Wednesday, September 17, 2008
04:15 pm
Ring spectra of finite global dimension
Speaker: Mark Hovey, Wesleyan University<br>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.
Exley Science Center - ESC 638
Wednesday, September 24, 2008
04:15 pm
Noncommutative geometry and compactifications of the moduli space of curves
Speaker: Alastair Hamilton, UCONN<br>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.
Exley Science Center - ESC 638
Wednesday, October 01, 2008
04:15 pm
C(X) is/is not a topological group
Speaker: A. Hager, Wesleyan University<br>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.
Exley Science Center - ESC 638
Wednesday, October 22, 2008
04:15 pm
The complexity of the structure of the knot concordance group
Speaker: Constance Leidy<br>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.
Exley Science Center - ESC 638
Wednesday, October 29, 2008
04:15 pm
The "small subgroup generating property" in minimally almost periodic topological groups
Speaker: Frank Gould, Wesleyan University<br>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.
Exley Science Center - ESC 638
Wednesday, November 19, 2008
04:15 pm
(Co)homology, Spectra and Linear Functions
Speaker: Stephen Miller (UCONN)<br>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.
Exley Science Center - ESC 638
Wednesday, December 03, 2008
04:15 pm
Extending Continuous Functions
Speaker: W. Wistar Comfort, Wesleyan University<br>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 <alpha-many regular-closed sets. Then (a) every continuous f:Y\rightarrow Z depends on <\alpha-many coordinates; and (b) every such f extends continuously over (X_I)_\kappa.[Note. Refreshments will be available in the Math Lounge after the talk.](This is joint work with Ivan Gotchev)
Exley Science Center - ESC 638
Wednesday, January 28, 2009
04:15 pm
Nontrivial Knotting in 3-Manifolds
Speaker: Prudence Heck, Indiana University<br>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.
Exley Science Center - ESC 638
Wednesday, February 18, 2009
04:15 pm
Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links
Speaker: D. Shea Vela-Vick, University of Pennsylvania<br>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.
Exley Science Center - ESC 638
Wednesday, February 25, 2009
04:15 pm
Categorified abelian groups and stable homotopy types
Speaker: Nick Gurski, Yale University<br>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.
Exley Science Center - ESC 638
Wednesday, March 04, 2009
04:15 pm
Symmetric monoidal structures on R-modules
Speaker: Mark Hovey, Wesleyan University<br>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.
Exley Science Center - ESC 638
Wednesday, March 25, 2009
04:15 pm
An approximation theorem for real-valued functions
Speaker: A. Hager, Wesleyan University<br>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.
Exley Science Center - ESC 638
Wednesday, April 01, 2009
04:15 pm
Localized Higher-Order Alexander Modules and Higher-Order Degrees
Speaker: John R. Burke<br>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.
Exley Science Center - ESC 638
Wednesday, April 15, 2009
04:15 pm
Polygons inscribed in simple closed curves
Speaker: Elizabeth Denne, Smith College<br>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.
Exley Science Center - ESC 638
Wednesday, April 22, 2009
04:15 pm
A Topological Solution to the A-infinity Deligne Conjecture
Speaker: Rachel Schwell, CCSU<br>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.
Exley Science Center - ESC 638
Wednesday, November 11, 2009
04:15 pm
- 05:30 pm
Cardinal Invariants for kappa-Box Products
Speaker: Wistar Comfort, Wesleyan University<br>Abstract: This derives from joint work with Ivan Gotchev. The symbols w, d and S denote density character, weight and Souslinnumber, respectively, this last defined as follows: for a space X, S(X) is theleast cardinal alpha such that X admits no family of alpha-many pairwisedisjoint 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 theirusual 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 andeach 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) = alphaif alpha is regular, S(X_I) = \alpha^+ otherwise. The authors generalize those and other familiar cardinality results about productspaces 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 <kappa-many coordinates, so X_I = (X_I)_omega). There is much ado here about "the arithmetic of infinite cardinals" andinfinitary combinatorics, since many of the results derive from theorems about thetopological product of discrete spaces. For example, Erdos-Rado theory and sucharrow relations as (2^alpha)^+ --> (alpha^+)^2_alpha play a prominent role. (Motivational combinatorial test question: Given a sequence of finite sets, is therea subsequence whose pairwise intersections coincide?)
