OSU - Topology Seminar

Topology Seminar Spring 2008 (last updated: May 2nd, 2008)
Time Location	Speaker	Title of Talk Abstract
Tuesday March 25, 2008
3:30 PM SM 3082	Chris Miller OSU	Geometric categories and o-minimal structures The theory of subanalytic sets is an excellent tool in various analytic-geometric contexts. We axiomatize the notion of "behaving like the category of subanalytic sets" by introducing the notion of analytic-geometric category. The objects of such a category share many of the hereditary and geometric finiteness properties of subanalytic sets. Proofs of the more difficult results of this nature, like the Whitney-stratifiability of sets and maps in such a category, often involve the use of charts to reduce to the case of subsets of R^n. For subsets of R^n, the theory of o-minimal structures on the real field, an abstraction of the theory of semialgebraic sets (and the subject of Math 949 in Sp08) provides an elegant and efficient setting in which to work. Certain fairly natural sets---like {(x,x^r):x>0} for positive irrational r, and {(x,e^{-1/x}): x>0}---are not subanalytic (at the origin) in R^2. Because there are o-minimal structures on the real field which include these sets, we now have analytic-geometric categories which include these sets among its objects.
Tuesday May 13, 2008
3:30 PM SM 3082	Tamal Dey OSU (Comp. Sci.)	Reconstructing surfaces from samples with topology and geometry guarantees In recent years, algorithms have been proposed which can reconstruct a piecewise linear surface T from a dense point sample of a surface S with provable guarantees. In this talk I shall go over this development. We will present an algorithm called Cocone, its analysis for theoretical guarantees, and results of its implementation. If time permits, I shall mention different extensions some of which are still part of active research.
Tuesday May 20, 2008
3:30 PM TBA	Mark Johnson Penn. State - Altoona	Pi-algebras and higher homotopy operations A $\Pi$-algebra is the formal analog of the graded homotopy groups $ \pi_X$ for a pointed topological space, together with its primary homotopy operations. I'll discuss joint work with David Blanc and Jim Turner, which began with constructing an obstruction theory to detect when diagrams of Pi-algebras arise by applying $\pi_$ to a commutative diagram of pointed spaces. These obstructions lie in certain Andre-Quillen cohomology groups, and our recent work has focused on identifying ``the last obstruction" to realizing finite, directed diagrams with a general definition of higher homotopy operation. As one example, Toda brackets arise by applying $\pi_*$ to a homotopy commutative diagram and then looking for an obstruction to finding a strictly commutative model, where the null composites are replaced by actual basepoint maps. We have also developed new computational tools for the Andre-Quillen cohomology of diagrams, building three types of spectral sequences from towers of fibrations between spaces of natural transformations. In the hopes of making all of this machinery more palatable, I'll also try to say a few words about where we hope to go next.
Thursday May 22, 2008
3:30 PM TBA	Graham Denham U. Western Ontario (Canada)	TBA
Tuesday May 27, 2008
3:30 PM SM 3082	Ian Biringer Univ. of Chicago	TBA
Friday May 30, 2008
3:30 PM TBA	Chris Leininger University of Illinois - U.-C.	TBA

Topology Seminar Winter 2008 (last updated: Jan. 14th, 2008)
Time Location	Speaker	Title of Talk Abstract
Tuesday Jan. 22, 2008
3:30 PM DU 027	Liviu Nicolaescu University of Notre Dame	Counting Morse functions on the two sphere We say that a Morse function on the 2-sphere is excellent if there are no two critical points on the same level set, while two Morse functions are called equivalent if we can obtain one from another via a global orientation preserving change in coordinates on S^2 and a global orientation preserving change of coordinates of the real axis. We denote by g(n) the number of equivalence classes of excellent Morse functions on the 2-sphere with n saddle points. V.I. Arnold conjectured that the ratio log(g(n))/2n log(n) tends to 1 as n tends to infinity. We will express the exponential generating series G(t)=\sum_{n\geq 0}\frac{g(n)t^{2n+1}}{(2n+1)!} in terms of elliptic integrals and then use this to confirm Arnold's prediction.
