Mathematics & Computer Science

Seminars and Colloquia

Topology et al. Seminar

Wednesday, September 26, 2012

04:15 pm - 06:00 pm

Topology Seminar, Biao Wang, Wesleyan: "Minimal Surfaces in Quasi-Fuchian 3-Manifolds"

Abstract: Let M be a quasi-Fuchsian 3-manifold which is homotopic to a closed surface S with genus greater than or equal to 2. It is well known that M admits at least one incompressible minimal surface that is homotopic to S, and at most finitely many of them. In this talk, I will discuss the connection between the short geodesics with large rotation and the number of incompressible minimal surfaces in M.

ESC 638

Monday, May 21, 2012

04:15 pm - 05:15 pm

Topology Seminar, Sergio Estrada, Universidad de Murcia: Derived algebraic geometry via Cotorsion pairs

Abstract: In the talk we present a general construction of model category structures on the category Ch(𝔔𝔠𝔬(X)) of unbounded chain complexes of quasi-coherent sheaves on a scheme X. The construction is based on making compatible the filtrations of individual modules of sections at open affine subsets of X. One of the main advantages of the construction is that it does not require closure under direct limits as previous methods. We apply it to describe the derived category 𝒟(𝔔𝔠𝔬(X)) via various model structures on Ch(𝔔𝔠𝔬(X)). As particular instances, we recover recent results on the flat model structure for complexes of quasi-coherent sheaves. Our approach also includes the case of (infinite-dimensional) vector bundles, and of restricted flat Mittag-Leffler quasi-coherent sheaves, as introduced by Drinfeld. We will comment that the unrestricted case does not induce a model category structure as above in general. Finally, we will discuss on possible extensions of the results to categories of quasi-coherent sheave son stacks.

ESC 638

Wednesday, May 09, 2012

04:15 pm - 06:00 pm

Topology Seminar Chris Schommer-Pries, MIT: The Structure of fusion categories via higher categories and topology.

Abstract: Fusion categories arise in several areas of mathematics and<br/>physics - conformal field theory, operator algebras, representation<br/>theory of quantum groups, and others. They have a rich an fascinating<br/>structure. In this talk we will explain recent work, joint with<br/>Christopher Douglas and Noah Snyder, which ties this structure to the<br/>structure of 3-dimensional topology and homotopy theory. We will show<br/>that fusion categories are the so-called fully-dualizable objects in a<br/>certain natural 3-category, and as such they inherit a homotopy<br/>O(3)-action. The existence of this action gives new topological proofs<br/>of old results. There are also connections to 3-dimensional<br/>topological field theories, and hence 3-manifold invariants. <br/>

ESC 638

Wednesday, May 02, 2012

04:15 pm - 06:00 pm

Topology Seminar Marcy Robertson, University of Western Ontario: Spaces of Operad Structures

Abstract: A multicategory, also known as a colored operad, is simply a generalized non-commutative algebra. In this talk we focus on studying maps between multicategories enriched in simplicial sets. We show that the homotopy function complex of maps between any two multicategories can be computed as the moduli space of a certain small category of (operatic) bimodules. As an application, we show how this description leads to several important decompositions which allow one to compute various geometric invariants.

ESC 638

Wednesday, April 25, 2012

04:15 pm - 06:00 pm

Topology Seminar Vesna Stojanoska, MIT: ON THE STRUCTURE OF tmf˄ tmf

Abstract: The mod-2 homology of tmf ˄ tmf splits as a direct sum of Brown-Gitler modules; however, this splitting is not realizable by spectra as is the analogous one for bo ˄ bo. Mahowald has conjectured that some of these Brown-Gitler modules glue together to give spectra related to tmf , which do split off. In an ongoing work in progress with Mark Behrens and Kyle Ormsby, we hope to show that Mahowald's conjecture holds, and along the way understand as much as we can about the structure of tmf ˄ tmf. We will use Gerd Laures's results on cooperations in tmf-homology modulo torsion, together with an approximation by level structures.

