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2017 Lloyd Roeling UL Lafayette Mathematics Conference: Topology

10-12 November 2017

Maxim Doucet Hall

University of Louisiana at Lafayette, Lafayette, Louisiana

The conference will provide an opportunity for researchers and students in topology (including both geometric topology and homotopy theory) and related fields to come together, present their research, and learn from each other.

The conference will begin on Friday at 1:20 p.m. (Sign-in is just before this, 12:30-1:20 p.m.) and end at 12:15 p.m. on Sunday.

The conference budget is modest: our annual tradition since 1970 is that, while there isn't funding for the travel and lodging expenses of speakers and participants, there will be nice (perhaps "classical and contemporary") Cajun-style dinners on Friday and Saturday nights (respectively) provided for by the Lloyd Roeling Conference Fund and the UL Lafayette Mathematics Department.

The invited speakers are:

  • Ettore Aldrovandi (Florida State)
  • Scott Bailey (Clayton State)
  • Jeremy Brazas (West Chester University)
  • Saul Glasman (Minnesota)
  • Jennifer Hom (Georgia Tech)
  • Michael Kelly (Loyola University New Orleans)
  • Rafal Komendarczyk (Tulane)
  • Samuel Lisi (Mississippi)
  • Mona Merling (Johns Hopkins)
  • Mentor Stafa (Tulane)
  • Nathaniel Stapleton (Kentucky/Regensburg)
  • David Shea Vela-Vick (LSU)

The Ph.D. student speakers are:

  • Haldun Ozgur Bayindir (University of Illinois at Chicago)
  • Jacobson Blomquist (Ohio State)
  • Sudipta Kolay (Georgia Tech)
  • Miriam Kuzbary (Rice)
  • Christina Osborne (Virginia)
  • Fang Sun (Tulane)
  • Yaineli Valdes (Florida State)

The student talks will be 20 minutes long and 3 prizes, in the amounts of $125, $100, and $75, will be awarded.

The Organizing Committee members are: Scott Bailey (Clayton State), Daniel Davis (UL Lafayette), Robin Koytcheff (UL Lafayette), and Peter Oman (Southeast Missouri State).


Maps


All talks will be in room 208 of Maxim Doucet Hall

Maxim D. Doucet Hall faces Johnston Street. Map links are posted above.


Registration

There is no registration fee. However, to aid in planning please complete the registration form as soon as possible.


Schedule

The schedule is here.


Titles and Abstracts

Biextensions of stable modules and presentations of bimonoidal categories
Ettore Aldrovandi
Florida State

There is an increasing interest about bimonoidal, or "rig" categories, as possible inputs of various machines outputting $K$-theory spectra. The underlying 1-types of such categories are represented by stable modules. We show that the rig structure can be described in terms of so-called biextensions of such modules.

Modules and splittings
Scott Bailey
Clayton State

Computations involving the root invariant prompted Mahowald and Shick to develop the slogan: "the root invariant of $v_n$-periodic homotopy is $v_n$-torsion." While neither a proof, nor a precise statement, of this slogan appears in the literature, numerous authors have offered computational evidence in support of its fundamental idea. In this talk, we will discuss the modules and splittings involved in this computational evidence, and provide yet another example in support of the slogan.

Topological equivalences of E-infinity DGAs
Haldun Ozgur Bayindir
University of Illinois at Chicago

To capture more information about a topological space, one considers the co-chain complex of a space with extra multiplicative structure instead of just the co-chain complex itself. One of these structures is called the E-infinity structure. An E-infinity DGA is a chain complex with a multiplication that is associative and commutative up to coherent higher homotopies. Co-chain complexes of topological spaces are examples of E-infinity DGAs.
Weak equivalences of E-infinity DGAs are maps of E-infinity DGAs that induce an isomorphism in homology, these are called quasi-isomorphisms. In our work, we use stable homotopy theory to construct new equivalences between E-infinity DGAs which we call E-infinity topological equivalences. E-infinity DGAs are called E-infinity topologically equivalent when the corresponding commutative ring spectra are weakly equivalent. Quasi-isomorphic E-infinity DGAs are topologically equivalent. We show that the converse to this statement is not true, i.e. we construct examples of E-infinity DGAs that are E-infinity topologically equivalent but not quasi-isomorphic. This means that there are more equivalences to consider between E-infinity DGAs than just quasi-isomorphisms. Also, we show that for co-chain complexes of spaces with integer coefficients, E-infinity topological equivalences and quasi-isomorphisms agree.