Tuesday Feb. 5, 2008
3:30 PM DU 027	Dan Burghelea OSU	On the topology and the geometric analysis of a generalized Morse function (after Hon Kit Wai) Generalized Morse functions are smooth functions whose critical points are either non degenerate or "birth death". Two Morse functions $f_1$ and $f_2$ can not be, in general, the ends of a smooth family of Morse functions $f_t$, with $t$ between 1 and 2. They can however be the ends of a smooth family of generalized Morse functions with all but finitely many Morse functions. The topology in the title refers to the elementary Morse theory of a pair $(f,g)$ with $f$ a Morse function and $g$ a generic Riemannian metric. The geometric analysis refers to the so called Witten-Hellfer-Sjostrand theory associated with such pair. This talk reports on the results in the thesis of of Hon Kit Wai (OSU-1995) about the extension of both elementary Morse theory and of Witten Hellfer Sjostrand (WHS) theory to the case of a generalized Morse function. WHS theory for generalized Morse functions is surprising and beautiful. The mathematics in the case of Morse function is based on the harmonic oscillator (i.e. the differential operator $-d^2/dx^2 + x^2 $ while for generalized Morse function involves in addition the differential operator $-d^2/dx^2 +x^4-2x .$ My (and S Haller) contribution to this study is a collection of more conceptual proofs and some not obvious generalizations which are essential for our recent work.
Tuesday Feb. 19, 2008
3:30 PM DU 027	Christophe Pittet University of Aix-Marseille I (France)	Bounded 2-cocycles on Lie groups, flat bundles, and a theorem of Dupont Milnor showed that there are flat SL(2,R)-bundles with non-zero Euler number. Dupont then explained Milnor's results in terms of bounds on universal classes. We extend Dupont's result to reductive and solvable Lie groups.
Tuesday Feb. 26, 2008
3:30 PM DU 027	Ivonne Ortiz Miami University	On the lower algebraic K-theory of Gamma_4 Let Gamma_4 be the group of integral, positive, Lorentzian 5 x 5 matrices; this group is a non-cocompact, 4-simplex, hyperbolic reflection group. We will present a preliminary report on the lower algebraic K-theory of the integral group ring of Gamma_4.
Tuesday March 4, 2008
3:30 PM DU 027	Qayum Khan Vanderbilt University	The Nil-Nil theorem in algebraic K-theory Bass defined an exotic Nil-summand of the algebraic K-theory of a polynomial extension. Later, Waldhausen extended the definition to tensor algebras and defined an exotic Nil-summand of the algebraic K-theory of an injective amalgam of groups. The Nil-Nil theorem states, under a certain finiteness condition, that there is a natural isomorphism from the amalgam Nil to a tensor Nil. An important application is that the Farrell-Jones conjecture in algebraic K-theory can be sharpened from the family of virtually cyclic subgroups to the family of finite-by-cyclic subgroups of a discrete group G. This is joint work with J.F. Davis and A.A. Ranicki.

Topology Seminar Autumn 2007 (last updated: Oct. 1st, 2007)
Time Location	Speaker	Title of Talk Abstract
Tuesday Sept. 25, 2007
3:30 PM BO 317	Stefan Haller University of Vienna (Austria)	Exponential growth and the Novikov complex Consider a vector field with Morse type zeros on a closed manifold. In addition assume that this vector field admits a Lyapunov 1-form, ie. with respect to some Riemannian metric the vector field corresponds to a closed 1-form. In a generic situation, such vector field gives rise to a Novikov complex over a Novikov ring, a formal completion of the group ring of the fundamental group. If the vector field satisfies an exponential growth condition then the `Laplace transform' of this Novikov complex has a non-trivial domain of absolute convergence. This Laplace transformed complex is related to the deRham complex by an absolutely converging integration homomorphism which, generically, induces an isomorphism in cohomology. Its torsion is related to the torsion of the underlying manifold and a contribution associated with the closed trajectories of the vector field. This is joint work with Dan Burghelea.
Tuesday Oct. 30, 2007
3:30 PM BO 317	Stephan Stolz University of Notre Dame	Quantum field theories and generalized cohomology This is a talk on an ongoing joint project with Peter Teichner (Berkeley). Based on Graeme Segal's axiomatization of quantum field theories, we incorporate super symmetry into the picture and define the notion of a super symmetric d-dimensional field theory over a manifold X for d=0,1,2. It turns out that 0- dimensional QFT's over X can be identified with closed differential forms, and a vector bundle with connection leads to a 1-dimensional QFT over X, while 2-dimensional QFT's over X seem to be genuinely new geometric objects that don't have a classical description. The topological information these objects carry is revealed by passing to concordance classes of QFT's over X, where we call two QFT's concordant if there is a QFT over the product of X with the interval that restricts to the given QFT's on the boundary. The above description of 0-dimensional QFT's implies that their concordance classes can be identified with ordinary cohomology with real coefficients; we can show that concordance classes of d-dimensional QFT's gives K-theory for d=1 and conjecture that we obtain elliptic cohomology for d=2 (more precisely, the `topological modular form theory' of Hopkins-Miller). Evidence for the latter is provided by our result that we can show that the `partition function' of a 2- dimensional QFT is an integral modular function.