ESC 638

Wednesday, April 18, 2012

04:15 pm - 06:00 pm

Topology Seminar, Michael Warren Institute for Advanced Study: Types, homotopy types and univalence

Abstract: Recent research has revealed a number of interesting connections between homotopy theory and type theory, where the latter is a branch of mathematical logic. Type theory has traditionally found numerous applications in computer science and this new connection promises interesting applications to both homotopy theory and type theory. In this talk we will give an introduction to type theory, its connection with homotopy theory and Voevodsky's (related) program of "univalent foundations". It is our goal to explain this material in a way that is accessible to topologists and, as such, no prior familiarity with type theory on the part of the audience will be assumed.

ESC 638

Wednesday, April 11, 2012

04:15 pm - 06:00 pm

Topology Seminar, Kathryn Lesh, Union College: Bredon homology of partition complexes

Abstract: I'll discuss joint work with Greg Arone and Bill Dwyer on the computation of the homology groups of partition complexes with certain Bredon coefficient systems. It turns out that if n is not a power of a prime, then these are trivial in a strong sense, while if n is a prime power, one gets a description involving a Tits building. The calculation uses techniques of homology approximations due to Dwyer, applied in the context of more general coefficient systems.

ESC 638

Wednesday, March 28, 2012

04:15 pm - 06:00 pm

Topology Seminar, Ismar Volic, Wellesley: Configuration space integrals and the cohomology of knot and link spaces

Abstract: Inspired by the linking number, Bott and Taubes, following work of Bar-Natan, Kontsevich, and others, defined certain integrals over configuration spaces which produce invariants of classical knots and links as well as higher cohomology classes on spaces of knots and links in Euclidean space of dimension >3. The main goal of this talk will be to describe this construction and its many interesting and unexpected connections to Vassiliev and Milnor invariants, calculus of functors, and rational homotopy theory.

ESC 638

Wednesday, March 07, 2012

04:15 pm - 06:00 pm

(Topology Seminar) Michael Ching, Amherst: A Classification of Taylor Towers

Abstract: I'll give an introduction and overview of the Goodwillie's calculus of homotopy functors. Then I'll describe joint work with Greg Arone to understand structures on the Taylor coefficients of a functor from which the full Taylor tower can be reconstructed. For functors from spaces to spectra, the relevant structure is related to right modules over the sequence of En -operads.

ESC 638

Wednesday, February 29, 2012

04:15 pm - 06:00 pm

(Topology Seminar) Clark Barwick, MIT: The Unicity of the Theory of Higher Categories

Abstract: I will report on joint work with Chris Schommer-Pries, in which we show that the space of theories of (∞ , n )-categories is a product of n copies of RP ∞.

ESC 638

Wednesday, December 07, 2011

04:15 pm - 06:00 pm

Megan Heenehan, Wesleyan: Graph Immersions and Minimum Degree (Topology Seminar)

Abstract: A classic question in graph theory is: does a graph with chromatic number n behave, in some way, like a complete graph on n vertices? A possible answer to this question is that an n-chromatic graph contains an immersion of a complete graph. We will look at an attempt to prove this by looking at a stronger statement involving minimum degree. We will consider examples that show a graph with minimum degree <br/>n - 1 need not contain an immersion of a complete graph on n vertices for n greater than or equal to 8.<br/>

ESC 638

Wednesday, November 09, 2011

04:15 pm - 06:00 pm

Topology Seminar: On Legendrian Graphs

Speaker: Danielle O'Donnol, Smith College<br/><br/>Abstract: We investigate Legendrian graphs in R3 with the standard contact structure. We extend the classical invariants, Thurston-Bennequin number and rotation number to Legendrian graphs. We prove that a graph can be Legendrian realized with all its cycles Legendrian unknots with tb=-1 and rot=0 if and only if it does not contain K_4 as a minor. We show that the pair (tb, rot) does not characterize a Legendrian graph up to Legendrian isotopy if the graph contains a cut edge or a cut vertex. This is joint work with Elena Pavelescu.

