# Seminars and Colloquia

## Topology et al. Seminar

Wednesday, September 16, 2015

04:15 pm - 05:15 pm

Topology Seminar, Dan Licata (Wes): "Structural Proof Theory of Adjoint Functors"

Abstract: Adjoint functors are a central tool in category theory, because an adjunction provides a well-behaved way to compare two objects that belong to two different categories. They are related to modal logic, because every adjunction gives rise to both a comonad, which is like a necessitation modality, and a monad, which is like a possibility modality. In this talk, I will describe some work, joint with Mike Shulman (University of San Diego), investigating the structural proof theory of adjoint functors. An application is integrating the synthetic homotopy theory of homotopy type theory with the synthetic topology of Lawvere's axiomatic cohesion, which opens up new opportunities for proofs and formalizations in type theory.

Exley Science Center (Tower)

Wednesday, April 22, 2015

04:15 pm - 05:30 pm

Topology Seminar, Chris Arretines (CUNY): 'Combinatorial and algorithmic questions in low dimensional topology'

Abstract: I will discuss how several geometric and topological questions related to curves on surfaces can be presented in terms of combinatorial data, and I will outline algorithmic methods for answering these questions. In particular, I will discuss how to determine intersection numbers for collections of curves, and how to determine whether or not a collection of curves is filling, which means that the complement of the curves is a disjoint union of topological disks. The study of these properties is motivated in part by the role of curves on surfaces in proving several powerful theorems related to Teichmueller space, which I will try to convey. Time permitting, I will also showcase some interesting constructions motivated by exploring the above algorithms.

ESC 638

Wednesday, April 15, 2015

04:15 pm - 05:30 pm

PhD Defense/Topology Seminar, Brett Smith (Wes): 'On Minimality of Planar Graphs with Respect to Treewdith'

Abstract: Robertson and Seymour developed the parameter, treewidth, in order to prove the Graph Minors Theorem, which says that the class of finite, undirected graphs is well-quasi-ordered by the minor relation. The treewidth of a graph can be defined using structures called brambles. A clique is a simple example of a bramble, but brambles can be much more intricate. The primary result in our work is a characterization of a family brambles in planar graphs, called 3-nets, along with an algorithm for determining the order of any bramble in this family. The 3-nets lead us to several results on the minimality of planar graphs with respect to tree width.

ESC 638

Wednesday, March 25, 2015

04:15 pm - 05:30 pm

Topology Seminar, Ann Trenk (Wellesley): 'Unit Interval Orders of Open and Closed Intervals'

Abstract: The class of unit interval orders arises in scheduling problems and has a lovely forbidden poset characterization. The characterization remains the same whether the intervals used in the representation are all open intervals or all closed intervals. In this talk we consider the class of posets that arise when both open and closed unit intervals are allowed in the same representation. We give a structural characterization of this class of posets and an efficient algorithm for recognizing the class. The algorithm takes a poset P as input and either produces a representation or returns a forbidden poset induced in P.

ESC 638

Wednesday, March 04, 2015

04:15 pm - 05:30 pm

Topology Seminar, Anthony Hager (Wes): 'The sigma property in C(X)'

Abstract: A vector lattice is a real linear space with a compatible lattice-order.Examples are C(X) (continuous functions from the topological space X to the reals ),and any abstract 'measurable functions mod null functions'.The sigma property of a vector lattice A is (s) For each sequence (a(n)) in A+,there are a sequence (p(n)) of positive reals and a in A,for which p(n)a(n) < a for each n.Examples:C(X) for compact X (trivial);Lebesgue Measurable mod Null (not trivial;connected with Egoroff's theorem).An application:If a quotient A/I has (s),the the quotient map lifts countable disjoint sets to disjoint sets.We consider which C(X) have (s),for example: For discrete X,C(X) has (s) iff the cardinal of X < the bounding number b. For metrizable X ,C(X) has (s) iff X is locally compact and each open cover has a sub cover of size <b. (This much studied'b' is the minimum among cardinals m for which any family of functions from the positive integers N to N of size < m is bounded in the order of eventual domination for such functions.It is uncountable,no bigger than c,and regular.In ZFC,not much more can be said.)

ESC 638

Wednesday, November 19, 2014

04:15 pm - 05:15 pm

Topology Seminar, W. W. Comfort (Wesleyan Emeritus): 'Counting Compact Group Topologies'