ESC 638
Wednesday, November 18, 2009
04:15 pm
- 05:30 pm
Topology Seminar
Speaker: A. Hager, Wesleyan University<br>Title: Baire functions and frames of ideals
ESC 638
Wednesday, December 02, 2009
04:15 pm
- 05:30 pm
Topology Seminar
Speaker: Jennifer French, MIT<br>Title: Localizations of spaces; examples and construction
ESC 638
Wednesday, April 14, 2010
04:15 pm
- 06:00 pm
Bordered Heegaard Floer Homology and Knot and Link Concordance
Speaker: Adam Levine, Columbia UniversityAbstract: Using Bing and Whitehead doubling, we construct a family of links that are topologically but not smoothly slice. We also show that the positive Whitehead double of the Borromean rings is not smoothly slice; whether it is topologically slice remains unknown. the proof uses the new theory of bordered Heegaard Floer homology, illustrating its applicability to the study of satellite knots.
ESC 618
Wednesday, April 21, 2010
04:15 pm
- 06:00 pm
A Universal Coefficient Theorem for Twisted K-Theory
Speaker: Mehdi Sarikhani Khorami, Wesleyan UniversityAbstract: In this talk, we introduce twisted K-theory for a space X equipped with a three dimensional cohomology class. We will discuss the existence of a spectral sequence that converges to twisted K-theory of X, with certain Tor groups in its E2 page. As we will discuss in this talk, all these Tor groups are zero, giving rise to a "Universal Coefficient" isomorphism for twisted K-theory.
ESC 618
Wednesday, April 28, 2010
04:15 pm
- 06:00 pm
An Algorithm for Bivariate Singularity Analysis
Speaker: Timothy Devries, University of PennsylvaniaAbstract: In combinatorics, one often wishes to find a formula for a sequence that has been defined in some combinatorial manner. What is the n^th Fibonacci number, or the n,m^th Delannoy number? A common technique is to embed the sequence as the coefficients of a formal power series, known as a generating function. When this function is locally analytic, we hope that the analytic properties of this function may help us to extract asymptotic formulae for its coefficients. We will explore this technique, known as singularity analysis, in the case that the generating function is bivariate rational. We then sketch a new algorithm that, for many such generating functions, automatically produces these asymptotic formulae. In spite of its combinatorial origins, this algorithm is quite geometric in nature (touching on topics from homology theory, Morse theory, and computational algebraic geometry).
ESC 618
Wednesday, May 05, 2010
04:15 pm
- 06:00 pm
Remembering Mel Henriksen and (Some of) His Theorems
Speaker: Wistar Comfort, Wesleyan UniversityAbstract: Mel Henriksen died October 15, 2009. He was an active and internationally visible set-theoretic topologist, a good personal friend to several members of the department, and a frequent Van Vleck visitor. This talk derives from an address of 5/27/10 to a memorial conference in Mel's honor at Harvey Mudd College in California. For present purposes I will truncate the personal + social reminiscences to a set of small but positive measure, focusing instead on three of his theorems and some consequences (some new, I believe). These are the three general settings. 1. [1958, MH + John Isbell] Perfect functions. Definition, characterizations, uses and consequences. 2. [1961, MH + Isbell + Donald Johnson] Theorems of the form: If a space X has property P, then every Baire set in X has property Q. Is a closed Baire set necessarily a zero-set? 3. [2000, MH + a committee of co-authors] When is |C(XxY)|=|C(X)||C(Y)|? Remark. Mel's work had a definitive flavor, but it generated nevertheless some attractive unsolved problems which seem to this day to be susceptible to solution. Some of these will be mentioned, as time permits.