Tuesday Nov. 6, 2007
3:30 PM BO 317	Dan Ramras Vanderbilt University	Deformation K-theory and the Atiyah-Segal theorem Classically, the Atiyah-Segal theorem relates the representation ring of a compact Lie group G to the topological K-theory of the classifying space BG. Carlsson's deformation K-theory spectrum may be seen as the homotopy theoretical analogue of R(G), and this spectrum provides a new context in which to study representations of infinite discrete groups. After introducing deformation K-theory, I will explain how Yang-Mills theory can be used to prove an analogue of the Atiyah-Segal theorem for fundamental groups of aspherical surfaces M. This theorem relates deformation K-theory of the fundamental group of M to topological K-theory of M and, when combined with work of Tyler Lawson, leads to concrete information about the stable moduli space of flat connections over surfaces.
Tuesday Nov. 13, 2007
3:30 PM BO 317	Anna Wienhard Princeton University	Bounded Toledo invariants and rotation numbers The classical Toledo invariant is a characteristic number associated to a representation of the fundamental group of a closed surface into a Lie group G of Hermitian type (e.g. G=Sp(2n,R)). For G=PSL(2,R) the Toledo invariant is just the Euler number. We give a (bounded) cohomological description of the Toledo invariant which leads to a (bounded) Toledo invariant which is also defined for representations of fundamental groups of surfaces with boundary. This new invariant is not a characterstic number, but its integrality properties relate to rotation numbers and allow to retrieve interesting information about the structure of the space of representation with maximal Toledo invariant.
Tuesday Nov. 20, 2007
3:30 PM BO 317	Damian Osajda University of Wroclaw (Poland)	On some combinatorial non-positive curvature We will define local conditions (in terms of small balls) on a simplicial complex that imply non-positive-curvature-like properties of it. The class of groups acting geometrically by automorphisms on such complexess contains in particular CAT(0) cubical and systolic groups.

Topology Seminar Spring 2007 (last updated: March 28th, 2007)
Time Location	Speaker	Title of Talk Abstract
Tuesday April 3, 2007
4:30 PM BO 317	Moon Duchin University of California - Davis	Divergence in Teichmuller space and the mapping class group The rate of divergence of geodesic rays is one of several Gromovian ways to get a handle on curvature. In many settings, there is a gap between linear and exponential rates of divergence. We show that Teichmuller space (with the Teichmuller metric) and the mapping class group both have intermediate divergence, in fact quadratic, by exploiting the product region structure in each case. We can be quite explicit about how to travel around efficiently. This is joint work with Kasra Rafi.
Thursday April 5, 2007
2:30 PM JR 0221	Evgenij Troitsky Moscow State Univ. (Russia)	Geometry and analysis of Reidemeister classes (partially joint with A.Felshtyn and A.Vershik) The talk is about some conjecture related to the Reidemeister classes (twisted conjugacy classes) of an automorphism of a discrete group. The twisted Burnside-Frobenius theorem (conjecture, problem) is to identify the number of Reidemeister classes (the Reidemeister number of an automorphism) and the number of fixed points on an appropriate dual object. This is important for Dynamical applications etc. References: math.RT/0606155, "Non commutative Geometry and Number Theory", 141--154, 2006, Braunschweig, Vieweg. math.RT/0606161, "Math. Res. Lett.", 13 (2006) 719--728 math.GR/0606179, "Crelle's Journal", accepted math.GR/0606191, "Funct. Anal. Appl." 40 (2006), No. 2, 117--125 math.GR/0606725, "Geom. Dedicata", submitted Note: special time and place, this lecture is joint with the Geometric Analysis seminar.
Friday April 20, 2007
4:30 PM CH 218	Tom Farrell Binghamton University	The space of negatively curved Riemannian metrics, Part II Let M be a closed smooth manifold which supports a Riemannian metric whose sectional curvatures are all negative. Suppose also that M has dimension greater than 9. Then Pedro Ontaneda and I prove that the space of all negatively curved Riemannian metrics on M is disconnected. This is a continuation of last week's colloquium talk, and will pick up where the colloquium left off. Note: special time and place.
Tuesday May 1, 2007
4:30 PM BO 317	Chris Dwyer Binghamton University	Completion in Twisted Equivariant K-theory I will discuss extending proof of the Atiyah-Segal completion theorem in equivariant K-theory to twisted equivariant K-theory and some of the issues that arise.