Higher stabilization and completion with respect to stable homotopy
Jacobson Blomquist
Ohio State University

We will explain how estimates from a higher stabilization theorem show that the stable homotopy completion studied by Carlsson, and subsequently in Arone-Kankaanrinta (S-localization), fits into a derived adjunction via the Arone-Ching theory that can be turned into a derived equivalence by restricting to simply connected spaces. If time permits, the analogous results from finite suspensions of spaces, and their analogs and duals in structured ring spectra, will be discussed.

Detecting Local Properties of Fundamental Groups
Jeremy Brazas
West Chester University

Recent advances in the understanding of fundamental groups of spaces with wild local structure have significant connections and applications to infinite group theory and topological algebra. For instance, Katsuya Eda's remarkable homotopy classification of 1-dimensional spaces implies that homotopy type of a 1-dimensional Peano continuum, e.g. the Hawaiian earring, Sierpinski carpet, and Menger curve, is completely determined by the isomorphism type of its fundamental group. When considering such spaces, the algebraic structure of $\pi_1(X,x_0)$ often depends heavily on the local structure of $X$. Roughly speaking, to verify local properties that characterize this kind of dependence, it is necessary to detect the existence of a specific homotopy given a certain, possibly infinite, arrangement of paths. In this talk, I'll discuss joint work with Hanspeter Fischer that introduces a unified approach to characterizing and comparing a number of these properties by constructing closure operators on the $\pi_1$-subgroup lattice in terms of maps from a fixed “test" domain.​

Stratified topology and the yoga of orbital categories
Saul Glasman
Minnesota

In joint work with a number of collaborators, particularly C. Barwick, E. Dotto, D. Nardin and J. Shah, we have developed a formalism that "takes the G out of 'genuine'" by substituting the orbit category O_G for a category with similar properties in the machinery of unstable and stable equivariant homotopy theory. I'll give an overview of some of the successes of this theory, which puts G-spectra on the same footing as seemingly dissimilar objects coming from functor calculus. I'll then discuss a new connection between the theory of orbital categories and stratified topology, linking perverse sheaves to the equivariant slice filtration used in Hill, Hopkins and Ravenel's solution of the Kervaire invariant one problem. If time permits I'll end with some musings about stratified $\infty$-topoi and their shapes.

Heegaard Floer homology and homology cobordism
Jen Hom
Georgia Tech

Heegaard Floer homology is an invariant of closed three-manifolds. We consider three-manifolds up to a weaker notion of equivalence known as homology cobordism. Using additional data from the involutive Heegaard Floer homology package of Hendricks and Manolescu, we discuss applications of Heegaard Floer homology to homology cobordism. This is joint work with Kristen Hendricks and Tye Lidman.

Fixed point index bounds and aspherical 2-complexes
Michael Kelly
Loyola University - New Orleans

Given a self-map of a compact, connected topological space we consider the problem of determining upper and lower bounds for the fixed point indices of the map. One can not expect to have bounds in general, so we need to restrict attention to the class of spaces considered and also the class of self-maps. Motivated by an elementary result in the case of a 1-dimensional complex this talk will focus attention to the setting of 2-complexes. Some past results and related examples will be presented, leading to some current joint work with D. L. Goncalves (U. Sao Paulo, Brasil).