ESC 618
Wednesday, September 08, 2010
04:15 pm
- 06:00 pm
Epimorphisms and Projectives in Subcategories of Compact Hausdorff Spaces
Speaker: Anthony Hager, Wesleyan UniversityAbstract: (This is joint work with B.Banaschewski.) Familiar to algebraists are the notions in a category of monomorphism (monic),essential extension,injective and injective hull. It's not too hard to show (with a few assumptions): (*) If the category C has all injective hulls,and if V is a full subcategory closed under homomorphic images and coproducts,then V has all injective hulls IF m(v,C):every V-monic is C-monic. The hard part is understanding m(V,C). Towards that,we (1) dualize to epimorphism (epic),cover,projective and projective cover,and (2)restrict to the simple (?) category Comp (compact Hausdorff spaces),where: epic=surjective,cover means [surjection with no proper closed subspace mapping onto],projective =extremally disconnected,and Comp has all projective covers;the dual of the properties of V above is [R is closed under formation of closed subspaces and products],the dual of (*) holds,and the dual of m(V,C) is e(R,Comp):every R-epic is surjective.Then,it's (really) easy to show:e(R,Comp) iff R has a non-void projective;then the Comp-projective covers are R-projective covers. We note that this holds for R=ZDComp (zero-dimensional),and describe an R where this fails (in fact,a proper class of such R's (if there is no measurable cardinal)). We do not know if there is Any R other than Comp and ZDComp with e(R,Comp).
ESC 638
Wednesday, September 29, 2010
04:15 pm
- 06:00 pm
Knots and Operads
Speaker: Constance Leidy, Wesleyan UniversityAbstract: In a recent preprint, Ryan Budney introduces a new topological operad called the splicing operad. This is essentially the satellite operation on knots. We will review this operad and introduce a generalization to genetic infection by a string link. We will discuss how this operad can provide additional information about the knot concordance group.
ESC 638
Wednesday, October 06, 2010
04:15 pm
- 06:00 pm
Exceptional Points of Cocompact Fuchsian Groups
Speaker: Joe Fera, Wesleyan UniversityAbstract: A hyperbolic surface S is the quotient of the hyperbolic plane H by a Fuchsian group G. The action of G can be studied using a convex hyperbolic polygon based at any point in H called the Dirichlet region. When S is compact of genus g, G is called cocompact and each of its associated Dirichlet region has at most 12g-6 sides; this upper-bound is attained for Dirichlet regions centered at almost every point in H. Points which admit Dirichlet regions with less than 12g-6 sides are called exceptional and comprise a zero-measure subset of H. In this talk, we prove that exceptional points always exist for cocompact Fuchsian groups. We also define and discuss the existence of higher order exceptional points.
ESC 638
Wednesday, October 13, 2010
04:15 pm
- 06:00 pm
The String Topology Loop Project As a Twisted Tensor Product
Speaker: Micah Miller, CUNYAbstract: String Topology is the study of the free loop space of a manifold LM. The loop product, defined on the homology of LM, is described intuitively as a combination of the intersection product on M and loop concatenation in the based loop space of M. However, since the intersection product is well-defined only on transversally intersecting chains, this description is incomplete. Brown's theory of twisting cochains provides a chain model of a bundle in terms of the chains on the base and chains on the fiber. We extend this theory so that it can be applied to String Topology. We give a precise definition of the loop product defined at the chain level. In doing so, we will also see that the loop product is the universal enveloping algebra of a Lie algebra.
ESC 638
Wednesday, October 20, 2010
04:15 pm
- 06:00 pm
Forcing on perfect matchings in plane bipartite graphs
Speaker: Zhongyuan Che, Wesleyan and Penn StateAbstract: A perfect matching of a graph is a set of a disjointedges that covers all vertices of the graph. A perfect matching ofa benzenoid graph is also called a Kekule structure in chemistry.The forcing number of a perfect matching M is the cardinality ofa smallest subset of M which completely determines M. The rootof this concept came from the innate degree of freedom of a Kekulestructure in a benzenoid graph, which is a 2-connected plane bipar-tite graph whose each interior face is a unit hexagon. If a Kekulestructure has a forcing bond, then its innate degree of freedom isat most 1. In this talk, we will introduce the concept of forcinghexagons of a benzenoid, characterize those benzenoids with forc-ing hexagons, give the coexistence property of forcing edges andforcing hexagons in a benzenoid graph, and explain their relatedchemical properties. Then we extend this concept to forcing facesof general plane bipartite graph. This is a joint work with my col-league Zhibo Chen from Penn State University, Greater AlleghenyCampus.
