Topology et al. Seminar Archive

December 02, 2009

Topology Seminar

Speaker: Jennifer French, MIT
Title: Localizations of spaces; examples and construction

November 18, 2009

Topology Seminar

Speaker: A. Hager, Wesleyan University
Title: Baire functions and frames of ideals

November 11, 2009

Cardinal Invariants for kappa-Box Products

Speaker: Wistar Comfort, Wesleyan University
Abstract: This derives from joint work with Ivan Gotchev. The symbols w, d and S denote density character, weight and Souslin number, respectively, this last defined as follows: for a space X, S(X) is the least cardinal alpha such that X admits no family of alpha-many pairwise disjoint nonempty open subsets. [Remark: Always d(X) <= w(X), S(X) <= (d(X))^+.] Now, given a set {X_i : i in I} of nontrivial spaces and denoting by X_I their usual topological product, consider these basic results from General Topology. 1. w(X_I) = max{|I|, sup{w(X_i) : i in I}}. 2. [Hewitt-Marczewski-Pondiczery] If alpha >= omega, |I| <= 2^alpha and each d(X_i) <= alpha, then d(X_I) <= alpha. 3. If \alpha >= omega and each d(X_i) <= alpha, then S(X_I) <= alpha^+. 4. Let \alpha := sup{S(X_F : F subseteq I, F is finite}. Then S(X_I) = alpha if alpha is regular, S(X_I) = \alpha^+ otherwise. The authors generalize those and other familiar cardinality results about product spaces X_I to spaces of the form (X_I)_kappa, which is X_I with the kappa-box topology (basic open sets are restricted in (alpha^+)^2_alpha play a prominent role. (Motivational combinatorial test question: Given a sequence of finite sets, is there a subsequence whose pairwise intersections coincide?)

April 22, 2009

A Topological Solution to the A-infinity Deligne Conjecture

Speaker: Rachel Schwell, CCSU
Abstract: An operad is an algebraic structure whose use and behavior are best observed in the way it acts upon algebras (akin to representations in group theory). In particular, an A-infinity algebra}, often though of as a ``homotopy-associative" algebra, is an algebra acted upon by the operad of associahedra, which are polytopes whose cells are determined by the possible associations in an n-letter multiplication. The A-infinity Deligne conjecture, proposed in 1993 by Pierre Deligne, relates A-infinity algebras to another operad of interest (the little discs operad). It was first proved in 2000, but has since been reexamined by many others using different techniques and methods, most of which have turned out to be as interesting as the actual result of the conjecture itself. We will discuss a new proof of the A-infinity case of Deligne's conjecture based on a reconstruction of the associahedra (and cyclohedra, which are similar to the associahedra), accomplished by indexing the cells of the above polytopes with trees with weighted edges.

Joint work with Ralph Kaufmann.

April 15, 2009

Polygons inscribed in simple closed curves

Speaker: Elizabeth Denne, Smith College
Abstract: In 1911 Toeplitz asked whether a simple closed curve in the plane has four points that form the vertices of a square. The answer is yes, provided the curve belongs to certain regularity classes, but in the general case the question remains open. In this talk I present a new approach to this problem by using configuration spaces. I also present work on a related problem: given a simple closed curve, C, in Euclidean space and a fixed point p_1 on C, can we find a sequence of points p_2,..., p_n inscribed in C so that the n distances p_1p_{i+1} are in a prescribed ratio? This is joint work with Jason Cantarella and John McCleary.

April 01, 2009

Localized Higher-Order Alexander Modules and Higher-Order Degrees

Speaker: John R. Burke
Abstract: In this talk, we will define localized higher-order Alexander modules of CW-complexes which are derived from the integral higher-order Alexander modules by using the Ore localization of certain rings. We will then show that these localized higher-order Alexander modules are right modules over P.I.D.s and with certain conditions are torsion modules. We will also define higher-order Alexander polynomials and then define higher-order degrees of these polynomials. We will show that these higher-order degrees are all significant finite integral valued invariants of certain CW-complexes.

