Mathematics & Computer Science

Seminars and Colloquia

Topology et al. Seminar

Wednesday, November 30, 2016

04:20 pm - 05:30 pm

Topology Seminar

Boris Gutkin, University Duisberg-Essen: Pairings between periodic orbits in hyperbolic coupled map lattices. Abstract: Upon quantization, hyperbolic Hamiltonian systems generically exhibit universal spectral properties effectively described by Random Matrix Theory. Semiclassically this remarkable phenomenon can be attributed to the existence of pairs of classical periodic orbits with small action differences. So far, however, the scope of this theory has, by and large, been restricted to small-dimensional systems. I will discuss an extension of this program to hyperbolic coupled map lattices with a very large number of sites. The crucial ingredient is a two-dimensional symbolic dynamics which allows an effective representation of periodic orbits and their pairings. I will illustrate the theory with a specific model of coupled cat maps, where such a symbolic dynamics can be constructed explicitly. The core of the talk is based on the joint work with V. Osipov: Nonlinearity 29, 325 (2016) and a work in progress with P. Cvitanovic, R. Jafari, L. Han, A. Saremi.

Exley Science Center Tower ESC 121

Wednesday, September 28, 2016

04:20 pm - 05:30 pm

Topology Seminar

Dave Constantine, Wes: " Hausdorff dimension and the CAT(K) condition for surfaces" Abstract: A geodesic metric space satisfies the CAT(K) condition if its geodesic triangles are all `thinner' than triangles with the same side lengths in the model space of constant Riemannian curvature K. This condition allows one to extend many arguments relying on an upper curvature bound from Riemannian geometry to the metric space setting. How `strange' can a metric be while still satisfying the CAT(K) property? One way to measure this is with the difference between the topological dimension of the space and its Hausdorff dimension with respect to the metric. In this talk I'll show that, at least for surfaces, a CAT(K) metric is tame in the sense that it yields Hausdorff dimension 2. I'll also provide some motivation for this question by showing how results like this allow one to extend volume entropy rigidity statements to the CAT(-1) setting.

Exley Science Center Tower ESC 638

Wednesday, March 23, 2016

04:15 pm - 05:30 pm

Topology Seminar

Scott Taylor, Colby College: Neighbors of knots in the Gordian graph Abstract: Switching a crossing on a knot diagram is one of the simplest methods for converting one type of knot into another type of knot. The Gordian graph is the graph which keeps track of which knot types can be converted into which other knot types by a single crossing change. Its vertex set is the set of knot types and its edge set consists of pairs of knots which have a diagram wherein they differ at a single crossing. Bridge number is a classical knot invariant which is a measure of the complexity of a knot. It can be re_ned by another, recently discovered, knot invariant known as \bridge distance". We show, using arguments that are almost entirely elementary, that each vertex of the Gordian graph is adjacent to a vertex having arbitrarily high bridge number and bridge distance. This is joint work with Ryan Blair, Marion Campisi, Jesse Johnson, and Maggy Tomova.

Exley Science Center Tower ESC 638

Wednesday, March 02, 2016

04:15 pm - 05:30 pm

Topology Seminar

Katherine Raous, Brandeis University: "Rational knots, Rational Seifert surfaces and genus bounds" Abstract: Let K be a knot in a 3-manifold Y that represents a torsion class in the first homology of Y. Since K is torsion, it has finite order, p, and unless p=1, K does not bound a surface in Y. However, we can always find a surface which wraps p times around K. Using this construction, Ni showed that K defines a filtration of the Heegaard Floer chain complex of Y indexed by the rationals. We will use this filtration to define analogues of the Ozsvath-Szabo tau-invariants for such knots and show that when Y bounds a rational homology ball, these invariants give a lower bound for the genus of a surface with boundary K.

Exley Science Center Tower ESC 638

Wednesday, November 11, 2015

04:15 pm - 05:15 pm

Topology Seminar, Alyson Hildum (Wes): "Right-angled Artin groups with tame cohomology"

Abstract: In this talk we will discuss certain group cohomological conditions arising in the study of 4-manifolds with right-angled Artin fundamental groups. While investigatinga 4-manifold classification problem, Ian Hambleton and I discovered an interesting question about the cohomology of right-angled Artin groups (RAAGs) with group-ring coefficients. We call a $G$-module A a torsion module if $Hom_{ZG}(A,ZG)=0$ (where ZG is the group-ring). For any group $G$, the group cohomology group $H^i(G;ZG)$ is a $G$-module, and one can ask under which conditions these cohomology groups are torsion modules. Certain conditions on the cohomology groups (which we call "tame cohomology") allow for a better understanding of the structure of the second homotopy group of a 4-manifold $M$, $\pi_2(M)$, as a $\pi_1(M)$-module, which is necessary for tackling our classification problem.

Exley Science Center (Tower)

Wednesday, October 21, 2015

04:15 pm - 05:00 pm

Topology Seminar: John Schmitt (Middlebury): "Two tools from the polynomial method toolkit"

Abstract: The polynomial method is an umbrella term that describes an evolving set of algebraic statements used to solve problems in arithmetic combinatorics, combinatorial geometry, graph theory and elsewhere by associating a set of objects with the zero set of a polynomial whose degree is somehow constrained. Algebraic statements about the zero set translate into statements about the set of objects of interest. We will examine two tools from the polynomial method toolkit, each of which generalizes the following, well-known fact: a one-variable polynomial over a field can have at most as many zeros as its degree. The first generalization which we will discuss is Alons Non-vanishing Corollary, a statement for a multivariate polynomial introduced in the 1990s that follows from his celebrated Combinatorial Nullstellensatz. The second generalization is the Alon-Furedi Theorem, a statement which gives a lower bound on the number of non-zeros of a multivariate polynomial over a Cartesian product. We give an application for each of these tools. For the first we show how to apply it to a combinatorial problem of the polymath Martin Gardner known as the minimum no-three-in-a-line problem. For the second we show how it quickly proves a number-theoretic result from the 1930s due to Ewald Warning, a statement which gives a lower bound on the number of common zeros of a polynomial system over a finite field.

Exley Science Center (Tower)

Wednesday, October 14, 2015

04:15 pm - 05:00 pm

Topology Seminar: Tue Ly (Brandeis): "Diophantine approximation on number fields, homogenous dynamics and Schmidt game"

Abstract: In 1960's, Wolfgang Schmidt used his (\alpha, \beta)-game to prove that the set of badly approximable numbers has countable intersection property. In this talk, I will discuss about extending Schmidt's result to the set BA_K of vectors badly approximable by elements of a fixed number field K using the connection to homogeneous dynamics and recent developments of Schmidt's game. This strengthens a recent result by Anish Ghosh, Beverly Lytle and Manfred Einsiedler concerning the intersection of BA_K with curves. Joint work with Dmitry Kleinbock.

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'

Abstract: Budney showed that the space of (long) knots has a little 2-cubes operad action and moreover is freely generated over the 2-cubes operad by the space of prime knots. This generalizes classical results on isotopy classes of knots to the whole space of knots. He then constructed an operad which encodes splicing of knots and proved a theorem decomposing the space of knots over this operad. Infection of knots (or links) by string links is a generalization of splicing from knots to links and is useful for studying concordance of knots. In joint work with Robin Koytcheff, we have constructed a colored operad that encodes this infection operation. By a prime decomposition theorem for 2-string links (proven by us and Ryan Blair), this operad encodes all the relations in the 2-string link monoid. We can also show that a certain subspace of 2-string links is freely generated over a suboperad of our infection colored operad (homeomorphic to the 1-cubes operad) by its subspace of prime links.

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.