The Topology / Geometry seminar meets at 3:00 p.m. on Tuesdays in 210 Deady, except as noted. See also the University of Oregon Mathematics Department webpage.
## Fall 2016
## Winter 2017
## Spring 2017
## Abstracts## Fall 2016Abstract: There are different ways to define the convergence of knots. For example, the diagram graphs of a sequence of knots might converge to another graph, using ideas from graph theory, or the geometric structures on the knot complements might converge to a metric space, using ideas from geometry. In this talk, we will discuss both notions of convergence of knots, some consequences, and open questions.
Abstract: This talk will start from an expository introduction to the smooth knot concordance group. Then we will review knot Heegaard Floer theory and survey a few recent results on concordance as applications. Most of these results will be about families of infinitely many linearly independent knots, revealing fine structure of the concordance group.
Abstract: Algebraic topologists love to find perfect algebraic reflections of topological phenomena, for example how subgroups of the fundamental group correspond to covering spaces. Quillen found that rational homotopy theory was perfectly modeled by differential graded Lie or commutative algebras, remarkably implying that those theories are equivalent. Sullivan extended the theory and showed how de Rham theory makes this calculable. In his thesis, Mandell showed that p-adic homotopy theory is perfectly reflected in (but not equivalent to) singular cochains as an E-infinity algebra. I will give progress on geometric models of this E-infity algebra for manifolds, where intersection and linking play the role that differential forms play in de Rham theory.
Abstract: Special Lagrangian cones play a central role in the understanding of the SYZ conjecture, an important conjecture in mathematics based upon mirror symmetry and certain string theory models in physics. According to string theory, our universe is a product of the standard Minkowsky space with a Calabi-Yau 3-fold. Strominger, Yau, and Zaslov conjectured that Calabi-Yau 3-folds can be fibered by special Lagrangian 3-tori with singular fibers. To make this idea rigorous one needs control over the singularities, which can be modeled by special Lagrangian cones. In this talk, we discuss special Lagrangian cones, the difficulties involved in defining and computing invariants of them, and the hope that these invariants may offer in understanding the SYZ conjecture.
Abstract: A central question of algebraic topology is to understand homotopy classes of maps between finite cell complexes. The Nilpotence Theorem of Hopkins-Devinatz-Smith together with the Periodicity Theorem of Hopkins-Smith describes non-nilpotent self maps of finite spectra. The Morava K-theories K(n)
_{∗} are extraordinary cohomology theories which detect whether a finite spectrum X supports a v_{n}-self map. Such maps are known to exist for each finite spectrum X for an appropriate n but few explicit examples are known. Working at the prime 2, we use a technique of Palmieri- Sadofsky to produce algebraic analogs of vn maps that are easier to detect and compute. We reproduce the existence proof of Adams’s v_{1}^{4} map on the Mod 2 Moore spectrum, and work towards a v_{2}^{i} map for a small value of i.November 8, 2016. Tian Yang (Stanford), "Volume conjectures for Reshetikhin-Turaev and Turaev-Viro invariants".Abstract: In joint work with Qingtao Chen, we conjecture that at the root of unity exp(2πi/r) instead of the usually considered root exp(πi/r), the Turaev-Viro and Reshetikhin-Turaev invariants of a hyperbolic 3-manifold grow exponentially with growth rates respectively the hyperbolic and the complex volume of the manifold. This reveals a different asymptotic behavior of the relevant quantum invariants than that of Witten's invariants (that grow polynomially by the Asymptotic Expansion Conjecture), which may indicate a different geometric interpretation of the Reshetikhin-Turaev invariants than SU(2) Chern-Simons gauge theory. Recent progress toward these conjectures will be summarized, including joint work with Renaud Detcherry and Effie Kalfagianni.
November 15, 2016. Demetre Kazaras (UO), "Minimal hypersurfaces with free boundary and psc-bordism".Abstract: There is a well-known technique due to Schoen-Yau from the late 70s which uses (stable) minimal hypersurfaces to study the topological implications of a (closed) manifold's ability to support positive scalar curvature metrics. In this talk, we describe a version of this technique for manifolds with boundary and discuss how it can be used to study bordisms of positive scalar curvature metrics.
November 22, 2016. Nathan Dunfield (UIUC), "A tale of two norms".Abstract: The first cohomology of a hyperbolic 3-manifold has two natural norms: the Thurston norm, which measure topological complexity of surfaces representing the dual homology class, and the harmonic norm, which is just the L^2 norm on the corresponding space of harmonic 1-forms. Bergeron-Sengun-Venkatesh recently showed that these two norms are closely related, at least when the injectivity radius is bounded below.