Braided Embeddings
Sudipta Kolay
Georgia Tech

The theory of braids has been very useful in the study of classical knot theory. One can hope that higher dimensional braids will play a similar role in higher dimensional knot theory. In this talk we will introduce the concept of braided embeddings, and discuss existence, lifting and isotopy problems for braided embeddings.

Knot and link invariants for vector fields
Rafal Komendarczyk
Tulane University

In 1979, V. I. Arnold showed that the fundamental invariant of 2--component links, namely the linking number, can be generalized to an invariant of volume preserving vector fields. In this talk, Arnold's construction will be outlined, together with various applications in mathematical physics and geometric knot theory. Further, more recent results concerning generalizations of this construction to Vassiliev invariants of knots, and Milnor higher linking numbers will be presented.

A new concordance group of links
Miriam Kuzbary
Rice University

The knot concordance group has been the subject of much study since its introduction by Ralph Fox and John Milnor in 1966. One might hope to generalize the notion of a concordance group to links; however, the immediate generalization to the set of links up to concordance does not form a group since connected sum of links is not well-defined. In 1988, Jean Yves Le Dimet defined the string link concordance group, where a link is based by a disk and represented by embedded arcs in D^2 × I. In 2012, Andrew Donald and Brendan Owens defined groups of links up to a notion of concordance based on Euler characteristic. However, both cases expand the set of links modulo concordance to larger sets and each link has many representatives in these larger groups. In this talk, I will present joint work with Matthew Hedden where we define a link concordance group based on the "knotification" construction of Peter Ozsváth and Zoltan Szabó, giving a definition of a link concordance group where each link has a unique group representative. I will also present invariants for studying this group coming from Heegaard-Floer homology as well as a new group theoretic invariant for studying concordance of knots inside certain types of 3-manifolds.

Symplectic homology for complements of smooth divisors
Samuel Lisi
Mississippi

Symplectic Homology is a kind of Floer homology defined for a class of non-compact symplectic manifolds including cotangent bundles and smooth affine algebraic varieties. In joint work with Luis Diogo, we have developed a method of computing SymplecticHomology for the complement of a smooth divisor in a projective variety in terms of the Gromov-Witten invariants of the divisor and of the variety. I will provide some background on symplectic homology, including a discussion of some of its applications, and will then discuss some of the ingredients of the proof of our theorem.

Equivariant A-theory
Mona Merling
Johns Hopkins

Waldhausen's introduction of A-theory of spaces revolutionized the early study of pseudo-isotopy theory. Waldhausen proved that the A-theory of a manifold splits as its suspension spectrum and a factor Wh(M) whose first delooping is the space of stable h-cobordisms, and its second delooping is the space of stable pseudo-isotopies. I will describe a joint project with C. Malkiewich aimed at telling the equivariant story if one starts with a manifold M with group action by a finite group G.

Investigations of the classifying and classification diagrams
Christina Osborne
Virginia

One can classify categories by using the nerve construction. But the nerve cannot determine the difference between certain types of categories. For example, the nerve cannot distinguish the difference between the trivial category and a category with two objects and one nontrivial morphism between the objects. Rezk's classifying and classification diagrams are generalizations of the nerve construction and can distinguish the difference between these categories. In this talk, we will discuss applying the appropriate diagram to the category of finite sets and the category of graphs as well as the relationship between these diagrams. Also, we will describe the classification diagram of a category where all of the morphisms are weak equivalences.

The rational cohomology of representation spaces
Mentor Stafa
Tulane University

Let $\pi$ be a discrete group and $G$ be a Lie group. We study the topology and in particular cohomology of the space of representations $Hom(\pi,G)$. For $\pi$ a nilpotent group or a free abelian group we describe the rational cohomology of the representation space in terms of the invariants of finite reflection groups. Moreover, we describe stability properties for the cohomology of representation spaces and character varieties. This is joint work with Dan Ramras.