ESC 638
Wednesday, October 27, 2010
04:15 pm
- 06:00 pm
Doubling Operators and Concordance
Speaker: John Burke, Wesleyan UniversityAbstract: In this talk, we will define the concordance group of knots and discuss the n-solvable filtration of this group defined by Cochran, Orr, and Teichner. Satellite operations will be introduced followed by a generalization called infection by string links. We will then discuss some of the previous results about the structure of the concordance group uncovered with satellite operations. In particular, we will discuss the structure of the abelian quotient groups, G_n, of n-solvable knots modulo n.5-solvable knots. We will end by discussing how one must use genetic infection with string links and not satellite constructions alone if one wishes to fully understand the structure of the knot concordance group. In particular, we will construct knots in G_n which are linearly independent from previously studied knots.
ESC 638
Wednesday, November 03, 2010
04:15 pm
- 06:00 pm
Cabling, Concordance and the Four-Ball Genus
Speaker: Jen Hom, University of PennsylvaniaAbstract: The Ozsvath-Szabo concordance invariant, tau, gives a homomorphism from the knot concordance group to the integers and also a lower bound on the four-ball genus of knot. We will give a formula for tau of the (p, q)-cable of a knot K in terms of p, q, tau(K), and a new concordance invariant, epsilon(K), associated to the knot Floer complex. We will discuss various properties of the invariant epsilon; in particular, epsilon is strictly stronger than tau at obstructing sliceness, and in certain cases, gives a better lower bound on the four-ball genus than tau alone.
ESC 638
Wednesday, November 10, 2010
04:15 pm
- 06:00 pm
An Algebraic Approach to Knot Floer Homology
Speaker: Allison GilmoreAbstract: Ozsvath and Szabo gave the first completely algebraic description of knot Floer homology via a cube of resolutions construction. Starting with a braid diagram for a knot, one singularizes or smooths each crossing, then associates an algebra to each resulting singular braid. These can be arranged into a chain complex that computes knot Floer homology. After introducing knot Floer homology in general, I will explain this construction, then outline a fully algebraic proof of invariance for knot Floer homology that avoids any mention of holomorphic disks or grid diagrams. I will close by describing some potential applications of this algebraic approach to knot Floer homology, including potential connections with Khovanov-Rozansky's HOMFLY-PT homology.
ESC 638
Wednesday, November 17, 2010
04:15 pm
- 06:00 pm
Graph Coloring and Immersions of Complete Graphs
Speaker: Megan Heenehan, Wesleyan UniversityAbstract: One of the interesting open questions in graph coloring is: if a graph is t-chromatic does it contain (in some way) a complete graph on t vertices? Attempts to solve this problem have included looking for subdivisions of complete graphs, minors of complete graphs, and immersions of complete graphs. This talk will focus on graph immersion. We say a graph H is immersed in a graph G if and only if there exists an injection from the vertices of H to the vertices of G for which the images of adjacent elements in H are connected in G by edge disjoint paths. In 2003 Abu-Khzam and Langston conjectured that if a graph G has chromatic number greater than or equal to t, then there is a complete graph on t vertices immersed in G. We will look at the progress that has been made towards proving this conjecture by considering the connectivity of t-immersion-critical graphs. We will also discuss why immersions may be the right approach to this problem.
ESC 638
Wednesday, December 01, 2010
04:15 pm
- 06:00 pm
Ideals in Homotopy Theory
Speaker: Mark Hovey, Wesleyan Abstract: In homotopy theory, there is a good notion of a ring; namely, a cohomology theory with cup products. So this includes ordinary cohomology, K-theory, and cobordism. But these rings do not have elements in the usual sense, so there has been no corresponding theory of ideals. About 2005, Jeff Smith gave a talk in which he outlined how a theory of ideals might be created. In this preliminary report, we offer a different approach to Smith's ideals, putting many of his ideas on a firm foundation, although many questions remain open.