Tuesday May 15, 2007
4:30 PM BO 317	Thang Le Georgia Tech	On the A-polynomial and the Jones polynomial of knots We will discuss a conjecture (due to Garoufalidis) concerning an interesting relation between the Jones polynomial and the A-polynomial of a knot. The latter is defined through representations of the fundamental group, while the Jones polynomial has very few direct interpretations in terms of classical topology. We also sketch a proof of the conjecture for the class of twist knots. Note: This lecture is joint with the Geometric Analysis seminar.
Tuesday May 22, 2007
4:30 PM BO 317	Jean Lafont OSU	Algebraic K-theory of 3-dimensional hyperbolic reflection groups I will discuss some recent work on computing the lower algebraic K-theory for (the integral group ring of) lattices in hyperbolic space. This was joint work with Ivonne Ortiz.
Tuesday May 29, 2007
4:30 PM BO 317	Richard Hind Univ. of Notre Dame	Estimates on symplectic embeddings Extending Gromov's nonsqueezing theorem, we will give restrictions on which domains in Euclidean space can be moved into others under Hamiltonian flows.

Topology Seminar Autumn 2006 (last updated: October 6th, 2006)
Time Location	Speaker	Title of Talk Abstract
Tuesday September 26, 2006
3:30 PM MA 417	Stephen Haller University of Vienna (Austria)	Complex valued Ray-Singer torsion We will discuss a complex valued analogon of the Ray-Singer torsion, constructed with the help of non-selfadjoint Laplacians. In many situations we can show that this non-vanishing complex number coincides with the Reidemeister torsion, up to a simple correction term. We expect that this is always the case, and regard it as a complex valued version of the Cheeger-Muller theorem. This is joint work with Dan Burghelea.
Wednesday September 27, 2006
3:30 PM MA 417	Rainer Vogt University of Osnabrueck (Germany)	On the multiplicative structure of topological Hochschild homology In 1993 Deligne asked whether the Hochschild cochain complex of an associative is an E_2 algebra, more precisely, has a canonical action of the singular chains of the little 2-cubes operad. Affirmative answers have been found by Kontsevich, Tamarkin and Voronov. A more general approach to the conjecture by McClure and Smith also proves the topological version of the Deligne Conjecture: The topological Hochschild cohomology spectrum of a ring spectrum is an E_2 ring spectrum. The talk addresses the dual problem. We prove the dual of the higher Deligne Conjecture: The topological Hochschild homology spectrum THH(R) of an E_{n+1} ring spectrum is an E_{n} ring spectrum. The result is joint work with Morten and Zig Fiedorowicz. Multiplicative structures on THH(R) provide tools for its calculation. Many of the latest calculations of algebraic K-groups are based on calculations of topological cyclic homology, which is derived from fixed points data of THH(R).
Tuesday October 3, 2006
3:30 PM JR 143	Andrei Pajitnov Universite de Nantes (France) and OSU	Circle-valued Morse theory This generalization of the classical Morse theory was initiated by Novikov in 1980s. We introduce the basic construction of the theory (the Novikov complex) and discuss some of the recent developments, in particular, applications to knots and links.
Tuesday October 10, 2006
3:30 PM JR 139	Pat Hooper Northwestern University	Stable Billiards in Acute and Obtuse Triangles The orbit-type of a periodic billiard path in a triangle is the sequence of edges the billiard ball hits. We say a periodic billiard path in a triangle T is "stable" if given any small enough perturbation of T to a new triangle T', we can find a periodic billiard path in T' with the same orbit-type. We will discuss the proof of the following theorem: No right triangle admits stable periodic billiard paths. Moreover, a stable periodic billiard path in an acute triangle never has the same orbit-type as a stable periodic billiard path in an obtuse triangle. Surprisingly, the proof is essentially topological.
Tuesday October 17, 2006
3:30 PM CL 102	Craig Westerland University of Wisconsin-Madison	Free loop spaces and Koszul duality The Chas-Sullivan theory of "string topology" encodes certain multiplicative structures defined on the free loop space of a manifold. This theory has been extended to classifying spaces of compact Lie groups by Gruher-Salvatore. Recently Gruher has shown that this structure is, in some sense, Spanier-Whitehead dual to the fusion product in the Verlinde algebra, as described by Freed-Hopkins-Teleman. In this talk we take another approach to this relationship, using Koszul duality in place of Spanier-Whitehead duality.
