Seminars and Colloquia

Topology et al. Seminar

Apr 18

Topology Seminar

04:20 pm

Melissa Zhang, Boston College Symmetries in topological spaces and homology-type invariants Abstract:Topologists often encounter spaces with interesting symmetries. By analyzing the symmetries of an object through the regularities of its algebraic invariants, we are able to learn more about the object and its relationship with smaller, less complex objects. For example, by using the right tools, we can easily see that for a topological space Xequipped with a cyclic action, the rank of the singular homology of Xis at least that of the fixed point set. In low-dimensional topology, knots and links are ubiquitous and far-reaching in their associations. One particular interesting algebraic invariant of links is Khovanov homology, a combinatorial homology theory whose graded Euler characteristic is the Jones polynomial. In this talk, we consider links exhibiting 2-fold symmetry and prove a rank inequality for a variant of Khovanov homology.

Feb 21

Topology Seminar

04:20 pm

Kyle Hayden, Boston College From algebraic curves to ribbon disks and back Abstract: There is a rich, symbiotic relationship between knot theory and the study of algebraic curves in complex spaces. In particular, intersecting an algebraic curve in C^2 with a three-sphere of constant radius yields a knot or link that contains useful information about the algebraic curve. We'll look at a thirty-year-old conjecture about the relationship between different knots associated to the same algebraic curve. Using a simple construction, we'll show that the conjecture is false and that counterexamples are quite common.

Nov 29

Topology Seminar

04:20 pm

Patrick Devlin, Yale University Topological Methods in Combinatorics Abstract: It is sometimes remarked that combinatorics is not the study of structures or theorems, so much as the study of techniques. Several techniques are well known and evidently quite fruitful (e.g., linear programming, the probabilistic method, Fourier analysis, entropy, et cetera). But another tool that should be in every discrete mathematician's backpocket is topology. In this talk, we will discuss some clever applications of topology to combinatorics providing quick primers on any relevant topological (and combinatorial) concepts along the way. We will focus on work related to graph colorings and boolean function complexity and introduce several open problems and conjectures in the area.

Oct 25

Topology Seminar

04:20 pm

Speaker: Arie Levit, Yale Title: Local rigidity of uniform lattices Abstract: A lattice is topologically locally rigid (t.l.r) if small deformations of it are isomorphic lattices. Uniform lattices in Lie groups were shown to be t.l.r by Weil [60']. We show that uniform lattices in any compactly generated topological group are t.l.r. A lattice is locally rigid (l.r) if small deformations arise from conjugation. It is a classical fact due to Weil [62'] that lattices in semi-simple Lie groups are l.r. Relying on our t.l.r results and on recent work by Caprace-Monod we prove l.r for uniform lattices in the isometry groups of certain CAT(0) spaces, with the exception of SL_2(R), which occurs already in the classical case. In the talk I will explain the above notions and results, and present some geometric ideas from the proofs. This is a joint work with Tsachik Gelander.

Oct 11

Topology Seminar

04:20 pm

Speaker: Noelle Sawyer, Wes Title: The specification property and closed orbit measures Abstract: Given a compact metric space X, and a homeomorphism T on X, the dynamical system (X,T) has the specification property if we can approximate distinct orbit segments with the orbit of a periodic point. I will present a result by Sigmund showing that if (X,T) has the specification property, any T-invariant measure on X can be approximated by a closed orbit measure.

Oct 4

Topology Seminar

04:20 pm

Speaker: Shelly Harvey, Rice University Title: Braids, gropes, Whitney towers, and solvability of links Abstract: The n-solvable filtrations of the knot/link concordance groups were defined as a way of studying the structure of the groups and in particular, the subgroup of algebraically slice knots/links. While the knot concordance group C^1 is known to be an abelian group, when m is at least 2, the link concordance group C^m of m-component (string) links is known to be non-abelian. In particular, it is well known that the pure braid group with m strings is a subgroup of C^m and hence when m is at least 3, this shows that C^m contains a non-abelian free subgroup. We study the relationship between the derived subgroups of the the pure braid group, n-solvable filtration of C^m, links bounding symmetric Whitney towers, and links bounding gropes. This is joint with with Jung Hwan Park and Arunima Ray.

Sep 27

Topology Seminar

04:20 pm

Speaker: Samuel Lin, Dartmouth Title: Curvature Free Rigidity of Higher Rank Three- manifolds Abstract: Fixing K=-1, 0, or 1, a Riemannian manifold (M, g) is said to have higher hyperbolic, spherical, or Euclidean rank if every geodesic in M admits a normal parallel field making curvature K with the geodesic. Rank rigidity results, which usually involve a priori sectional curvature bounds, characterize locally symmetric spaces in terms of these geometric notions of rank. After giving a short survey of historical results, Ill discuss how rank rigidity holds in dimension three without a priori sectional curvature bounds.

Nov 30

Topology Seminar

04:20 pm

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.

Sep 28

Topology Seminar

04:20 pm

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.

Mar 23

Topology Seminar

04:15 pm

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.

Mar 2

Topology Seminar

04:15 pm

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.

Nov 11

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

04:15 pm

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.

Oct 21

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

04:15 pm

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.

Oct 14

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

04:15 pm

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.

Sep 16

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

04:15 pm

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.