ESC 638
Wednesday, February 23, 2011
04:15 pm
- 06:00 pm
Lattice Embeddings and the Slice-Ribbon Conjecture
Speaker: Josh Greene, Columbia UniversityAbstract: A knot in the three-sphere is called *slice* if it bounds a smoothly embedded disk whose interior may pass into the interior of a four-ball. I will discuss an obstruction to a knot being slice coming from Floer homology that takes the form of a simple lattice embedding condition. Furthermore, this obstruction is sufficient to distinguish the slice knots amongst the odd three-stranded pretzels, settling the slice-ribbon conjecture for this family of knots. This is joint work with Slaven Jabuka.
ESC 638
Wednesday, March 02, 2011
04:15 pm
- 06:00 pm
Homological Dimensions of Ring Spectra
Speaker: David White, WesleyanAbstract: Cohomology theories are examples of ring-like objects called S-algebras in homotopy theory. They are ring-like when viewed through a diagrammatic lens, but have no points with which to do traditional ring theory. To measure the complexity of these objects one would like to generalize the usual notions of dimension in ring theory. Ill discuss how to do this, give numerous examples, and discuss when the analogy with traditional ring theory is nice and when it is not.
ESC 638
Wednesday, March 23, 2011
04:15 pm
- 06:00 pm
Topology Seminar
Speaker: Dustin Mulcahey, CUNYAbstract: We give a comanadic interpretation of Morava's Change of Rings theorem, which is part of the construction of the chromatic spectral sequence. We then go over an approach to formulating an unstable version of the Morava change of rings theorem in terms of this more abstract formulation.
ESC 638
Wednesday, April 20, 2011
04:15 pm
- 06:00 pm
The stable derived category of a ring via model categories
Speaker: Daniel Bravo-Vivallo, WesleyanAbstract: Let R be a ring. The stable derived category S(R) is defined asthe full subcategory of exact complexes of the homotopy category ofchain complexes of injective R-modules, in the case that R is aNoetherian ring. We define a model structure over Ch(R), thecategory of chain complexes of R-modules, such that its homotopycategory is precisely S(R). This construction allows us to removethe Noetherian condition on the ring and gives us a better and moretransparent understanding of the properties of S(R).
ESC 638
Wednesday, April 27, 2011
04:15 pm
- 06:00 pm
On infection by string links and new structure of the knot concordance group
Speaker: John Burke, PhD DefenseAbstract: In this talk, we will define the concordance group of knots and examine the structure of this group via the n-solvable filtration. Our discussion will include some of the previous results about the structure of the concordance group. In particular, we will discuss the structure of the abelian quotient groups of n-solvable knots modulo n.5-solvable knots. This will be followed by presenting generalizations of techniques used by Cochran, Harvey, Leidy in their study of the n-solvable filtration. We will define infection by a string link and define when a multivariable polynomial is strongly coprime to another. This will culminate with the result that there is indeed new structure in the n-solvable filtration which is revealed by considering infection by string links and is distinct from nearly all previously known structure that was determined by infection by knots only.
ESC 638
Wednesday, May 04, 2011
04:15 pm
- 06:00 pm
On countable dense and strong $n$- homogeneity
Speaker: Jan van MillAbstract: We prove that if a space $X$ is countable densehomogeneous and no set of size $n{-}1$ separates it,then $X$ is strongly $n$-homogeneous. Our main resultis the construction of an example of a Polish space$X$ and a (separable metrizable) topological group$G$ such that (1) $G$ acts on $X$ and makes $X$strongly $n$-homogeneous for every $n$, (2) $X$is not countable dense homogeneous. The group cannotbe chosen to be Polish. This example shows thatthe assumption on local compactness in Ungar'shomogeneity theorems is essential.
ESC 638
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