Abstract: Given a group K let 𝔠𝔤𝔱 (K ) be the set of Hausdorff compact group topologies on K. The authors ask: when |K | = κ ≥ ω, what are the possible cardinalities of a pairwise homeomorphic subset (h) ⊆ 𝔠𝔤𝔱 (K ) [resp., pairwise nonhomeomorphic subset 𝕍(n) ⊆ 𝔠𝔤𝔱 (K )]? Revisiting (sometimes improving) therorems of Halmos, Hulanicki, Fuchs, Hawley, Chuan/Liu and Kirku, the authors show inter alia:Always |𝕍 (h )| ≤ 2κ and |𝕍 (h )| ≤ κ.If K is abelian and some 𝖳 ∈ 𝔠𝔤𝔱 (K ) is connected, then |𝕍 (h )| = 2κ does occur. In particular for λ ≥ ω and K = ℝλ or K = 𝕋λ, |𝕍 (h )| = 2(2λ) does occur.[K not necessarily abelian] If some 𝖳 ∈ 𝔠𝔤𝔱 (K ) is connected and the connected component Z0 ( K, 𝖳) of the center of ( K, 𝖳) satisfies π1(Z0 ( K, 𝖳)) ≠ {0}, then |𝔠𝔤𝔱 (K)| ≥ 2𝔠.Corollary to 3: Every nonsemisimple compact connected Lie group admits exactly 2𝔠-many compact group topologies.For K = 𝕋: |𝕍(n)| = 𝔠 occurs in ZFC.For K = ℝ: |(n)| = ω occurs in ZFC; |𝕍(n)| = ω is best possible in ZFC+CH; and |𝕍(n)| > ω is consistent with ZFC.*Joint work with Dieter Remus

Wednesday, November 05, 2014

04:15 pm - 05:15 pm

Topology Seminar, Sara Maloni (Brown): 'Polyhedra inscribed in quadrics, anti-de Sitter and half-pipe geometry.'

Abstract: In this talk we will show that a planar graph is the1-skeleton of a Euclidean polyhedron inscribed in a hyperboloid if and only if it is the 1-skeleton of a Euclidean polyhedron inscribed in a cylinder if and only if it is the 1-skeleton of a Euclidean polyhedron inscribed in a sphere and has a Hamiltonian cycle. This result follows from the characterisation of ideal polyhedra in anti-de Sitter and half-pipe space in terms of their dihedral angles and induced metric on its boundary.(This is joint work with J Danciger and J-M Schlenker.)

Wednesday, October 08, 2014

04:15 pm - 05:15 pm

Topology Seminar, Jen Hom (Columbia): 'An infinite rank summand of topologically slice knots'

Abstract: Let T denote the subgroup of the smooth knot concordance group generated by topologically slice knots. Endo showed that T contains an infinite rank subgroup, and Livingston and Manolescu-Owens showed that T contains a summand of rank three. We show that in fact T contains an infinite rank summand. The proof relies on the knot Floer homology package of Ozsvath-Szabo and the concordance invariant epsilon.

Wednesday, September 24, 2014

04:15 pm - 05:15 pm

Topology Seminar, Arumina Ray (Brandeis): 'Satellite operations and knot concordance'

Abstract: Satellite operations are a natural generalization of the connected sum operation on knots, and are of interest both within and beyond knot theory. I will discuss a number of recent results about satellite operations on topological and smooth knot concordance classes, particularly winding number one satellite operators, and how they contribute to the conjecture that the concordance groups are a fractal space. Some of the results I will talk about are joint work with Tim Cochran and/or Christopher Davis.

Wednesday, September 17, 2014

04:15 pm - 05:15 pm

Topology Seminar, Matthew Willis (Conn College): 'Local Condition Sets for DemazureTableaux'

Abstract: In a Lie theoretic setting, choosing a dominant weight ⋋ and aWeyl group element 𝓌 determines a Demazure module. This module yields aDemazure character obtained by summing formal exponentials over its set ofweights. In type A, this character can be produced via the weights of a certainset of semistandard tableaux, called Demazure tableaux, using the notion of\right keys introduced by Lascoux and Schutzenberger. In this talk we willpresent a new method to compute the right key of a semistandard tableau.We will then use it to provide \local conditions for each value in a givenDemazure tableau, analogous to those of semistandard tableaux. If timeallows we will discuss further applications such as Demazure atoms, convexpolytopes, and connections to Flag-Schur functions.

Wednesday, April 16, 2014

04:15 pm - 05:30 pm

Topology Seminar, Peter Horn (Syracuse): 'Computing higher order alexander polynomials of knots'

Abstract: The classical Alexander polynomial of a knot can be defined in several ways, one of which is via covering spaces. Using higher covering spaces, Cochran defined the higher-order Alexander polynomials. It is known that the degree of the classical Alexander polynomial gives a lower bound for the genus of a knot, and so do the degrees of the higher-order Alexander polynomials. These higher-order bounds are known to be stronger than the classical bound for satellite knots, but little is known about low crossing knots. We will present an algorithm to compute the degree of the first higher-order Alexander polynomial of any knot, and we will discuss some interesting computations and results.