March 25, 2009

An approximation theorem for real-valued functions

Speaker: A. Hager, Wesleyan University
C(X) is the group and lattice (l-group) of continuous real-valued functions on the space X,and C*(X) the sub-l-group of bounded functions.Let H be a divisible sub-l-group of C*(X) containing the constant 1,and separating points from closed sets in X. A version of the Stone-Weierstrasse theorem says (SW) [If X is compact,then H is uniformly dense in C(X)].Now, let locH consist of all f in C(X) which are in H locally on X. [Theorem.If X is locally compact and sigma-compact,then locH is uniformly dense in C(X).] This uses,and includes (SW). Some applications to more general lattice-ordered algebra will be described.

March 04, 2009

Symmetric monoidal structures on R-modules

Speaker: Mark Hovey, Wesleyan University
Abstract: One of the basic questions in mathematics is: Given a set S, how many ways are there to make S into an abelian group? We categorify this by asking: given a category C, how many ways are there to make C into a closed symmetric monoidal category? So we are asking how many ways there are to put some kind of well-behaved tensor product functor on C. We are most interested in this for categories that arise in algebraic topology, but those are beyond our reach for now. This will therefore be more of an "et al" Seminar, concentrating on categories of modules over a ring R. The answer of course depends on R; we give examples where there are no tensor products at all, exactly 1 tensor product, exactly 7 tensor products, and an infinity of tensor products so large that it can't be a set. General theorems are a bit hard to come by so far, but we do have a few.

February 25, 2009

Categorified abelian groups and stable homotopy types

Speaker: Nick Gurski, Yale University
Abstract: One guiding principle of higher category theory is that there is a correspondence between n-dimensional versions of groupoids and homotopy n-types, or spaces whose homotopy groups all vanish above dimension n. For low values of n, this is easy to see, but for high values of n we can use this principle to check that a possible notion of n-dimensional groupoid is reasonable by comparing with homotopy theory. A little less well-known is the stable version of this statement. While more difficult to state precisely, it is actually quite easy to motivate this correspondence by studying some variations on the extremely simple category of 1-dimensional vector spaces. This is joint work with Mikhail Kapranov.

February 18, 2009

Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links

Speaker: D. Shea Vela-Vick, University of Pennsylvania
Abstract: Three-component links in the 3-dimensional sphere were classified up to link homotopy by John Milnor in his senior thesis. A complete set of invariants is given by the pairwise linking numbers p, q and r of the components, and by the residue class of one further integer mu, which is well-defined modulo the greatest common divisor of p, q and r.

To each such link L we associate a geometrically natural characteristic map g_L from the 3-torus to the 2-sphere in such a way that link homotopies of L become homotopies of g_L. Maps of the 3-torus to the 2-sphere were classified up to homotopy by Pontryagin in 1941. A complete set of invariants is given by the degrees p, q and r of their restrictions to the 2-dimensional coordinate subtori, and by the residue class of one further integer nu, an "ambiguous Hopf invariant" which is well-defined modulo twice the greatest common divisor of p, q and r.

We show that the pairwise linking numbers p, q and r of the components of L are equal to the degrees of its characteristic map g_L restricted to the 2-dimensional subtori, and that twice Milnor's mu-invariant for L is equal to Pontryagin's nu-invariant for g_L.

When p, q and r are all zero, the mu- and nu-invariants are ordinary integers. In this case we use J. H. C. Whitehead's integral formula for the Hopf invariant, adapted to maps of the 3-torus to the 2-sphere, to provide an explicit integral formula for nu, and hence for mu.

January 28, 2009

Nontrivial Knotting in 3-Manifolds

Speaker: Prudence Heck, Indiana University
Abstract: Substantial progress has been made toward understanding knots in the 3- sphere. However, not much is known about knotting in arbitrary 3- manifolds. While considering the special case of knots in the Heisenberg manifold, we will investigate knotting that only occurs in manifolds with nontrivial fundamental group. This talk assumes only a basic background in topology and will be accessible to graduate students.

December 16, 2008

On certain classes of minimally almost periodic groups

Speaker: Franklin R. Gould

December 03, 2008

Extending Continuous Functions

Speaker: W. Wistar Comfort, Wesleyan University
Abstract: Given infinite \kappa and given spaces X_i (i \in I), the symbol (X_I)_\kappa denotes the set X_I:=\Pi_{i \in I} X_i with the <\kappa-box topology. The authors give conditions sufficient to ensure for certain subspaces Y\subseteq (X_I)_\kappa and certain spaces Z that every continuous f:Y\rightarrow Z extends continuously over (X_I)_\kappa. Sample theorem, notation to be explained: Let \kappa<<\alpha with \alpha regular, each d(X_i)<\alpha, and \pi_J[Y]=X_J for each non-empty J\subseteq I such that |J|<\alpha. Let the diagonal in Z be the intersection of

[Note. Refreshments will be available in the Math Lounge after the talk.]