Tuesday October 24, 2006
3:30 PM JR 139	Andrei Pajitnov Universite de Nantes (France) and OSU	Circle-valued Morse theory This generalization of the classical Morse theory was initiated by Novikov in 1980s. We introduce the basic construction of the theory (the Novikov complex) and discuss some of the recent developments, in particular, applications to knots and links. [This is a continuation of the seminar talk on Oct. 3rd]
Tuesday October 31, 2006
3:30 PM CL 102	Vladimir Chernov Dartmouth College	Some applications of the Graded Poisson algebras on bordism groups of garlands Fix a manifold M and a set $\frak N$ consisting of closed manifolds. Roughly speaking, the space $G_{\frak N, M}$ of $\frak N$-garlands in M is the space of mappings into M of singular manifolds obtained by gluing manifolds from $\frak N$ at some marked points. We define a bordism group $\hat \Omega_(G_{\frak N,M})$ and operations $\star$ and $[\cdot, \cdot]$ on $\hat \Omega_(G_{\frak N,M})\otimes \mathbb Q.$ For $\frak N$ consisting of odd-dimensional manifolds, $\hat \Omega_*(G_{\frak N,M})\otimes \mathbb Q$ is a graded Poisson algebra. The mod 2 analogue of $[\cdot, \cdot]$ for one element sets $\frak N$ and some other operations were introduced in our previous work with Rudyak. We discuss the applications of the algebra to minimizing the intersection of loops on a surface. The Gauss linking number is defined for zero homologous linked submanifolds. Our algebra allows one to generalize the linking number to the case of submanifolds realizing arbitrary homology classes.
Tuesday November 7, 2006
3:30 PM CL 102	Craig Jensen University of New Orleans	Brownstein-Lee Conjecture The pure symmetric automorphism group of a free group consists of those automorphisms which send each generator to a conjugate of itself. Another way to view this group is as the group of motions of n unknotted, unlinked circles where each circle returns to its original position. In 1993, Alan Brownstein and Ronnie Lee calculated the first and second cohomology groups (including the cup product structure going from the first to the second) of the group of pure symmetric automorphisms of a free group of finite rank. They further conjectured what the entire cohomology ring should be. Jon McCammond, John Meier and I verified this conjecture.
Tuesday November 9, 2006
3:30 PM TBA	B. Botvinick University of Oregon	Homotopy groups of the moduli space of positive scalar curvature metrics The problem of determining when a smooth compact manifold admits a positive scalar curvature (psc) Riemannian metric is comparatively well understood. However, even for the n-sphere, surprisingly little work has been done to date concerning the topological stucture of the space of all psc-metrics. In this talk, I will present some new results concerning the rational homotopy groups of this space for the n-sphere with n>4. My approach uses results on higher analytical/topological torsion due to Hatcher, Igusa, and Goethe.
Tuesday November 21, 2006
3:30 PM CL 102	Peter Linnell Virginia Tech	L^2-Betti numbers and the Atiyah conjecture I will survey some results on the Atiyah conjecture and its relation to L^2-Betti numbers and embeddings of group rings into Artinian rings. There are various versions of the Atiyah conjecture; here is one of them. Let N(G) denote the group von Neumann algebra of the group G. There is a well defined dimension dim_{N(G)} associated to every N(G)-module, which is either a non-negative real number or infinity. Suppose G is a group for which the finite subgroups have bounded order and d is the least common multiple of the finite subgroups. Then one version of the Atiyah conjecture states that d(dim_{N(G)} M) is either an integer or infinity for every CG-module M. The case G is a congruence subgroup (groups of matrices congruent to the identity matrix modulo some prime) will especially be considered.
Tuesday November 28, 2006
3:30 PM CL 102	Georgi Khimshiashvili Razmadze Mathematical Institute (Georgia) and OSU	Stable holomorphic curves in loop spaces We'll be concerned with several classes of loop spaces which can be endowed with natural almost complex structures and (pseudo)holomorphic maps of Riemann surfaces into such spaces. The main attention will be given to the (germs of) holomorphic maps of the unit disc into the Brylinski loop space of a closed orientable Riemannian 3-fold. Local geometric description of stable germs of such maps will be presented and their relation to Seifert fibrations will be outlined. It will be explained that explicit examples of holomorphic curves in loop spaces of round 3-sphere can be constructed from isolated singularities of algebraic plane curves. In particular, it will be shown that the "inverse of Hopf fibration" is holomorphic as a map of Riemann sphere into Brylinski loop space of round 3-sphere. Examples of holomorphic solid tori in Euclidean space will be constructed using the Villarceau circles on round torus and wavefronts of closed convex curves in the plane. A number of open problems will be also formulated.
