Andy Nicol
APR
22
2014
3:00PM
Quasi-isometries of graph manifolds do not preserve non-positive curvature
In this talk, we will see the definition of high dimensional graph manifolds and see that there are examples of graph manifolds with quasi-isometric fundamental groups, but where one supports a locally CAT(0) metric while the other cannot. We will use properties of the Euler class as well as various results on bounded cohomology.

Past talks

Jim Fowler
OCT
05
2010
3:30PM
A first talk on surgery
Some graduate students have asked me about surgery theory and what it can do; this talk is an extraordinarily brief, high-level introduction.
Dan Burghelea
OCT
12
2010
3:30PM
Persistence, an invitation to “Topology for Data Analysis”
Persistence is an important new topic in Computational Topology. In this talk I will explain what “Persistence” is, what is this good for, how can it be calculated and what are the new invariants involved in the measuring of persistence. This is a summary of work of Edelsbrunner, Letcher, Zomorodian, Carlsson. To the extent the time permits, or in a follow up lecture, a more refined version of persistence, whose calculation has the same degree of complexity but carry considerably more information will be described (joint work with Tamal Dey). The exposition is elementary and needs only basic concepts of simplicial complexes and homology.
Rob Kirby
OCT
19
2010
3:30PM
Wrinkled fibrations for 4-manifolds
(This is joint work with David Gay) I will discuss the existence and uniqueness theorems for wrinkled fibrations of arbitrary orientable, smooth $n$-manifolds ($n=4$ is the most interesting case) over orientable surfaces. Existence sometimes holds, and there is a natural set of moves relating different wrinkled fibrations for a given $n$-manifold. A wrinkled fibration is one in which the rank of the differential is 2 or is a curve of points of rank 1 which look locally like an arc cross an indefinite $k$-handle (and the curve is the arc cross the critical point of the k-handle). Furthermore, fibers are always connected.
Bruce Williams
OCT
26
2010
3:30PM
Family Hirzebruch Signature Theorem with Converse
Let $X$ be a space which satisfied $4k$-dimensional Poincaré Duality, and let $\sigma(X)$ be the signature of $X$. If $X$ is a manifold, then $\sigma(X)$ can be “disassembled”, i.e. $\sigma(X)$ is determined by a local invariant, the Hirzebruch $L$-polynomial. In this talk I’ll give an enriched version of $\sigma(X)$ which is defined in all dimensions, and for dim >4, the enriched version can be disassembled if and only if $X$ admits manifold structure. There is also a family version of this for fibrations
Ian Leary
NOV
02
2010
3:30PM
Infinite Smith groups and Kropholler’s hierarchy I
This talk concerns actions of (discrete) groups on finite-dimensional contractible simplicial complexes. I call a group G a ‘Smith group’ if every action of G on a finite-dimensional contractible simplicial complex has a fixed point. (The P A Smith theorem tells us that every finite p-group is a Smith group; there are no other finite Smith groups.)
Kropholler’s hierarchy assigns an ordinal to a group, describing how simply it can be made to act on a finite-dimensional contractible simplicial complex. Finite groups are at stage 0 of the hierarchy and stage 1 contains all groups that act on a finite-dimensional contractible simplicial complex without a fixed point. Until our work, no group was known to lie in the hierarchy beyond stage 3.
We construct an infinite Smith group, and construct groups that show that for countable groups, Kropholler’s hierarchy is as long as it possibly could be.
In the talks, I will describe some fixed-point theorems and explain some aspects of our group constructions. The first talk will focus on Smith groups and the second on Kropholler’s hierarchy.
Joint with G Arzhantseva, M Bridson, T Januszkiewicz, P Kropholler, A Minasyan and J Swiatkowski.
Ian Leary
NOV
09
2010
3:30PM
Infinite Smith groups and Kropholler’s hierarchy II
Igor Kriz
NOV
16
2010
3:30PM
Homotopy and Reality
The Galois action of $\mathbb{Z}/2$ on the field of complex numbers plays prominent role in algebraic topology. Its significance in various contexts was discovered by Atiyah (in $K$-theory), Karoubi (Hermitian $K$-theory) and Landweber (real cobordism MR). It also played an important role in the development of equivariant stable homotopy theory by Araki, Adams, May and others. In 1998, Po Hu and I did extensive work on MR, including a complete calculation of its coefficients, and development of what we called Real homotopy theory. Our work was discovered 10 years later by Hill, Hopkins and Ravenel, and played a central role in their recent solution of the Kervaire invariant 1 problem. Meanwhile, there is a baffling parallel between Real and motivic homotopy theory which was used by Morel and Voevodsky, Levine and others in their investigation of algebraic cobordism. Recently, Hu,Ormsby and myself combined Real and algebraic techniques in a solution of the homotopy completion problem for Hermitian K-theory for fields of characteristic 0. I hope to touch on the different aspects of this amazing story in my talk.