(This is joint work with Ivan Gotchev)

November 19, 2008

(Co)homology, Spectra and Linear Functions

Speaker: Stephen Miller (UCONN)
Abstract: We will discuss equivalences between cohomology theories, spectra and linear functors. We will introduce some basic tools of homotopy theory, quote Brown Representability and mention functor calculus. This talk should be accessible to graduate students.

October 29, 2008

The "small subgroup generating property" in minimally almost periodic topological groups

Speaker: Frank Gould, Wesleyan University
Abstract: We examine a set of sufficient conditions, exhibited by some classical examples, that a topological group be "minimally almost periodic", that is, that it have no non-trivial continuous homomorphism to a compact group. Then we focus on a generalization of one of these properties we call the "small subgroup generating property" or SSGP. Finally, we show how we can generate some new examples of Abelian topological groups with this property by using some techniques developed in a paper by the Hungarian mathematicians Ajtai, Havas and Komlss.

October 22, 2008

The complexity of the structure of the knot concordance group

Speaker: Constance Leidy
Abstract: In 1997, T. Cochran, K. Orr, and P. Teichner defined a filtration of the classical knot concordance group. The filtration is defined in terms of gropes or Whitney towers and is connected to the classification of topological 4-manifolds. We will discuss some joint work with Tim Cochran and Shelly Harvey that establishes explicit families of knots that generate infinite rank subgroups of each filtration quotient.

October 01, 2008

C(X) is/is not a topological group

Speaker: A. Hager, Wesleyan University
Abstract: C*(X) is the set of bounded continuous real-valued functions on the topological space X. With addition defined pointwise, its an abelian group. There is a certain Hausdorf topology t on C*(X), which has arisen from considerations in the theory of lattice-ordered groups, in which + is separately continuous. We address the question: When is + jointly continuous, i.e., when is (C*(X) ,+,t) a topological group? Sometimes it is, and sometimes it isnt. The situation is quite set-theoretic. (In the talk, I will be more specific.)

This is joint work with R. Ball, V. Gochev, S. Tedorcevic, S. Zoble.

September 24, 2008

Noncommutative geometry and compactifications of the moduli space of curves

Speaker: Alastair Hamilton, UCONN
Abstract: There is a theorem, due to Kontsevich, which states that the homology of the moduli space of curves can be expressed as the homology of a certain Lie algebra. In this talk I will explain how the homology of a certain compactification of the moduli space, introduced by Kontsevich in his study of Wittens conjectures, can be expressed as the homology of a certain differential graded Lie algebra by deforming Kontsevichs original Lie algebra using a Lie bialgebra structure considered by many authors.

September 17, 2008

Ring spectra of finite global dimension

Speaker: Mark Hovey, Wesleyan University
Abstract: We give an introduction to work with Lockridge on the subject of the title. A ring spectrum is a sort of generalized ring that appears in algebraic topology; basically it is a cohomology theory with a cup product. A good setup for the study of ring spectra is only about 10 years old, and much of the ring theory of ring spectra remains undeveloped. A ring theory has associated to it an ordinary ring (its coefficient ring), and much of our work concerns the difference between the behavior of the ring spectrum and the behavior of its coefficient ring.

May 07, 2008

Ungar's Theorems on Countable Dense Homogeneity Revisited

Speaker: Jan van Mill, Vrije Universiteit, Amsterdam
Abstract: We introduce a slightly stronger form of countable dense homogeneity that for Polish spaces can be characterized topologically in a natural way. Along the way, we generalize theorems obtained by Bennett and Ungar on countable dense homogeneity. We show for example that if the group $G$ makes the space $X$ countable dense homogeneous and no set of size $n{-}1$ separates $X$, then $G$ makes $X$ strongly $n$-homogeneous.

April 23, 2008

Functional dependence on small sets of indices

Speaker: Ivan Gotchev, Central Connecticut State University and Wesleyan University

Title: Functional dependence on small sets of indices

Abstract:

Joint work with W. W. Comfort In this talk we present some new results that improve, generalize, extend to the k-box topology some known results of W. W. Comfort and S. Negrepontis, A. Mischenko, and M. Husek.