Wednesday, April 09, 2014

04:15 pm - 05:30 pm

Topology Seminar, John Baldwin (Boston College): 'From knots invariants to bordered Floer homology '

Abstract: I'll describe in this talk some motivation for a recent construction of bordered monopole Floer homology (joint work with Jon Bloom). I'll start with a review of the classical Jones polynomial of a knot in the 3-sphere. I'll then introduce Khovanov homology, a more sophisticated homological knot invariant which encodes the Jones polynomial but is a stronger invariant in general. In particular, Khovanov homology detects the unknot, whereas the analogous question for the Jones polynomial remains open. This fact was proven using a relationship between Khovanov homology and an even more sophisticated invariant of knots called instanton Floer homology. I'll survey some known relationships between the Khovanov homology of knots and Floer homology theories of knots and 3-manifolds. I'll then describe how these sorts of relationships, combined with work of Khovanov on tangle invariants, has motivated a recent construction of bordered monopole Floer homology, which provides invariants of 3-manifolds with parametrized boundary and a pairing formula for computing the monopole Floer homology of a closed 3-manifold from the invariants associated to its pieces.

Wednesday, March 26, 2014

04:15 pm - 05:30 pm

Topology Seminar, Karen L. Collins (Wes): 'Split graphs and counting NG-graphs'

Abstract: A graph G is an NG-graph} if it satisfies the Nordhaus-Gaddum inequality, that its chromatic number plus the chromatic number of its complement is less than or equal to its number of vertices plus 1, with equality. In this talk, we will explore connections between NG-graphs and split graphs and count the number of NG-graphs on n vertices.This is joint work with Ann N. Trenk (Wellesley College).

Wednesday, March 05, 2014

04:15 pm - 05:30 pm

Topology Seminar, John Burke PhD '11 (Rhode Island College): 'A colored operad for string link infection'

Wednesday, February 26, 2014

04:15 pm - 05:30 pm

Topology Seminar, Dan Licata (Wes): 'Eilenberg-MacLane Spaces in Homotopy Type Theory

Abstract: Homotopy type theory is an extension of Martin-Lof type theory with principles inspired by category theory and homotopy theory. With these extensions, type theory can be used to construct proofs ofhomotopy-theoretic theorems, in a way that is very amenable to computer-checked proofs in proof assistants such as Coq and Agda. In this talk, we describe a construction of Eilenberg-MacLane spaces. For an abelian group G, an Eilenberg-MacLane space K(G,n) is a space (type) whose n^th homotopy group is G, and whose homotopy groups are trivial otherwise. Their construction in type theory is an illustrative example, which ties together many of the constructions and methods that have been used in homotopy type theory so far. The construction has been formalized using the Agda proof assistant.

Wednesday, February 19, 2014

04:15 pm - 05:30 pm

Topology Seminar, Biao Wang (Wes): 'Stability of Spherical Minimal Catenoids in Hyperbolic 3-Space

Abstract: In this talk, we use the ball model for hyperbolic 3-space, i.e. a unit ball in three dimensional Euclidean space with hyperbolic metric.A minimal spherical catenoid in hyperbolic 3-space is a surface of revolution whose mean curvature is identically equal to zero. Each minimal spherical catenoid is symmetric about some hyperbolic plane, so we may consider all minimal spherical catenoids which are symmetric about the xy-plane and whose rotation axis is the z-axis. Then all catenoids in this family can be parametrized by one parameter, i.e. the (hyperbolic) distance from the catenoid to the origin. We will show that there exists a constant ac≈ 0.49577389 such that the minimal catenoid is stable if its distance to the origin is grater than or equal to ac, and its unstable if the distance is less than ac.

Wednesday, February 12, 2014

04:15 pm - 05:30 pm

Topology Seminar, Lew Robertson (Wes): 'Prime number density in the intervals (0, x] and (x 2x]'

Abstract: Assuming the validity of the Riemann hypothesis, it follows that lim┬(x→∞)⁡〖(logx-x/(π(x)))^ 〗≔ K∞ exists and equals an interesting number. However, this limit result is valid without RH and hence 1/(-K∞+logx) is the optimal (π (x))/x approximator in the Legendre family 1/(C+logx). The limit 〖lim〗┬(x→∞)(2 - (π(2x))/(π(x))) logx also exists, and once again the limit value is an interesting number. A combination of the limit results yields the possibly surprising conclusion 1/log2x < (π(2x)- π(x))/x < 1/logx , valid for x sufficiently large. The quotients (li (x))/x of the logarithmic integral function can be accurately approximated in terms of logx. These bounding approximtors are a key ingredient from proving that □(1/(- 1+lnx)) + □(1/(〖ln〗^(3 ) x)) < □((π (x))/x) < □(1/(-1+lnx)) + □(1.4/(〖ln〗^(3 ) x))is eventually valid.

Wednesday, February 05, 2014

04:15 pm - 05:30 pm

Topology Seminar, Dan Licata (Wes): 'Eilenberg-MacLane Spaces in Homotopy Type Theory'

Abstract: Homotopy type theory is an extension of Martin-Lof type theory with principles inspired by category theory and homotopy theory. With these extensions, type theory can be used to construct proofs ofhomotopy-theoretic theorems, in a way that is very amenable to computer-checked proofs in proof assistants such as Coq and Agda. In this talk, we describe a construction of Eilenberg-MacLane spaces. For an abelian group G, an Eilenberg-MacLane space K(G,n) is a space (type) whose n^th homotopy group is G, and whose homotopy groups are trivial otherwise. Their construction in type theory is an illustrative example, which ties together many of the constructions and methods that have been used in homotopy type theory so far. The construction has been formalized using the Agda proof assistant.

