H-almost fixed points. We prove that there exists a constant K depending only on S so that if the diameter of the set of H- almost fixed points is greater than K then the centralizer of H in MCG(S) is infinite. I will start by explaining the proof of the analogous statement for hyperbolic groups, then I will explain the extra ingredients needed for the case of mapping class groups.

Thursday, April 19, 2012

Group Theory
1:00 pm in 347 Altgeld, Thursday, April 19, 2012

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Submitted by nmd.

Pere Menal Ferrer (Universitat Autonoma de Barcelona)
Reidemeister torsion for hyperbolic 3-manifolds
Abstract: Reidemeister torsion is an invariant defined for a CW-complex and a linear representation of its fundamental group. It was first defined in the 1930s by Reidemeister, de Rham and Franz to classify lens spaces in dimension 3, and since then it has proven to be a powerful invariant. In this talk, I will first give a brief review of Reidemeister torsion, and how to define it for a hyperbolic 3-manifold. Then I will introduce a certain class of invariants ${ T_n (M) }$ attached to a hyperbolic manifold $M$, which are defined as the Reidemeister torsion of $M$ with respect to the composition of the holonomy representation of $M$ and the $n$-dimensional fundamental representation of $\mathrm{SL}(n, C)$. I will show that the sequence $\{ \log |T_n(M)| / n^2 \}$ converges to $-\mathrm{Vol}(M)/ 4\pi$ (this is an extension of a result by W. M�ller which deals with closed manifolds). Finally, I will discuss how the sequence $\{ T_n (M) \}$ determines and is determined by the complex length spectrum of $M$. This is joint work with Joan Porti.

Tuesday, April 24, 2012

Group Theory (note unusual day)
2:00 pm in 241 Altgeld Hall, Tuesday, April 24, 2012

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Submitted by jathreya.

Shubojoy Gupta (Yale)
Asymptoticity of grafting and Teichmuller rays
Abstract: We shall discuss a result showing that any grafting ray in Teichmuller space is (strongly) asymptotic to a Teichmuller geodesic ray. Our method involves constructing quasiconformal maps between the underlying Thurston metric of a complex projective surface on one hand, and the singular flat metric induced by a holomorphic quadratic differential on the other. As a consequence we can show that the set of points in Teichmuller space obtained by integer graftings on any hyperbolic surface projects to a dense set in moduli space.

Thursday, April 26, 2012

Group Theory Seminar
1:00 pm in Altgeld Hall 347, Thursday, April 26, 2012

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Submitted by kapovich.

Pekka Pankka (University of Helsinki)
From Picard's theorem to quasiregular ellipticity
Abstract: In the quasiconformal geometry of Riemannian manifolds the classical Picard theorem from complex analysis turns into an existence question for non-constant quasiregular mappings from Euclidean spaces into Riemannian manifolds. In this talk, I will discuss the role of the fundamental group in these questions and a class of metrics, introduced by Semmes, that connect these quesiregular ellipticity questions to questions on quasiconformal geometry of decomposition spaces. This talk is based on joint works with Kai Rajala and Jang-Mei Wu.

Thursday, August 30, 2012

Thursday, September 6, 2012

Thursday, October 11, 2012

Thursday, October 18, 2012

Group Theory Seminar
1:00 pm in Altgeld Hall 347, Thursday, October 18, 2012

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Submitted by kapovich.

Alex Furman (University of Illinois at Chicago)
Classifying lattice envelopes for (many) countable groups
Abstract: Let $\Gamma$ be a given countable group. What locally compact groups $G$ contain a lattice (not necessarily uniform) isomorphic to $\Gamma$ ? In a joint work with Uri Bader and Roman Sauer we answer this question for a large class of groups including Gromov hyperbolic groups and many linear groups. The proofs use a range of facts including: recent work of Breuillard-Gelander on Tits alternative, works of Margulis on arithmeticity of lattices in semi-simple Lie groups, and a number of quasi-isometric rigidity results.

Wednesday, October 24, 2012

Math 499: Introduction to Graduate Mathematics
4:00 pm in 245 Altgeld Hall, Wednesday, October 24, 2012

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Submitted by laugesen.