April 16, 2008

Stabilizing and flipping Heegaard splittings

Speaker: Jesse Johnson, Yale University

Title: Stabilizing and flipping Heegaard splittings.

Abstract: Every Heegaard splitting of a closed 3-manifold has a stabilization such that there is an isotopy of the 3-manifold that interchanges the handlebodies. I will describe a combinatorial proof that, the genus of the smallest such stabilization is bounded below by the smaller of twice the genus or half the Hempel distance of the original splitting. Similar methods imply that for certain 3-manifolds with boundary, there are pairs of Heegaard surfaces for which the minimal genus of a common stabilization is very high. The proof is inspired by the recent paper by Hass, Thompson and Thurston, which proves similar results using hyperbolic geometry, but without a precise bound in terms of the Hempel distance of the Heegaard splitting.

April 09, 2008

Equational Classes of f-rings

Speaker: James J. Madden (Wesleyan Ph.D 1983), Louisiana State University
Abstract: Rings are assumed commutative, with unit. The smallest equational class in the lattice of the title is HSP(Z), the equational class generated by the integers. Theorem: HSP(Z) is properly smaller that the class of "discrete f-rings", i.e., the class whose t.o.rings have no elements between 0 and 1. Theorem: Yet larger is the class of "nearly discrete f-rings", which is the class whose t.o.rings, modulo their infinitesimals, are discrete, and there are infinitely many different equational classes of nearly discrete f-rings. Another landmark in the lattice is HSP(Q), the class generated by the rationals (called by Henriksen and Isbell "formally real"). It is easy to see that the discrete f-rings--but not the nearly discrete f-rings--are a subclass of HSP(Q). Describing the classes between HSP(Q) and the class of all f-rings involves syzygies in Z[X, Y, ... ]-modules in an interesting way. Describing the classes between HSP(Z) and the discrete f-rings involves diophantine geometry, and a complete classification appears to require solutions to unsolved diophantine problems.

April 02, 2008

Topology and combinatorics

Speaker: Tom Braden, UMASS
Abstract: Toric varieties are algebraic varieties whose geometry is determined by a convex polyhedron, while hypertoric varieties are "quaternionic analogs" of toric varieties whose geometry is determined by a hyperplane arrangement. In each case calculating various topological invariants of the variety gives interesting connections with the combinatorics of the associated polyhedron or arrangement. I will survey some of these connections, particularly focusing on "strange dualities": relations between the topology of toric varieties defined by dual polyhedra and between hypertoric varieties defined by Gale dual arrangements.

March 26, 2008

The distinguishing chromatic number

Speaker: Karen Collins, Wesleyan University
Abstract: This is joint work with Ann Trenk, Wellesley College and Mark Hovey, Wesleyan University.

Collins and Trenk introduced the distinguishing chromatic number of a graph G, \chi_D(G), as the minimum number of colors needed to color the vertices so that

(1) the coloring is a proper graph coloring and (2) the only automorphism of the graph which preserves colors is the identity.

Thus \chi_D(G) is closely related to both the chromatic number and the distinguishing number of a graph. A naive approach to finding an upper bound for the distinguishing chromatic number of a graph would be to expect that it would be less than the sum of the distinguishing number plus the chromatic number. In this talk, we provide a tight bound on the distinguishing chromatic number of graphs with abelian automorphism group \Gamma, where the difference between \chi_D (G) − \chi (G) can be any nonnegative integer, depending on the number of prime power factors of \Gamma; however, D(G) for graphs with abelian automorphism groups is always less than or equal to 2.

March 05, 2008

Some extremal lattice-ordered groups

Speaker: A. Hager, Wesleyan University
Abstract: D(X) is the lattice of all continuous functions on the space X to the extended reals,which are real on a dense set. D(X) is a group (thus l-group) iff X is quasi-F (each dense cozero-set is C*-embedded). Each archimedean l-group G with weak unit e has the canonical Yosida representation in a certain D(YG),e becoming the constant function 1. The convex sub-l-group of G generated by e is denoted G*; it coincides with those elements bounded in the representation. Call G *-maximum if [H* isomorphic to G* implies H embeds in G]. An obvious example is D(X) for X quasi-F. We characterize the *-max G in several ways,show that every G has a minimum essential *-max extension,and write down all examples G which have YG zero-dimensional and quasi-F. We don't know if every example has YG quasi-F.