Sergei Chmutov
NOV
30
2010
3:30PM
Polynomials of graphs on surfaces
The Jones polynomial of links in 3-space is a specialization of the Tutte polynomial of corresponding plane graphs. There are several generalizations of the Tutte polynomial to graphs embedded into a surface. Some of them are related to the theory of virtual links. Although virtual link theory predicts some relations between these generalizations. I will report about the results obtained in this direction during my summer program "Knots and Graphs".
In particular I will compare three polynomials of graphs on surfaces and a relative version of the Tutte polynomial of planar graphs. The first polynomial, defined by M.Las Vergnas, uses a strong map of the bond matroid of the dual graph to the circuit matroid of the original graph. The second polynomial is the Bollobas-Riordan polynomial of a ribbon graph, a straightforward generalization of the Tutte polynomial. The third polynomial, due V.Krushkal, is defined using the symplectic structure in the first homology group of the surface.
Zbigniew Fiedorowicz
JAN
04
2011
3:30PM
Interchange of monoidal structures in homotopy theory
Guido Mislin
JAN
11
2011
3:30PM
Borel cohomology and large-scale geometry in Lie groups
The Borel cohomology groups $H_B^\star(G, \mathbb{Z})$ of a Lie group $G$ are based on cocycles, which are Borel maps. These Borel cohomology groups are known to be naturally isomorphic to the singular cohomology groups $H^\star(BG,\mathbb{Z})$ of the classifying space $BG$ of $G$, the domain of primary characteristic classes. We discuss the relationship between boundedness properties of cocycles in $H_B^\star(G, \mathbb{Z})$ and subgroup distortion in $G$ (joint work with Indira Chatterji, Yves de Cornulier and Christophe Pittet).
Louis Kauffman
JAN
18
2011
3:30PM
Virtual Knot Theory
Virtual knot theory studies knots in thickened surfaces and has a combinatorial representation that is similar to the diagrams for classical knot theory. This talk is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial and categorifications of the arrow polynomial. The arrow polynomial (of Dye and Kauffman) is a natural generalization of the Jones polynomial, obtained by using the oriented structure of diagrams in the state sum. We will discuss a categorification of the arrow polynomial due to Dye, Kauffman and Manturov and will give an example (from many found by Aaron Kaestner) of a pair of virtual knots that are not distinguished by Khovanov homology (mod 2), or by the arrow polynomial, but are distinguished by a categorification of the arrow polynomial.
Brian Munson
JAN
25
2011
3:30PM
Linking numbers, generalizations, and homotopy theory
The linking number was first defined by Gauss in 1833, who wrote it as an integral which is supposed to compute the number of times one circle wraps around another in space. I will begin by discussing the classical linking number using a much simpler definition by taking a planar projection of the link and counting the number of times one component lies over the other. From this we will see exactly what it is the linking number counts, and this leads to two things. The first is the realization that the linking number is really a "relative" invariant. The second is a generalization, due to the speaker, of a linking "number" for arbitrary manifolds in an arbitrary manifold. I will also discuss Milnor’s higher-order linking numbers, which detect, for example, that classical links such as the Borromean rings are linked (despite being pairwise unlinked). This was generalized by Koschorke to higher-order linking of arbitrary spheres in Euclidean space, and the speaker generalized this to arbitrary manifolds. These higher-order invariants are also relative invariants in the same way the linking number is, and they admit a number of interesting geometric interpretations. Along the way, we will observe that the classical linking number is related to the stable homotopy groups of spheres, whereas the higher-order generalizations are related to a certain filtration of the unstable homotopy groups of spheres.
Niles Johnson
FEB
01
2011
3:30PM
Complex Orientations and p-typicality
This talk will describe computational results related to the structure of power operations on complex oriented cohomology theories (localized at a prime $p$), making use of the amazing connection between complex orientations and the theory of formal group laws. After introducing the relevant concepts, we will describe results from joint work with Justin Noel showing that, for primes $p$ less than or equal to 13, orientations factoring non-trivially through the Brown-Peterson spectrum cannot carry power operations, and thus cannot provide $MU_{(p)}$-algebra structure. This implies, for example, that if E is a Landweber exact $MU_{(p)}$-ring whose associated formal group law is $p$-typical of positive height, then the canonical map $\mathrm{MU}_{(p)} \to E$ is not a map of $H_\infty$ ring spectra. It immediately follows that the standard $p$-typical orientations on $\mathrm{BP}$, $E(n)$, and $E_n$ do not rigidify to maps of $E_\infty$ ring spectra. We conjecture that similar results hold for all primes.
Martin Frankland
FEB
04
2011
1:30PM
Moduli spaces of 2-stage Postnikov systems
It is a classic fact that any graded group (abelian above dimension 1) can be realized as the homotopy groups of a space. However, the question becomes difficult if one includes the data of primary homotopy operations, known as a Pi-algebra. When a Pi-algebra is realizable, we would also like to classify all homotopy types that realize it.