Wednesday, January 29, 2014

04:15 pm - 05:30 pm

Topology Seminar, Wis Comfort (Wes): The Density Nucleus of a Topological Group

Abstract: Given a topological group G (usually compact abelian), the authors study the poset 𝒟 = (G) of dense subgroups of G and its impact on the algebraic structure of G. A key tool for this is the subgroup den(G) ≔ ⋂𝒟.Definition. For a cardinal 𝜅 ≥ 1, a topological group is in the class ℱf (𝜅) [resp., ℱ (𝜅); resp., ℱ2(𝜅); resp., ℱad (𝜅)] if some family of 𝜅-many dense supgroups of G is independent and freely generated; [resp., independent; resp., pairwise independent; resp., pairwise almost disjoint].[Note for each 𝜅 ≥ 1 the class-theoretic inclusions ℱf (𝜅) ⊆ ℱ (𝜅) ⊆ ℱ2 (𝜅) ⊆ ℱad (𝜅)].Let K be a compact abelian group. Then:1.K ∈ ℱad (2).2.There are D0, D1 ∈ 𝒟 (K) such that den (K) = D0 ∩ D1.3.K ∈ ℱ (2) ⇔ either r (K) >0 or each leading Ulm-Kaplansky invariant of K is infinite.4.K ∈ ℱ (𝜅) ⇔ K ∈ ℱ2 (𝜅).5.If K is torsion and K ∈ ℱ (2), then K ∈ ℱ (𝜅) ⇔ 𝜅 ≤ each leading Ulm-Kaplansky invariant of K.6.If r(K) > 0, then K ∈ ℱ(𝜅) ⇔ 𝜅 ≤ r (K); if in addition r(K) ≥ d(K), then K ∈ ℱf (𝜅) ⇔ K ∈ ℱ (𝜅).This is joint work with Dikran Dikranjan.

Wednesday, December 04, 2013

04:15 pm - 05:30 pm

Topology Seminar, Brett Smith (Wesleyan): 'Planar Grids as Trees'

Abstract: In their series of papers, Graph Minors, Robertson and Seymour introduce a tree decomposition of a graph. This definition leads to a useful graph property called tree width. The n x n-grid is the classical example of a planar graph of tree width n. We will prove this graph is not minimal in the sense that it contains a proper minor which also has tree width n. We will also characterize the edges in the n x n-graph whose removal reduces the tree width and show that the (n + 1) -triangular-grid also has tree width n. These graphs provide insight into the family of minor-minimal planar graphs of tree width n.

Wednesday, November 20, 2013

04:15 pm - 05:30 pm

Topology Seminar, Tobias Barthel (Harvard): 'Completed power operations for Morava E-theory'

Abstract: Morava E-theory is an important cohomology theory in the chromatic perspective on stable homotopy theory. After a brief introduction to chromatic homotopy theory, we discuss joint work with Martin Frankland on an algebraic theory of power operations for Morava E-theory. Our main theorem extends previous work of Charles Rezk by taking completions into account. Along the way, we study the category of L-complete modules, an abelian category with some exotic properties of independent interest.

Wednesday, November 13, 2013

04:15 pm - 05:30 pm

Topology Seminar, Cameron Hill (Wes): 'Category, Measure, and Expansions of Countably Categorical Structures.'

Abstract: A natural way to approach expansions of a fixed countably-categorical structure is through the compact metric space of such expansions. In this talk, I will discuss some ways to recover typical'' but constrained expansions of a fixed structure from the points of view of measure and category. Regarding measure, I will discuss a slight generalization, and a new proof, of a result of Ackerman-Freer-Patel on concentrating invariant measures on the isomorphism type of a structure. And for category, I will discuss using certain kinds of ultrafilters to extract a generic profile'' of a given expansion. Time permitting, I will share some applications to zero-one laws, structural Ramsey theory, and/or a weak form of the Lang-Weil estimates.

Wednesday, November 06, 2013

04:15 pm - 05:30 pm

Topology Seminar, Emily Riehl (Harvard): 'Algebraic model categories'

Abstract: One of the most successful axiomatizations for abstract homotopy theory is Quillen's model categories. In this talk, I'll introduce a more structured variant, called algebraic model categories, and show that the vast majority of model categories can be algebraicized. If a simple cellularity condition is satisfied, a monoidal model category structure can be algebraicized as well. As time permits, I'll discuss implications of this new theory for recognizing cofibrations, finding natural solutions to lifting problems, and constructing functorial factorizations in non-cofibrantly generated settings (at least as this notion is classically understood).