February 27, 2008

Finding an algorithm to study a problem in the intersection of algebraic geometry, low-dimensional topology, and non-commutative algebra

Speaker: Connie Leidy, Wesleyan University
Abstract: We begin with a complex curve. In order to understand the topology of the affine complement, we are led to studying modules over localized noncommutative group rings. Only a handful of computations of the related invariants have been done by hand. The (still open) problem is finding an algorithm that hopefully can be programmed into a computer to compute these invariants. I hope this talk will be of interest to many people (beyond just topologists) in this department, and perhaps will result in some collaborations.

February 20, 2008

Stable splitting of Generalized Moment-Angle Complexes

Speaker: Martin Bendersky, Hunter College (CUNY)
ABSTRACT: Toric varieties are an important family of spaces occurring in algebraic geometry and topology. In their seminal paper, Davis and Januszkiewicz define the moment-angle complex associated to a toric variety which in turn is associated to a simplicial complex. It was then generalized by Strickland. The cohomology of the generalized moment-angle complex was computed by Goresky and MacPherson in terms of the underlying simplicial complex. In joint work with A. Bahri, F. Cohen and S. Gitler we show that the generalized moment angle complex stably splits into pieces corresponding to the Goresky MacPherson theorem. As a consequence there is an analogous splitting for an arbitrary generalized homology theory.

February 13, 2008

Coherence for braided monoidal structures via topology

Speaker: Nick Gurski, Yale University
Title: Coherence for braided monoidal structures via topology

December 05, 2007

Subclasses of minimally almost periodic abelian groups

Speaker: Frank Gould, Wesleyan University
Abstract: The earliest known examples of abelian topological groups which admit no non-trivial continuous homomorphism to a compact group were found in the 1940s and 1950s. More recently, there has been a lot of attention focused on a stronger property called the fixed point on compacta property or f.p.c.: every continuous action by the group on a compact space has a fixed point. By now, numerous abelian f.p.c. groups have been found, but it is still not known if f.p.c. abelian groups are a proper subclass of m.a.p. abelian groups. We examine this and some other possible subclasses and we point out one easily defined subclass which can be shown to be proper.

November 28, 2007

Mehdi Khorami, Spin cobordism and K-theory

Speaker: Mehdi Khorami, Wesleyan University
Title: Spin cobordism and K-theory
Abstract: Considering the classifying space Bspin(n) for principal spin bundles and the map BSpin(n)
(BO(n), there is the associated Thom space of the induced bundle on BSpin(n) from the universal bundle on BO(n), denoted by MSpin(n). The cohomology obtained from the associated spectrum is called Spin Cobordism.

A work of M. Hopkins and M. Hovey shows that this theory determines K-theory. The main focus of this lecture will be on introducing some of the surrounding concepts leading up to such a determination.

October 10, 2007

About representing certain lattice-ordered groups

Speaker: A. Hager: Wesleyan University
Abstract: G is an archimedean lattice-ordered group. X is a Tychonoff space, D(X) is the set of all continuous functions from X to the extended reals, real on a dense set; for p in X, D(X,p) is all f in D(X) with f(p)=0. Every G has various representations in various D(X) (various authors).

We show: If (J) G has a representation in a D(X) which strongly determines the topology, and (C) G admits all bounded, and all disjoint, countable suprema, then G is isomorphic to a D(K,p) with K compact basically disconnected and P a non-isolated P-point; thus G admits a compatible multiplication.

Unanswered are: Does (C) imply (J)? Does every G satisfy (J)?

(Joint work with R. Ball, D. Johnson, A. Kizanis)

September 26, 2007

New examples of non-slice knots

Speaker: Constance Leidy, Wesleyan University

Abstract: A slice knot is a knot that bounds a smoothly embedded disc in the 4-ball. The set of all knots modulo slice knots can be given a group structure under the operation of connected sum. This group is known as the smooth knot concordance group. I will discuss some recent results of T. Cochran, S. Harvey, and myself that show that knots in a certain family whose slice status was previously unknown are in fact not slice. This result has implications about the structure of the Cochran-Orr-Teichner filtration of the knot concordance group.