Tuesday December 5, 2006
3:30 PM CL 102	Shaun Van Ault OSU	Introduction to Symmetric Homology, II The interpretation of cyclic homology as Tor functors of the category ?C lends itself to many generalizations, such as dihedral and quaternionic homology. In my thesis work with Zbigniew Fiedorowicz, I have been working with the category ?S, which allows one to define symmetric homology. In this talk I will provide some background as well as an introduction to my work in this subject. This is a continuation of the talk in the Low Dimensional topology seminar (last Thursday).

Topology Seminar Winter 2006 (last updated: January 8, 2006)
Time Location	Speaker	Title of Talk Abstract
Tuesday January 3, 2006
3:30 PM MW 154	Stratos Prassidis Canisius College	Detecting monomorphisms on poly-free groups We state and prove "homological" conditions that detect when a group homomorphism from certain poly-free groups is one-to-one. More specifically, we will show that if the induced map on the Lie algebra of the descending central series of the group is a monomorphism, then the map is a monomorphism. This criterion applies to pure braid groups, pure orbit braid groups and the McCool groups (subgroups of the group of automorphisms of free groups). At the end of the talk we will formulate a conjecture on the linearity of such groups. This is joint work with Dan Cohen and Fred Cohen.
Friday January 13, 2006
3:30 PM BO 316	Boris Okun University of Wisconsin - Milwaukee	L^2-cohomology of Coxeter groups
Tuesday January 17, 2006
3:30 PM DB 48	Chris Connell Indiana University	A volume gap theorem and smooth rigidity for manifolds with negatively curved targets We present volume and entropy conditions for a smooth compact manifold of dimension n>4 admitting a degree one map to a negatively curved manifold to be homotopic to a diffeomorphism. We discuss some applications including the distribution of critical points of C^1 maps and related finiteness theorems. These results do not require Anderson-Cheeger-Gromov type (pre)-compactness theorems.
Tuesday January 24, 2006
3:30 PM SM 1042	Alvaro Pelayo University of Michigan	Non-Hamiltonian symplectic torus actions An action of a torus T on a compact connected symplectic manifold M is Hamiltonian if it admits a momentum map, which is a map on M taking values in the dual of the Lie algebra of T. In 1982, Atiyah and Guillemin-Sternberg proved that the image of the momentum map is a convex polytope, now called the momentum polytope. In 1988, Delzant showed that if the dimension of T is half of the dimension of M, the momentum polytope determines M up to equivariant symplectomorphism. Moreover, Delzant proved that given a polytope with certain properties (now called Delzant polytopes), one can construct a manifold whose momentum polytope is precisely this one. In this talk, I will describe the classification for symplectic actions with coisotropic principal orbits, without assuming that the action is Hamiltonian. In this case, the polytope is one of six invariants of the manifold M. We also show that given a collection of such six ingredients, one can construct a manifold whose invariants are precisely these ingredients. This is joint work with J. J. Duistermaat from Utrecht University.
Tuesday January 31, 2006
3:30 PM SM 1042	Nathan Broaddus University of Chicago	Heegard splittings and the Johnson filtration of the mapping class group I will discuss a general method of producing calculable invariants of Heegard splittings of 3-manifolds using homomorphic images of the mapping class group. This method will then be applied to the specific homomorphisms of the mapping class group developed by Johnson and Morita. This is a work in progress. This is a joint project with Joan Birman and Tara Brendle.
Tuesday February 7, 2006
3:30 PM SM 1042	David Fisher Indiana University	Local rigidity and first cohomology In 1964, Andre Weil showed that a homomorphism $\pi$ from a finitely generated group $\Gamma$ to a Lie group G is locally rigid whenever $H^1(G, Lie(G))=0$. Here $\pi$ is locally rigid if any nearby homomorphism is conjugate to $\pi$ by a small element of G, and Lie(G) is the Lie algebra of G. The first aim of my talk is to describe an infinite dimensional analog of Weil's theorem: Let G be a finitely presented group, (M,g) a compact Riemannian manifold, and $\pi: G --> Isom(M,g)$ a homomorphism. Then if $H^1(G, Vect(M))=0$, the homomorphism $\pi$ is locally rigid as a homomorphism into Diff(M). Here Vect(M) is the space of infinitely differentiable vector fields on M, and Diff(M) is the group of infinitely differentiable diffeomorphisms of M. I will then state a more technical version that does not require that the original action be isometric. In joint work with Theron Hitchman, we use this more technical result to prove local rigidity of many affine actions of groups with property (T) of Kazhdan, particularly lattices in Sp(1,n) and $F_4^{-20}$.