Their work was motivated by the connection of the harmonic norm to the Ray-Singer analytic torsion and issues of torsion growth discussed in the first talk. After carefully introducing both norms and the connection to torsion growth, I will discuss new results that refine and clarify the precise relationship between them; one tool here will be a third norm based on least-area surfaces. This is joint work with Jeff Brock.
November 29, 2016. Ailsa Keating (Columbia), "On higher dimensional Dehn twists".Abstract: Given a Lagrangian sphere L in a symplectic manifold M, one can define a higher-dimensional Dehn twist in L, a diffeomorphism of M. This generalises the classical notion of a Dehn twist on a Riemann surface. After defining them, we will explore some of their properties, with an emphasis on comparing them with properties in the 2D case. No prior knowledge of symplectic topology will be assumed.
December 8, 2016. Vinicius Gripp Barros Ramos (IMPA), "Symplectic embeddings, toric domains and billiards".Abstract: Embedded contact homology capacities were defined by Michael Hutchings and they have been shown to provide sharp obstructions to many symplectic embeddings. In this talk, I will explain how they can be used to study symplectic embeddings of the lagrangian bidisk and how this space is related to the space of billiards on a round disk.
January 31, 2017, 11:00 a.m. Boris Botvinnik (UO), "On the topology of the space of Ricci-positive metrics".Abstract: This is joint work with David Wraith.
We study the space $\Riem^{\Ric+}(M)$ of metrics with
positive Ricci curvature on a closed spin manifold $M$ of $\dim M=d$.
There is a natural map $\iota: \Riem^{\Ric+}(M)\to \Riem^+(M)$ to the
space of metrics with positive scalar curvature. Let $g_0\in
\Riem^+(M)$ be any metric, then there is the index-difference map
$\mathrm{inddiff}_{g_0}: \Riem^+(M)\to \Omega^{\infty+d+1}KO$ defined
by Hitchin. Recently it was established by Botvinnik, Ebert and
Randal-Williams that the index-difference map $\mathrm{inddiff}_{g_0}$
detects non-trivial homotopy groups $\pi_q\Riem^+(M)$ for all $q$ such
that $KO_{d+q+1}\neq 0$, where $d\geq 6$. We show that for
any $\ell\geq 1$ and even integer $d\geq 6$, there exists a spin
manifold $W$, $\dim W = d$, together with a metric $g_0\in
\Riem^{\Ric+}(W)$, such that the composition $$\mathsf{inddiff}_{g_0}:
\Riem^{\Ric+}(W)\xrightarrow[ ]{\iota} \Riem^{+}(W) \xrightarrow[
]{\mathrm{inddiff}_{g_0}} \Omega^{n+1} KO$$ detects non-trivial
homotopy groups $\pi_q \Riem^{\Ric+}(W)$ for all $q\leq \ell$ and such
that $KO_{d+q+1}\neq 0$.
January 31, 2017, 3:00 p.m. David Pengelley (Oregon State), "How is a projective space fitted together?".Abstract: Projective spaces are among the most important geometric objects in mathematics. An example is n-dimensional real projective space, obtained from the n-sphere by identifying antipodal points.
We will investigate how the essential geometric cells of various dimensions in a projective space are glued to one another, as detected by cohomology operations that reflect specific geometric attachments.
We find a minimal set of generators and relations modulo two for the cells and attachments, that is, a minimal presentation for the cohomology of a real projective space as a module over the Steenrod algebra of cohomology operations.
The morning Homotopy Theory seminar will provide useful, but not necessary, hands-on preparation.
Abstract: Using Ramanujan identities and WDVV equations, we prove that the Gromov-Witten generating functions are quasi-modular forms when the target Calabi-Yau is a quotient of an elliptic curve. Furthermore, we apply Caylay transformation to relate the Gromov-Witten theory of these targets and their counterpart Fan-Jarvis-Ruan-Witten theory. This solves the LG/CY correspondence in these cases. The work is joint with Jie Zhou.
February 21, 2017. Paul Arnaud Songhafouo Tsopméné (University of Regina), "Cosimplicial models for manifold calculus".Abstract: Manifold calculus is a tool developped by Goodwillie and Weiss which enables to approximate a contravariant functor, F, from the category of m-manifolds to the category of spaces (or alike), by its ”Taylor approximation”, T_{\infty}F. I will explain how to construct a fairly explicit and computable cosimplicial model of T_{\infty}F(M) out of a simplicial model of the compact manifold M (i.e. out of a simplicial set whose realization is M, with extra tangential information if needed). This cosimplicial model in degree p is then equivalent to the evaluation of F on a disjoint union of as many m-disks as p-simplices in the simplicial model of M.
As an example, we apply this construction to the functor F(M) = Emb(M,W) of smooth embeddings in a given manifold W ; in that case our cosimplicial model in degree p is then just the configuration spaces of all the p-simplices of M in W product with a power of a Stiefel manifold. When dim(W ) > dim(M ) + 2, a theorem of Goodwillie-Klein implies that our explicit cosimplicial space is a model of Emb(M,W). (This generalizes Sinha’s cosimplicial model for the space of long knots which was for the special case when M is the real line.)