Wednesday, October 30, 2013

04:15 pm - 05:30 pm

Topology Seminar, Wistar Comfort (Wesleyan): 'The Density Nucleus of a Topological Group'

Abstract: Given a topological group G (usually compact abelian), the authors study the poset 𝒟 = 𝒟(G) of dense subgroups of G and its impact on the algebraic structure of G. A key tool for this is the subgroup den(G) ≔ ⋂𝒟.Definition. For a cardinal 𝜅 ≥ 1, a topological group is in the class ℱf (𝜅) [resp., ℱ (𝜅); resp., ℱ2(𝜅); resp., ℱad (𝜅)] if some family of 𝜅-many dense supgroups of G is independent and freely generated; [resp., independent; resp., pairwise independent; resp., pairwise almost disjoint].[Note for each 𝜅 ≥ 1 the class-theoretic inclusions ℱf (𝜅) ⊆ ℱ (𝜅) ⊆ ℱ2 (𝜅) ⊆ ℱad (𝜅)].Let K be a compact abelian group. Then:1.K ∈ ℱad (2).2.K ∈ ℱ (2) ⇔ either r (K) >0 or each leading Ulm-Kaplansky invariant of K is infinite.3.There are D0, D1 ∈ 𝒟 (K) such that den (K) = D0 ∩ D1.4. K ∈ ℱ (𝜅) ⇔ K ∈ ℱ2 (𝜅).5.If K is torsion and K ∈ ℱ (2), then K ∈ ℱ (𝜅) ⇔ 𝜅 ≤each leading Ulm-Kaplansky invariant of K.6.If r(K) > 0, then K ∈ ℱ(K) ⇔ 𝜅 ≤ r (K); if in addition r(K) ≥ d(K), then K ∈ ℱf (K) ⇔ K ∈ ℱ (K).This is joint work with Dikran Dikranjan

Wednesday, October 23, 2013

04:15 pm - 05:30 pm

Topology Seminar, Karol Szumilo (Universitat Bonn): 'Cofibration categories and quasicategories'

Abstract: Classically, homotopy theories are described using homotopical algebra, e.g. as model categories or (co)fibration categories. Nowadays, they are often formalized as higher categories, e.g. as quasicategories or complete Segal spaces. These two types of approaches highlight different aspects of abstract homotopy theory and are useful for different purposes. Thus it is an interesting question whether homotopical algebra and higher category theory are in some precise sense equivalent.In this talk I will concentrate on cofibration categories and quasicategories. I will discuss some basic features of both notions building up to a result that the homotopy theory of cofibration categories is indeed equivalent to the homotopy theory of cocomplete quasicategories.

Wednesday, October 16, 2013

04:15 pm - 05:30 pm

Topology Seminar, David Futer (Syracuse): 'Small volume link orbifolds'

I will discuss recent investigations of small volume hyperbolic 3-orbifolds whose singularlocus is a link. An orbifold of this type naturally arises as the quotient of a 3-manifoldunder a (nice) group action. Thus, lower bounds on the volume of these orbifolds lead torelations between the volume of a 3-manifold and the size of its symmetry group.In this talk, I will describe the unique smallest volume link orbifold whose singularlocus is a link in the 3-sphere. I will also identify the unique smallest volume link orbifoldwhose torsion order is n, for all suciently large n. Joint work with Chris Atkinson.

Wednesday, October 02, 2013

04:15 pm - 05:30 pm

Topology Seminar, Mehdi Khorami (Eastern CT State U): Higher Chromatic analogues of twisted K-theories

Abstract: Let Rn denote the homotopy fixed point spectrum E_n^hS𝔾n, whereas S𝔾n is the kernel of the determinant homomorphism det: 𝔾n ⟶ Z_p^⤬ with 𝔾n being the Morava group. Here En denotes the n-th Morava E-theory. We show that for a K (n)-local space X equipped with a K (ℤp, n + 1)-bundle P ⟶ X , the P-twisted Rn-theory of X , Rn∗(X, P ), is defined and there exist a universal coefficient isomorphismRn∗(X, P ) ≅ Rn∗(P ) ⊗ Rn∗(K ( Z_(p ), n +1)) ℝ n∗ .This extends an analogues result on twisted K-theory in the K (n)-local category.

Wednesday, September 25, 2013

04:15 pm - 05:30 pm

Topology Seminar, Mark Hovey (Wes): 'Finitenesss, flatness, and injectivity for modules'

Abstract: In this joint work with Wesleyan Ph. D.s Daniel Bravo and Jim Gillespie, we describe how different notions of finiteness for modules affect homological algebra as a whole and in particular the notion of flat and injective modules. These results clarify the duality between flat and injective modules and allow us to extend the subject of Gorenstein homological algebra from its usual home of Noetherian rings to all rings.