Tuesday February 14, 2006
3:30 PM SM 1042	Nicolau Saldanha PUC-Rio de Janeiro (Brazil)	Homotopy and cohomology of spaces of curves of bounded curvature in the sphere Let Q in SO(3) be an orthogonal matrix and k a real number. Let X_{Q,k} be the space of smooth regular curves gamma: [0,1] --> S^2 satisfying gamma(0) = e_1, gamma'(0) = e_2, gamma(1) = Qe_1, gamma'(1) = Qe_2, and such that the geodesic curvature of gamma is greater than k for all t in the interval [0,1]. The spaces X_{Q,k} have a surprisingly rich topological structure: in this talk we shall present several partial results concerning the homotopy and cohomology of these spaces. This work can be considered a continuation of the preprint "Homotopy and cohomology of spaces of locally convex curves in the sphere", and involves collaboration with Boris Shapiro.
Tuesday February 21, 2006
3:30 PM SM 1042	Tim Riley Cornell University	Filling length in groups and dual pairs of spanning trees in planar graphs In 2003 I gave a talk at Ohio State in which I described a graph theoretic question concerning duality and the diameters of spanning trees in planar graphs. In this talk I will explain recent work with W.P. Thurston answering this question. Also, I will discuss the significance the answer has for problems of Gromov and Stalling concerning group theoretic invariants (filling functions) that arise from the study of combinatorial homotopy discs (van Kampen diagrams) filling loops in Cayley 2-complexes.
Tuesday February 28, 2006
3:30 PM SM 1042	Ivonne J. Ortiz Miami University	Lower algebraic K-theory of hyperbolic 3-simplex reflection groups A hyperbolic 3-simplex reflection group is a Coxeter group acting geometrically on hyperbolic 3-space with fundamental domain a geodesic simplex (possibly with some ideal vertices). The classification of these groups is known, and there are exactly 9 cocompact examples, and 23 non-cocompact examples. We explicitly compute the lower algebraic K-theory of the Coxeter group [3,4,4] (one of the 23 non-cocompact examples). Part of this computation involves calculating certain Waldhausen Nil-groups for Z[D_2] and Z[D_3]. This is joint work with J.-F. Lafont.
Tuesday March 7, 2006
3:30 PM SM 1042	Carles Broto University of Chicago and Universidad Autonoma de Barcelona (Spain)	Chevalley p-local finite groups I will give an introduction to the theory of Chevalley p-local finite groups. These p-local finite groups arise from p-compact groups in a way similar to the way Chevalley finite groups arise from reductive algebraic groups.

Topology Seminar Autumn 2005 (last updated: November 18, 2005)
Time Location	Speaker	Title of Talk Abstract
Tuesday September 27, 2005
3:30 PM CC 302	Igor Belegradek Georgia Institute of Technology	Relative hyperbolization I shall discuss the properties of the aspherical manifolds obtained via relative strict hyperbolization of polyhedra, and show that in many ways these manifolds are close cousins of finite volume negatively pinched Riemannian manifolds. As an application I will show that any triangulated closed aspherical n-manifold with hyperbolic fundamental group is a retract of a triangulated closed aspherical (n+1)-manifold with hyperbolic fundamental group.
Tuesday October 4, 2005
3:30 PM BO 316	Mohamad Hindawi The Ohio State University	Large scale geometry of 4-dimensional closed nonpositively curved real analytic manifolds In this talk I will describe the asymptotic cones of 4-dimensional non-positively curved real analytic manifolds. As a result, I will prove that the existence of non-standard components in the Tits boundary, discovered by Christoph Hummel and Victor Schroeder in 1998, depends only on the quasi- isometry type of the fundamental group. Moreover, in the absence of any non-standard components in the Tits boundary, it follows that the fundamental group of the manifold is relatively hyperbolic with respect to the fundamental groups of maximal higher rank submanifolds. Also, I will discuss the relationship between these manifolds and CAT(0) spaces with isolated flats.
Tuesday October 11, 2005
3:30 PM BO 316	Satyan Devadoss Williams College and The Ohio State University	Graph-associahedra, Coxeter operads, and Moduli spaces We look at natural generalizations of the Deligne-Mumford compactification of the real moduli space of Riemann spheres. They inherit a tiling by the graph-associahedra convex polytopes. We obtain explicit configuration space models for the classical infinite families of finite and affine Weyl groups and demonstrate the appearance of the underlying Coxeter operad structure.