Ilya Kapovich [email] (UIUC Math)
Groups as geometric objects
Abstract: Geometric group theory is a vibrant and rapidly developing area of mathematics that lies at the juncture of group theory, low-dimensional topology, differential geometry and several other subjects. A crucial idea in the area is to view a finitely generated group as a geometric and not just as an algebraic object. One of the key tools for realizing this goal is the notion of the Cayley graph of a group, which is a metric space associated to a finitely generated group together with a finite generating set. Geometric group theory studies the connections between large-scale geometric properties of groups on one side and their algebraic and algorithmic properties on the other side. In this introductory talk we will explore the basic ideas and notions of the subject and demonstrate how the above connections manifest themselves in a number of representative results.

Thursday, October 25, 2012

Thursday, November 1, 2012

Group Theory Seminar
1:00 pm in Altgeld Hall 347, Thursday, November 1, 2012

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Submitted by kapovich.

Yael Algom-Kfir (Yale University)
Small dilatation automorphisms of the free group and their mapping tori
Abstract: We consider elements of Out(F_n) that can be represented by a self map of a graph which has the property that a high enough iterate of the map sends every edge over any other edge of the graph. Furthermore, we assume that positive iterates of the map send edges to immersed paths in the graph. These maps are called irreducible train-track maps. To each such automorphism \phi one can attach a real number \lam >1 called the dilatation of \phi. For every n, the set of real numbers realized as dilatations of elements in Out(F_n) is a discrete set however, letting n vary we can get dilatations arbitrarily close to 1. For a fixed n, the smallest dilatation of an element in Out(F_n) is on the order of 2^{1/n}. We define an element to be P-small if its dilatation is smaller than P^{1/n} (there are infinitely many such automorphisms). We prove that for a given P, there exist finitely many 2-complexes so that the mapping torus of any P-small automorphism is obtained by surgery from one of these 2-complexes. This is a direct analog of a theorem of Farb-Leininger-Margalit in the case of Mod(S) for a closed surface S. We also show that the fundamental group of such a mapping torus has a presentation with a uniformly bounded number of generators and relations. This is joint work with Kasra Rafi.

Thursday, November 8, 2012

Group Theory Seminar
1:00 pm in Altgeld Hall 347, Thursday, November 8, 2012

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Submitted by kapovich.

Paul Schupp (UIUC Math)
Multi-pass Automata and Group Word Problems
Abstract: After reviewing some well-known connections between group theory and formal language theory, I will address a question of Bob Gilman: Is there a ``reasonable'' class of formal languages which are more general than context-free languages, but much more restricted than linear bounded automata, which tells us something about of group word problems? It seems that the class of ``multi-pass'' languages is interesting from the point of view. Although starting from automata we will discuss some mapping tori and some flat manifolds. This is joint work with Tullio Ceccerini-Silberstein, Michel Coornaert and Francesa Fiorenzi.

Thursday, November 15, 2012

Thursday, December 6, 2012

Group Theory Seminar
1:00 pm in Altgeld Hall 347, Thursday, December 6, 2012

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Submitted by kapovich.

Brian Ray (UIUC Math)
General nonexistence of finite strongly relatively rigid sets in Culler-Vogtmann Outer Space.
Abstract: Given a subset $\Sigma$ of a finitely generated free group, we say that $\Sigma$ is (strongly) spectrally rigid if whenever $T, T'$ are trees in (the closure of) Culler-Vogtmann Outer Space for which $\| g \|_T = \| g \|_{T'}$ for every $g \in \Sigma$, then $T = T'$. Similarly, we say that $\Sigma$ is (strongly) relatively rigid at $T$ if given a tree $T'$ in (the closure of) C-V Outer Space for which $\| g \|_T = \| g \|_{T'}$ for every $g \in \Sigma$, then $T = T'$. It is well known that no finite spectrally rigid set exists. Recently, Carette, Francaviglia, Kapovich, and Martino proved that every $T$ in C-V Outer Space admits a finite relatively rigid set. We show the existence of a family of trees on the boundary of C-V Outer Space for which no finite strongly relatively rigid set exists. Time permitting, we will discuss how one can promote the result of CFKM and show that every tree in C-V Outer Space admits a finite \emph{strongly} relatively rigid set.