Wednesday, September 18, 2013

04:15 pm - 05:30 pm

Topology Seminar, Tony Hager (Wes): Some unusual countably complete vector lattices

Abstract. For M a vector lattice of measurable functions,with N an abstract ideal of null functions,the quotient M/N has the features (*) Any countable set which is either bounded above,or pairwise disjoint,has the supremum in M/N. Such M/N has also a compatible ring multiplication. This is not so special because,if E has (*) and a positive element e for which,for positive x, inf(x,e)=0 implies x=0, then E is isomorphic to some M/Nso E has a multiplication.Further,various E with (*),but lacking such e,also have a multiplication.So the question arises:Does (*) alone imply a multiplication? We show that a set-theoretic hypothesis weaker than the Continuum Hypothesis produces various examples of not within the vector lattice of all real-valued functions on an uncountable set.

Wednesday, September 11, 2013

04:15 pm - 05:30 pm

Topology Seminar, Dan Licata (Wes): Homotopy Theory in Type Theory

Abstract: Homotopy Type Theory is a logical calculus that describes structures called inifity-groupoids or homotopy types, which are studied in category theory and homotopy theory. This provides a new _synthetic_ way to work with these structures, by doing logical deductions in this calculus.Moreover, homotopy type theory can be implemented in a proof assistant, a computer program that verifies the correctness of proofs expressed in the logic, and thus it is possible to use homotopy type theory to check proofs in homotopy theory. In this talk, I will describe the basic definitions of Homotopy Type Theory, along with some computer-checked proofs of some basic theorems in algebraic topology, such as calculations of some homotopy groups of spheres (fundamental group of the circle, pi_n(S^n) and pi_3(S^2)), the Freudenthal suspension theorem, the Blakers-Massey theorem, and the van Kampen theorem. These proofs use some of the same ideas as the classical proofs of these theorems, but they also make use of some new type theoretic techniques.This talk is intended for a general mathematical audience; no prior background in type theory or proof assistants will be presumed.

Wednesday, September 04, 2013

04:15 pm - 05:30 pm

Topology Seminar, Biao Wang (Wes): Least area spherical catenoids in hyperbolic three space

Abstract: Let $y=f?(x?)$ be a smooth curve above the$x$-axis. If we rotate the curve along the$x$-axis, we get a surface of revolution inthree dimensional Euclidean space, and we call it $S$. Now suppose that the surface $S$is a minimal surface (In this case, $S$is called a (spherical) minimal catenoid), i.e.,the mean curvature of $S$ is zero at every point, it's well known that the curve $y=f?(x?)$ is a catenary. The equation of a catenary has the form?$y= a\cosh(x/a)$, here $a$ is a positive constant. In three dimensional hyperbolic space $\mathbb{H}^3$, we also have the similar spherical minimal catenoids. In this talk, I will talk about the equations ofthe catenaries and the properties of the sphericalminimal catenoids in $\mathbb{H}^3$.Besides, I will talk about how to determine whethera spherical minimal catenoidsin $\mathbb{H}^3$ is a least area surface.

Wednesday, April 24, 2013

04:15 pm - 06:00 pm

Topology Seminar, Viveca Erlandsson-CUNY Graduate Center: 'On Margulis Cusps of Hyperbolic 4-Manifolds'

Abstract: Consider a discrete subgroup G of the isometry group of hyperbolicn-space and a parabolic fixed point p. The Margulis region consists of allpoints in the space that are moved a small distance by an isometry in thestabilizer of p in G, and is kept precisely invariant under this stabilizer. Indimensions 2 and 3 the Margulis region is always a horoball, which givesthe well-understood picture of the parabolic cusps in the quotient manifold.In higher dimensions, due to the existence of screw-translations (parabolicisometries with a rotational part), this is no longer true. When the screw-translation has an irrational rotation, the shape of the corresponding regiondepends on the continued fraction expansion of the irrational angle. In thistalk we describe the asymptotic shape of the Margulis region in hyperbolic4-space corresponding to an irrational screw-translation. As a consequencewe show that the corresponding parabolic cusps are bi-Lipschitz rigid. Thisis joint work with Saeed Zakeri.

Wednesday, April 17, 2013

04:15 pm - 06:00 pm

Topology Seminar, Kyle Ormsby-MIT: 'Quadratic forms in motivic homotopy'

Abstract: Since its inception in the 1990s, motivic homotopy theory has melded algebraic topology and algebraic geometry to prove results about quadratic forms. In this talk, I will introduce the (stable) motivic homotopy category and survey how quadratic forms appear as endomorphisms of the motivic sphere spectrum (aka stable motivic π0). Time permitting, I will then describe how recent work with Paul Arne ?stv?r relates thehigher algebraic K-theory of quadratic forms to stable motivic π1.

Wednesday, April 03, 2013

04:15 pm - 06:00 pm

Topology Seminar, Ivan Gotchev-CCSU: 'Cardinal Functions on Topological Spaces'

Abstract: In 1969, answering a nearly fifty-year old question raised by Alexandroff and Urysohn, Arhangleski proved that for every Hausdorff topological space X, |X|≤2χ(X)L(X), where χ(X) is the character of X and L(X) is the Lindelf degree of X. There is a long list of theorems in the literature, each of which is either a generalization or a variation of Arhangelskis inequality. In this talk we will show how by introducing new cardinal functions some of these inequalites could be extended to be valid for larger classes of topological spaces. In particular, we will show how Arhangelskis inequality could be improved to become valid for all T1-topological spaces.