ESC 638

Wednesday, November 02, 2011

04:15 pm - 06:00 pm

Topology Seminar: Filtering Topologically Slice Knots

Speaker: Peter Horn, Columbia University<br/><br/>Abstract: We define a filtration of the smooth knot concordance group and restrict our attention to the induced filtration on the group of smooth concordance classes of topologically slice knots. This filtration is motivated by Cochran-Orr-Teichner's n-solvable filtration but is more amenable to smooth techniques. I will discuss how this filtration relates to known concordance invariants such as knot signatures, tau, s, and the d-invariants of branched covers. I will also discuss the relation to the Casson-Gordon invariants and von Neumann rho-invariants. I will prove that this filtration is non-trivial at certain levels. This is joint work with Tim Cochran and Shelly Harvey.

ESC 638

Wednesday, October 26, 2011

04:15 pm - 06:00 pm

Topology Seminar: Bounds on the Distinguishing Chromatic Number

Speaker: Karen Collins, Wesleyan University <br/><br/>Abstract: The distinguishing chromatic number of a graph is the smallest number of labels needed for a vertex labeling of the graph which satisfies: any two adjacent vertices are colored distinctly and only the trivial automorphism of the graph preserves the colors. In ordinary coloring, the only graphs which require every vertex to be distinctly colored are the complete graphs; for the distinguishing chromatic number, this role is filled by the complete multipartite graphs. We will show how this fact leads to a beautiful theorem with an elegant proof, and new questions about the distinguishing chromatic number.

ESC 638

Wednesday, October 12, 2011

04:15 pm - 06:00 pm

Topology Seminar: A relationship between representation-theoretic and Floer-theoretic braid invariants

Speaker: Elisendra Grigsby, Boston College<br/><br/>Abstract: Abstract: Given a braid, one can associate to it a collection of 3categorified2 braid invariants in two apparently different ways:<br/>3algebraically,2 via the representation theory of Uq(sl2) (using ideas of Khovanov and Seidel) and 3geometrically," via Floer theory (specifically, Ozsvath-Szabo's Heegaard Floer homology package as extended by Lipshitz-Ozsvath-Thurston). Both collections of invariants are strong enough to detect the trivial braid.<br/><br/>I will discuss what we know so far about the connection between these invariants, focusing on the relationship between the representation theory and the Floer theory. This is joint ongoing work with Denis Auroux and Stephan Wehrli.<br/>

ESC 638

Wednesday, October 05, 2011

04:15 pm - 06:00 pm

Volume of hyperbolic orbifolds

Speaker: Ilesanmi Adeboye, Wesleyan<br/><br/>Abstract: A hyperbolic orbifold is a quotient of hyperbolic space by a discrete group of orientation-preserving isometries. The concept of an orbifold generalizes that of a manifold.<br/><br/>For each dimension, there exists a hyperbolic orbifold of least volume.The identification of the smallest hyperbolic 2-orbifold is a classical result.<br/>The analogous result in dimension 3 is very recent. For higher dimensions, the problem remains open.<br/><br/>In this talk we construct an explicit lower bound for the volume of a hyperbolic orbifold dependent on dimension. We then show how the techniques developed extend to the complex, quaternionic and octonionic hyperbolic spaces. <br/>

ESC 638

Wednesday, September 28, 2011

04:15 pm - 06:00 pm

Non-commutative Filtrations

Speaker: Constance Leidy, Wesleyan University<br/><br/>Abstract: Knot Floer homology was defined by Ozvath-Szabo and independently by Rasmussen. Knot Floer homology was given a combinatorial description by Manolescu-Ozvath-Sarkar and Manolescu-Ozvath-Szabo-Thurston. Thus far, the development of knot Floer homology has been restricted to "abelian"<br/>information about the knot. I will talk about some work in progress to develop noncommutative filtrations that are finer than the Alexander filtration and that will result in homology theories that categorify the higher-order Alexander polynomials of a knot.<br/>

ESC 638

Wednesday, September 21, 2011

04:15 pm - 06:00 pm

The Arithmetic of Fuchsian Groups

Speaker: Ilesanmi Adeboye, Wesleyan<br/><br/>Abstract: The fundamental group of a hyperbolic surface is a *Fuchsian<br/>group*: a discrete group of isometries of the hyperbolic plane. The most important subclass of Fuchsian groups are those obtained by a specific number-theoretic construction.<br/><br/>In this talk we show how to construct *arithmetic Fuchsian goups* and discuss the key role arithmetic groups play in hyperbolic geometry.<br/><br/>This is the second in a series of two introductory and elementary talks on the mathematics I am interested in. Graduate students and non-topologists are particularly welcome.<br/>