Tuesday October 18, 2005
3:30 PM BO 316	Benjamin Schmidt University of Michigan	Simplicial volumes of locally symmetric spaces of non-compact type I will describe joint work with J.-F. Lafont, confirming Gromov s conjecture that closed locally symmetric spaces of non-compact type have positive simplicial volumes.
Tuesday October 25, 2005
3:30 PM BO 316	Donald Yau The Ohio State University	G-structure on the cohomology of Loday algebras I will talk about the existence of a Gerstenhaber algebra structure on the cohomology of various algebras defined by Loday and Loday-Ronco.
Tuesday November 1, 2005
3:30 PM BO 316	Chris Hruska University of Chicago	Cubulating relatively hyperbolic groups (Joint with Dani Wise) We study actions of groups on CAT(0) cube complexes. This subject generalizes the classical study of group actions on trees. The existence of such actions has implications regarding codimension-1 subgroups, Property (T), a-T-menability, and biautomaticity using work of Sageev, Niblo-Roller, and Niblo-Reeves. We give criteria for determining when a relatively hyperbolic group acts on a finite dimensional cube complex and when such an action is cocompact, generalizing a theorem of Sageev from the word hyperbolic setting. More generally, we describe a cusped cofinite structure in which cube complexes for the peripheral subgroups play the role of cusps. This structure is analogous to the cusped structure of a finite volume manifold with pinched negative curvature.
Tuesday November 8, 2005
3:30 PM BO 316	Jarek Kedra University of Szczecin (Poland) and University of Aberdeen (U.K.)	Metrics on groups of diffeomorphisms I will start with presenting two metrics. One on the group of all diffeomorphisms (isotopic to the identity) of a closed manifold, and the Hofer metric on the group of Hamiltonian diffeomorphisms of a symplectic manifold. I will prove their basic properties (mostly bad). Finally, I will present a construction of new metrics based on the fragmentation lemma. The natural representations of right angled Artin groups (which I will define) behave well with respect to these metrics.
Tuesday November 15, 2005
2:30 PM MA 417	Theron J. Hitchman Rice University	Cohomology, cocycles, and dynamical rigidity for lattices We shall describe a differential geometric approach to the study of rigidity properties of lattices in semisimple Lie groups. We shall provide several applications. First, new cohomology vanishing results for such lattices with coefficients in certain infinite dimensional representations. Second, a geometric proof of Zimmer's measurable cocycle superrigidity theorem which includes previously unknown groups. Finally, we shall discuss new local rigidity statements for actions of these groups on compact manifolds. This is joint work with David Fisher of Indiana University. NOTE: Special time and room number.
Tuesday November 22, 2005
3:30 PM BO 316	Dan Burghelea The Ohio State University	The geometric complex of a Morse-Bott function (joint work with Stefan Haller) As in the case of Morse theory (where a finite dimensional cochain complex quasi isomorphic to the DeRham complex can be associated to any Morse function), in the case of Morse-Bott theory a geometric complex quasi isomorphic to the DeRham complex can only sometimes be associated to a Morse-Bott function (and some extra data). Unfortunately this assignement (1) is not always possible, (2) the complex is infinite dimensional, and (3) traditional spectral geometry methods are apparently hard to pursue. In this talk, I will describe this complex and show that in relevant situations all these draw-backs can be overcome. A geometric complex can always be defined, it can be treated as a finite dimensional complex, and the analogue of the "spectrum of the Laplacians" can be defined. Moreover the complex can be used to prove new results and suggests new mathematics.
Tuesday November 29, 2005
3:30 PM BO 316	Alissa Crans The Ohio State University	Loop groups and Lie 2-algebras A major result in differential geometry is Ado's theorem, which states that every finite-dimensional Lie algebra comes from some Lie group. We seek to prove a higher-dimensional analog of this statement for Lie 2-groups and Lie 2-algebras, which are generalized versions of Lie groups and Lie algebras where we have replaced the associative law and the Jacobi identity, respectively, by natural isomorphisms called the `associator' and `Jacobiator.' While this problem remains open, we describe progress toward a solution by examining the Lie 2-algebras g_k which are constructed by starting with a simple Lie algebra g and equipping it with a Jacobiator proportional to a real number k and built from the Killing form. It seems that except for k=0, there does not exist a Lie 2-group whose Lie 2-algebra is isomorphic to g_k. However, we can construct for integral values of k an infinite-dimensional Lie 2-group whose Lie 2-algebra is equivalent to g_k. Moreover, these Lie 2-groups are closely related to the Kac-Moody central extensions of loop groups and the group String(n).