Wednesday, March 27, 2013

04:15 pm - 06:00 pm

Mathematics Colloquium/Topology Seminar, Dieter Remus-Universit?t Paderborn: 'On van der Waerden rings'

Abstract: In a paper from 2000 [J. Algebra 232, 21-47] W.W. Comfort and the author introduced the following notion: A van der Waerden ring (vdW-ring) is a compact ring on which every homomorphism to a compact ring is continuous. In that paper a structure theorem for semisimple vdW-rings was proven.In my talk I will report about joint results with Mihail Ursul. First I will give further examples of vdW-rings, and it will be shown that the class of vdW-rings is closed under extensions. Then I will present a structure theorem for associative, commutative vdW-rings.

Wednesday, March 06, 2013

04:15 pm - 06:00 pm

Topology Seminar, Don Larson-University of Rochester: ' A Computation in 3-Primary K(2)-Local Stable Homotopy Theory Using Topological Modular Forms'

Abstract: One way to obtain information about the 3-primary stable homotopy groups of spheres is to study the K(2)-local sphere; that is, the sphere spectrum localized with respect to the 2nd Morava K-theory at the prime 3. In this talk I will describe a method for computing the homotopy groups of a closely related spectrum Q(2), originally constructed by Mark Behrens in an effort to clarify and extend previous work on the K(2)-local sphere by Shimomura, Yabe, Goerrs, Henn, Mahowald, and Rezk. The spectrum Q(2) is built using isogenies of elliptic curves and spectra related to TMF, and as such, is conjectured to have interesting number-theoretic properties. I will discuss these conjectures as well as potential connections between Q(2) and the alpha and beta families in the 3-primary stable stems.

Wednesday, February 27, 2013

04:15 pm - 06:00 pm

Topology Seminar, Don Larson-University of Rochester: 'A computation in 3-primary K(2)-local stable homotopy theory using topological modular forms'

One way to obtain information about the 3-primary stable homotopy groups of spheres is to study the K(2)-local sphere; that is, the sphere spectrum localized with respect to the 2nd Morava K-theory at the prime 3. In this talk I will describe a method for computing the homotopy groups of a closely related spectrum Q(2), originally constructed by Mark Behrens in an effort to clarify and extend previous work on the K(2)-local sphere by Shimomura, Yabe, Goerrs, Henn, Mahowald, and Rezk. The spectrum Q(2) is built using isogenies of elliptic curves and spectra related to TMF, and as such, is conjectured to have interesting number-theoretic properties. I will discuss these conjectures as well as potential connections between Q(2) and the alpha and beta families in the 3-primary stable stems.

Wednesday, November 28, 2012

04:15 pm - 06:00 pm

Topology Seminar, Shawn Rafalski - Fairfiled University :'The Smallest Haken Hyperbolic Polyhedra'

Abstract: We determine the lowest volume hyperbolic Coxeter polyhedron whose corresponding hyperbolic polyhedral 3-orbifold contains an essential 2-suborbifold, up to a canonical decomposition along essential hyperbolic triangle 2-suborbifolds. This is joint work with Chris Atkinson (Temple University).

Wednesday, November 14, 2012

04:15 pm - 06:00 pm

Topology Seminar, Bia Wang - Wesleyan: 'Least area Spherical Catenoids in Hyperbolic Three-Dimensional Space'

Abstract: Let C be a curve in the hyperbolic plane, if we rotate it about a geodesic (which is disjoint from C), then we get a surface of revolution II in the hyperbolic three space. In this talk, Ill show how to determine the equation of C so that II is a minimal surface. I also try to discuss how to determine whether II is a least area minimal surface.

Wednesday, November 07, 2012

04:15 pm - 06:00 pm

Topology Seminar, Johanna Mangahas Kutluhan - Brown: ' Some Schottky subgroups of mapping class groups'

Abstract: Farb and Mosher defined a notion of 'convex cocompact' for subgroups of mapping class groups that models the original definition of convex cocompact for Kleinian groups; free groups of either kind are called Schottky. I'll describe a way to construct examples of Schottky mapping class subgroups that is (at least, a priori), different from the original abundant examples Farb and Mosher described. These examples grow out of one way, described by Clay, Leininger, and myself, to quasi-isometrically embed free groups (and more generally, right-angled Artin groups) into mapping class groups.