ESC 638

Wednesday, September 14, 2011

04:15 pm - 06:00 pm

The Geometry of Fuschian Groups

Speaker: Ilesanmi Adeboye, Wesleyan<br/><br/>Abstract: Orientable 2-dimensional manifolds are rich in mathematical structure. They connect complex analysis with geometry and group theory. A large subcategory of orientable 2-manifolds consists of those that admit a Riemannian metric with constant curvature -1, i.e., the hyperbolic surfaces.<br/>The fundamental groups of hyperbolic surfaces are *Fuchsian groups*:<br/>discrete groups of isometries of the hyperbolic plane.<br/><br/>The first half of the talk will cover the basic geometry of 2-dimensional hyperbolic space viewed as a subset of the complex plane. In the second half, we will define Fuchsian groups, discuss several examples, and describe the corresponding hyperbolic surfaces.<br/><br/>This is the first in a series of two introductory and elementary talks on the mathematics I am interested in. Graduate students and non-topologists are particularly welcome.<br/>

ESC 638

Wednesday, May 04, 2011

04:15 pm - 06:00 pm

On countable dense and strong $n$- homogeneity

Speaker: Jan van Mill<br/><br/>Abstract: We prove that if a space $X$ is countable dense<br/>homogeneous and no set of size $n{-}1$ separates it,<br/>then $X$ is strongly $n$-homogeneous. Our main result<br/>is the construction of an example of a Polish space<br/>$X$ and a (separable metrizable) topological group<br/>$G$ such that (1) $G$ acts on $X$ and makes $X$<br/>strongly $n$-homogeneous for every $n$, (2) $X$<br/>is not countable dense homogeneous. The group cannot<br/>be chosen to be Polish. This example shows that<br/>the assumption on local compactness in Ungar's<br/>homogeneity theorems is essential.<br/>

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 Defense<br/><br/>Abstract: 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, April 20, 2011

04:15 pm - 06:00 pm

The stable derived category of a ring via model categories

Speaker: Daniel Bravo-Vivallo, Wesleyan<br/><br/>Abstract: Let R be a ring. The stable derived category S(R) is defined as<br/>the full subcategory of exact complexes of the homotopy category of<br/>chain complexes of injective R-modules, in the case that R is a<br/>Noetherian ring. We define a model structure over Ch(R), the<br/>category of chain complexes of R-modules, such that its homotopy<br/>category is precisely S(R). This construction allows us to remove<br/>the Noetherian condition on the ring and gives us a better and more<br/>transparent understanding of the properties of S(R).<br/>

ESC 638

Wednesday, March 23, 2011

04:15 pm - 06:00 pm

Topology Seminar

Speaker: Dustin Mulcahey, CUNY<br/><br/>Abstract: 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, March 02, 2011

04:15 pm - 06:00 pm

Homological Dimensions of Ring Spectra

Speaker: David White, Wesleyan<br/><br/>Abstract: 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. I'll 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, February 23, 2011

04:15 pm - 06:00 pm

Lattice Embeddings and the Slice-Ribbon Conjecture

Speaker: Josh Greene, Columbia University<br/><br/>Abstract: 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, December 01, 2010

04:15 pm - 06:00 pm

Ideals in Homotopy Theory

Speaker: Mark Hovey, Wesleyan <br/><br/>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, November 17, 2010

04:15 pm - 06:00 pm

Graph Coloring and Immersions of Complete Graphs

Speaker: Megan Heenehan, Wesleyan University<br/><br/>Abstract: 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, November 10, 2010

04:15 pm - 06:00 pm

An Algebraic Approach to Knot Floer Homology

Speaker: Allison Gilmore<br/><br/>Abstract: 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 03, 2010

04:15 pm - 06:00 pm

Cabling, Concordance and the Four-Ball Genus

Speaker: Jen Hom, University of Pennsylvania<br/><br/>Abstract: 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, October 27, 2010