This allows one to make explicit computations. As an example, using this cosimplicial model we show that the rationnal Betti numbers of the space Emb(M,Rn) have an exponential growth when the Euler characteristic of M is < -1. (This is joint work with Pedro Boavida de Brito, Pascal Lambrechts, and Daniel Pryor).
February 28, 2017. Christian Millichap (Linfield College), "Commensurability of hyperbolic knot and link complements".Abstract: In general, it is a difficult problem to determine if two manifolds are commensurable, i.e., share a common finite sheeted cover. Here, we will examine some combinatorial and geometric approaches to analyzing commensurability classes of hyperbolic knot and link complements. In particular, we will discuss current work done with Worden to show that the only commensurable hyperbolic 2-bridge link complements are the figure-eight knot complement and the $6_{2}^{2}$ link complement. Part of this analysis also results in an interesting corollary: a hyperbolic 2-bridge link complement cannot irregularly cover a hyperbolic 3-manifold.
## Spring 2017Abstract: We will show that, for "almost" all arithmetic hyperbolic manifolds with dimension >3, their fundamental groups are not LERF. The main ingredient in the proof is a study of certain graph of groups with hyperbolic 3-manifold groups being the vertex groups. We will also show that a compact irreducible 3-manifold with empty or tori boundary does not support a geometric structure if and only if its fundamental group is not LERF.Abstract: In this talk, I will show that the limiting Khovanov chain complex of any inﬁnite positive braid categoriﬁes the Jones-Wenzl projector, extending Lev Rozansky's work with inﬁnite torus braids. I will also describe a similar result for the limiting Lipshitz-Sarkar-Khovanov homotopy types of the closures of such braids. Extensions to more general inﬁnite braids will also be considered. This is joint work with Gabriel Islambouli.Abstract: In the talk we will describe a new feature of the classical Schubert calculus which holds for all types of the classical Lie groups. As the main example we will use the type A Grassmanians. The usual definition of the Schubert cycles involves a choice of a parameter, namely a choice of a full flag. Studying the dependence of the construction of the Schubert cycles on these parameters in the equivariant cohomology leads to an interesting 1 cocycle on the permutation group or a solution to the quantum Yang Baxter equation. This connects the Schubert calculus to the theory of quantum integrable systems. We show the above cocycle is the 'Baxterization' ( the term introduced by V. Jones) of the natural action of the nil Coxeter algebra of Berstein Gelfand Gelfand Demazure difference operators in the equivariant cohomology of partial flag varieties. We will outline some applications of this connection as well.
May 9, 2017. Biji Wong (Brandeis), "Equivariant corks and Heegaard Floer homology". Abstract: A cork is a contractible smooth 4-manifold with an involution on its boundary that does not extend to a diffeomorphism of the entire manifold. Corks can be used to detect exotic structures; in fact any two smooth structures on a closed simply-connected 4-manifold are related by a cork twist. Recently, Auckly-Kim-Melvin-Ruberman showed that for any finite subgroup G of SO(4) there exists a contractible 4-manifold with an effective G-action on its boundary so that the twists associated to the non-trivial elements of G do not extend to diffeomorphisms of the entire manifold. In this talk we will use Heegaard Floer techniques originating in work of Akbulut-Karakurt to give a different proof of this phenomenon.May 30, 2017. Kirk McDermott (Oregon State), "Examples of 3-manifold spines arising from a family of cyclic presentations". Abstract: Cyclically presented groups arise naturally as the fundamental group of certain closed, orientable 3-manifolds. In this talk, we prove a particular family of cyclic presentations is a new collection of 3-manifold spines. A common approach is to take a spherical van Kampen diagram and then perform a classical face pairing technique using an Euler characteristic argument. Here, we instead work with a spherical picture- the dual to a diagram- and show, equivalently, when the picture represents a Heegaard diagram for a 3-manifold. The resulting 3-manifolds have cyclic symmetry, a consequence of the fact that each is a finite cyclic covering of a certain lens space. These examples include and extend earlier results of Cavicchioli, Repovs, and Spaggiari from 2003.June 6, 2017. Chris Scaduto (Stonybrook), "The mod 2 cohomology of some SU(2) representation spaces for a surface". Abstract: Consider the space of representations from the fundamental group of a punctured surface to SU(2) that are -1 around the puncture. I'll tell you about the 2-torsion in the cohomology of this space. This is a by-product of an investigation into the mod 2 cohomology ring of the space of representations modulo conjugation, which is in turn motivated by a problem in instanton homology. This is joint work with Matt Stoffregen. |