Wednesday, October 31, 2012

04:15 pm - 06:00 pm

Topology Seminar, Wistar Comfort - Professor Emeritus Wesleyan: 'Pseudocompact Topological Groups: Some Facts and Questions

Abstract: According to a definition of E. Hewitt (1948), a space X is pseudocompact if each continuous function from X to ℝ is bounded. (Exercise to ensure one's grasp of the concept: exhibit a non-compact, psuedocompact space. Slightly more challenging: exhibit such a topological group.) It is seen without difficulty (Comfort and Ross, 1966) that a pseudocompact topological group embeds as a dense subgroup of a (unique) compact group. Groups which so embed are called totally bounded.Question 1. Which (abelian) groups admit a pseudocompact group topology?[With apologies for the linguistic grotesquerie, we call such groups pseudocompactifiable].It is known (Comfort and Ross, 1964) that for every totally bounded abelian group (G, Ƭ ) there is a unique point-separating subgroup A of Hom(G,𝕋) such that Ƭ = ƬA (the symbol ƬA denoting the topology induced by A on G ). In fact, A = (G, Ƭ ).Question 2. Given a pseudocompactifiable abelian group G, for which such subgroups A is (G, ƬA) pseudocompact?With every pseudompactifiable (abelian) group G one naturally associates two intrinsically defined topologies: Ƭ# = Ƭ#(G ), the sup of all totally bounded group topologies on G; and Ƭ∨ = Ƭ∨(G ), the sup of all pseudocompact group topologies on G. Clearly, Ƭ∨ ⊆ Ƭ#.Question 3. For such G, is Ƭ∨ = Ƭ#?Other Facts and Questions will be offered, as time permits.

Thursday, October 25, 2012

04:15 pm - 06:00 pm

Math Colloquium, Rick Ball-University of Denver: 'Pointfree Integration'

Abstract: Lebesgue (1904) founded the theory of integration on a careful development of the notion of measure. Daniell (1914) reversed the procedure by showing that the measure theory drops out of the correct development of the notion of measurable function. This, then, opens the door for the application of ideas from pointfree topology, which usually gives penetrating insight into real-valued functions, to the theory of integration. In this talk we outline a pointfree development of integration. The crucial technical device is the pointfree version of pointwise convergence. After all, what drives the classical Daniell integral is the hypothesis that a monotone sequence of measurable functions which converges pointwise to 0 must also converge to 0 in measure. A couple of pictures help motivate the intuition behind the definition of pointfree pointwise convergence. In the end, what emerges is a definition which is surprisingly supple and free of clutter. Because integration is founded on the principle that certain bounded monotone sequences (of measurable functions) must have a limit, the ambient vector lattice is forced to have strong completeness properties. The classical development takes place in some vector lattice of the form ℝX, X a space, so these completeness properties of the vector lattice translate into strong separation properties for X. The same is true in the pointfree context; as it turns out, the relevant separation property is that the ambient frame must be a P-frame. Thus a satisfactory development of point-free integration had to await the recent discovery of the P-frame reflection.But achieving pointwise completeness only builds (the correct generalization of) the Baire measurable functions. The second, and final, step is a measure-sensitive version of the Dedekind-MacNeille completion by cuts. When concatenated, the two procedures do reproduce the classical Lebesgue integral. But the pointfree version of integration offers at least two significant advantages: the ideas and techniques of integration are greatly extended in scope, and the proofs become more conceptual, simpler and more transparent.

Wednesday, October 24, 2012

04:15 pm - 06:00 pm

Topology Seminar, Rick Ball-University of Denver: 'Truncated Vector Lattices'

Abstract: A sub-vector lattice A of ℝX is said to be closed under truncation if a ∧ 1 ∈ A whenever a ∈ A. The 1 here refers to the constant function on X, which need not be present in A. This property is of considerable importance. For instance, M. H. Stone observed that it is necessary in order for an integral on A to determine a measure on X. In fact, vector lattices with this property may very well have integrals which cannot be represented by any measure on X.When the integral is formulated in the broader pointfree context, the question of truncation inevitably arises. What is truncation, or more properly, what are its essential features? In this talk we will answer this question by developing the notion of a vector lattice with truncation, or truncated vector lattice, or trunk for short. The resulting category T, or, more properly, its full subcategory TA of archimedean objects, can be understood as a direct generalization of W, the familiar category of archimedean vector lattices with weak order unit introduced in the seminal paper of Hager and Robertson.The representation theory for W extends to the TA, with an interesting twist. Every W-object A is, up to quibbles, a subobject of D (X), the extended-real-valued functions on a compact Hausdorff space X which take on the values ?∞ on a nowhere dense set. The weak unit in D (X) is the constant 1 function, and it is present in A. On the other hand, every TA-object is a subobject of D∗ (X, ∗), the functions of D (X) which vanish at the designated point ∗ of X. The pointfree representation theory for TA likewise runs parallel to that of W, with a similar twist.

Wednesday, October 10, 2012

04:15 pm - 06:00 pm

Topology Seminar, Gonalo Tabuada, MIT: 'The Fundamental Theorem via Derived Morita Invariance, Localization and A1-Homotopy Invariance'

Abstract: I will prove that every functor defined on dg categories, which is derived Morita invariant, localizing, and A1-homotopy invariant, satisfies the fundamental theorem. As an application, we recover in a unified and conceptual way, Weibel and Kassel's fundamental theorems in homotopy algebraic K-theory, and periodic cyclic homology, respectively.