04:15 pm - 06:00 pm

Doubling Operators and Concordance

Speaker: John Burke, Wesleyan University<br/><br/>Abstract: 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, October 20, 2010

04:15 pm - 06:00 pm

Forcing on perfect matchings in plane bipartite graphs

Speaker: Zhongyuan Che, Wesleyan and Penn State<br/><br/>Abstract: A perfect matching of a graph is a set of a disjoint<br/>edges that covers all vertices of the graph. A perfect matching of<br/>a benzenoid graph is also called a Kekule structure in chemistry.<br/>The forcing number of a perfect matching M is the cardinality of<br/>a smallest subset of M which completely determines M. The root<br/>of this concept came from the innate degree of freedom of a Kekule<br/>structure in a benzenoid graph, which is a 2-connected plane bipar-<br/>tite graph whose each interior face is a unit hexagon. If a Kekule<br/>structure has a forcing bond, then its innate degree of freedom is<br/>at most 1. In this talk, we will introduce the concept of forcing<br/>hexagons of a benzenoid, characterize those benzenoids with forc-<br/>ing hexagons, give the coexistence property of forcing edges and<br/>forcing hexagons in a benzenoid graph, and explain their related<br/>chemical properties. Then we extend this concept to forcing faces<br/>of general plane bipartite graph. This is a joint work with my col-<br/>league Zhibo Chen from Penn State University, Greater Allegheny<br/>Campus.

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, CUNY<br/><br/>Abstract: 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.<br/> 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.<br/>

ESC 638

Wednesday, October 06, 2010

04:15 pm - 06:00 pm

Exceptional Points of Cocompact Fuchsian Groups

Speaker: Joe Fera, Wesleyan University<br/><br/>Abstract: 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, September 29, 2010

04:15 pm - 06:00 pm

Knots and Operads

Speaker: Constance Leidy, Wesleyan University<br/><br/>Abstract: 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, September 08, 2010

04:15 pm - 06:00 pm

Epimorphisms and Projectives in Subcategories of Compact Hausdorff Spaces

Speaker: Anthony Hager, Wesleyan University<br/><br/>Abstract: (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: <br/>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).<br/>

ESC 638

Wednesday, May 05, 2010

04:15 pm - 06:00 pm

Remembering Mel Henriksen and (Some of) His Theorems

Speaker: Wistar Comfort, Wesleyan University<br/><br/>Abstract: 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.<br/> 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.<br/> 1. [1958, MH + John Isbell] Perfect functions. Definition, characterizations, uses and consequences.<br/> 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?<br/> 3. [2000, MH + a committee of co-authors] When is |C(XxY)|=|C(X)||C(Y)|?<br/> 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.<br/>

ESC 618

Wednesday, April 28, 2010

04:15 pm - 06:00 pm

An Algorithm for Bivariate Singularity Analysis

Speaker: Timothy Devries, University of Pennsylvania<br/><br/>Abstract: 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, April 21, 2010

04:15 pm - 06:00 pm

A Universal Coefficient Theorem for Twisted K-Theory

Speaker: Mehdi Sarikhani Khorami, Wesleyan University<br/><br/>Abstract: 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 14, 2010

04:15 pm - 06:00 pm

Bordered Heegaard Floer Homology and Knot and Link Concordance

Speaker: Adam Levine, Columbia University<br/><br/>Abstract: 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 07, 2010

04:15 pm - 06:00 pm

Topology Seminar Mark Radosevich, '03

Spin Fillings of Contact 3-Manifolds

ESC 618

Wednesday, March 03, 2010

04:15 pm - 06:00 pm

Topology Seminar

ESC 618

Wednesday, December 02, 2009

04:15 pm - 05:30 pm

Topology Seminar

Speaker: Jennifer French, MIT<br/><br><br/>Title: Localizations of spaces; examples and construction

ESC 638

Wednesday, November 18, 2009

04:15 pm - 05:30 pm

Topology Seminar

Speaker: A. Hager, Wesleyan University<br/><br><br/>Title: Baire functions and frames of ideals

ESC 638

Wednesday, November 11, 2009

04:15 pm - 05:30 pm

Cardinal Invariants for kappa-Box Products

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

ESC 638