The character of the total power operation
Nathaniel Stapleton
University of Kentucky, Regensburg

In the 90's Goerss, Hopkins, and Miller proved that the Morava E-theories are E_\infty-ring spectra in a unique way. Since then several people including Ando, Hopkins, Strickland, and Rezk have worked on explaining the effect of this structure on the homotopy groups of the spectrum. In this talk, I will present joint work with Barthel that shows how a form of character theory due to Hopkins, Kuhn, and Ravenel can be used to reduce this problem to a combination of combinatorics and the GL_n(Q_p)-action on the Drinfeld ring of full level structures which shows up in the local Langlands correspondence.

Manifolds with the fixed point property and their squares
Fang Sun
Tulane University

We shall consider the Cartesian squares (powers) of manifolds with the fixed point property (f.p.p.). Examples of manifolds with the f.p.p. whose symmetric squares fail to have the f.p.p. will be given.

Multifunctor from Waldhausen Categories to the 1-type of their K-theory Spectrum
Yaineli Valdes
Florida State University

Zakharevich gave a proof of the fact that the category of Waldhausen categories is a closed symmetric multicategory and algebraic K-theory is a multifunctor from the category of Waldhausen categories to the category of spectra. By assigning to any Waldhausen category the fundamental groupoid of the 1-type of its K-theory spectrum, we get a 1-functor from the category of Waldhausen categories to the category of Picard groupoids (since stable 1-types are classified by Picard groupoids). We want to show this 1-functor is a multifunctor. We use the algebraic model defined by Muro and Tonks to define the multifunctor. This is useful because it will describe the algebraic structures on the 1-type of the K-theory spectra induced by the multiexactness pairings on the level of Waldhausen categories.

Transverse invariants, knot Floer homology and branched covers
David Shea Vela-Vick
Louisiana State University

Heegaard Floer theory provides a powerful suite of tools for studying 3-manifolds and their subspaces. In 2006, Ozsvath, Szabo and Thurston defined an invariant of transverse knots which takes values in a combinatorial version of this theory for knots in the 3—sphere. In this talk, we discuss a refinement of their combinatorial invariant via branched covers and discuss some of its properties. This is joint work with Mike Wong.


Lodging

A few hotels reasonably near campus are listed below.

Travelodge Lafayette
1101 West Pinhook Road
20-30 minute walk
Reservations: 337-234-7402
Rate: from $48.00

The Chateau Hotel
1015 W Pinhook Rd
Lafayette, LA, 70503
Reservations: 337-291-1088
Rate: from $97.00

DoubleTree by Hilton Hotel Lafayette
1521 West Pinhook Rd
Lafayette, LA 70503
Reservations: 337-235-6111
Rate: from $109.00

Hilton Garden Inn Lafayette/Cajundome
2350 West Congress Street
30-40 minute walk
Reservations: 337-291-1977
Rate: from $99.00 special conference rates Conference code: MATH17

Comfort Suites
(located off Pinhook Road behind Chili's Restaurant)
114 Rue Fernand
Lafayette, LA 70508
30-40 minute walk
Reservations: 337-456-8390
Rate: from $89.00

Wyndham Garden Lafayette, Lafayette
1801 W Pinhook Road
Lafayette, LA, 70508
Reservations: 337-233-8120
Rate: from $93.00

Best Western Plus Vermillion River Suites Hotel
125 East Kaliste Saloom Road
Lafayette, LA 70508
Reservations: 337-235-1367
Rate: from $90.00

Discovery Inn & Suites Lafayette
120 East Kaliste Saloom Road
Lafayette, LA 70508
Reservations: 337-235-0858
Rate: from $65.00


Parking

Friday On Friday you will need to park in the new parking garage. You will need a "coupon code" to park. (You may be able to park by Maxim Doucet Hall late in the afternoon.) (Use 1289 Girard Park Circle to locate the entrance to the parking garage online.) If you indicated that you need a coupon code for Friday on your registration form, you will receive an email with the code. Map links are posted above.
Saturday and Sunday Parking on Saturday and Sunday is not a problem; you can park by Maxim Doucet Hall. (Use 1401 Johnston Street to locate Maxim Doucet Hall online.)