Using an obstruction theory of Blanc-Dwyer-Goerss, we will describe the moduli space of realizations of certain 2-stage Pi-algebras. This is better than a classification: The moduli space provides information about realizations as well as their higher automorphisms.
Howard Marcum
FEB
08
2011
3:30PM
Hopf invariants in $W$-topology
Let ${\mathcal{T}\hspace{-0.3ex}op}_{\ast}$ denote the 2-category of based topological spaces, base point preserving continuous maps, and based track classes of based homotopies. Let $W$ be a fixed space or spectrum and consider the 2-functor on ${\mathcal{T}\hspace{-0.3ex}op}_{\ast}$ obtained by taking the smash product with $W$. The categorical full image of this functor is a 2-category denoted $W{\mathcal{T}\hspace{-0.3ex}op}_{\ast}$ and called the $W$-topology category. For $W$ a space the study of $W$-topology was initiated by Hardie, Marcum and Oda [1]. Of course $W$-topology and stable homotopy theory, while related, are distinct.
In the associated $W$-homotopy category the $W$-homotopy groups $\pi_{r}^W (X)$ have long been recognized as rather significant (but in other notation of course). For example, Barratt (1955) studied $\pi_{n}^W (S^{n})$ for $W=S^1 \cup_p e^2$ Toda (1963) considered the suspension order of a complex $Y_k$ having the same homology as the $(n-1)$-fold suspension $\Sigma^{n-1} P^{2k}$ of the real projective $2k$-space $P^{2k}$, namely the order of the identity class of $\pi_{1}^W (S^{1})$ when $W=Y_k$.
In [1] some non-trivial elements in $W$-homotopy groups were detected by making use of $W$-Hopf invariants. This talk focuses on a general proceedure for introducing Hopf invariants into $W$-topology. As an application, when $W$ is a mod $p$ Moore space, namely $W=S^1 \cup_p e^2$, we show that it is possible to detect elements in $\pi_{r+1}^W (\Omega S^{m+1})$ which have connection with known stable periodic families of the homotopy groups of spheres. In particular we prove nontriviality in $\pi_{r+1}^W (\Omega S^{m+1})$ of elements related to families discovered by Gray (1984) (for $p$ an odd prime) and by Oda (1976) (for $p=2$). This represents joint work with K. Hardie and N. Oda.
[1] K. Hardie, H. Marcum and N. Oda, The Whitehead products and powers in $W$-topology, Proc. Amer. Math. Soc. 131 (2003), 941–951.
Jim Fowler
FEB
15
2011
3:30PM
$\mathcal B$-bounded finiteness
Given a bounding class $B$, we construct a bounded refinement $BK(-)$ of Quillen’s $K$-theory functor from rings to spaces. $BK(-)$ is a functor from weighted rings to spaces, and is equipped with a comparison map $BK \to K$ induced by "forgetting control". In contrast to the situation with $B$-bounded cohomology, there is a functorial splitting $BK(-) \simeq K(-) \times BK^{rel}(-)$ where $BK^{rel}(-)$ is the homotopy fiber of the comparison map. For the bounding class $P$ of polynomial functions, we exhibit an element of infinite order in $PK^{rel}_0(Z[G])$ for $G$ the fundamental group of a certain 3-dimensional solvmanifold. This is joint work with Crichton Ogle.
Dan Burghelea
FEB
22
2011
3:30PM
New topological invariants in Morse Novikov theory (bar codes and Jordan cells)
Inspired by the idea of “persistence” (persistent homology) we introduce a new class of topological invariants for (tame) circle valued maps $f: X \to S^1$. They are Bar Codes and Jordan Cells. If $X$ is compact and $f$ is topologically tame (in particular a Morse circle valued function), they are algorithmically computable; moreover all topological invariants of interest in Novikov-Morse theory can be recovered from them; (for example the Novikov-Betti numbers of $(M, \xi)$, $\xi \in H^1(M;\mathbb{Z})$ representing the homotopy class of $f$, can be recovered from the bar codes while the usual Betti numbers of $M$ from bar codes and Jordan cells. A more subtle invariant like Reidemeister torsion is related to the Jordan cells. The definition of these invariants is based on representation theory of quivers ($=$oriented graphs). The above theory extends Novikov-Morse theory from Morse circle valued maps to tame maps $f:X \to S^1$ and even further to 1-cocycle which is the topological version of closed one form for smooth manifolds. This last extension is more elaborate and will be discussed later.
Crichton Ogle
MAR
01
2011
3:30PM
Finitely presented groups and the $\ell^1$ $K$-theory Novikov Conjecture
Using techniques developed for studying polynomially bounded cohomology, we show that the assembly map for $K_*^t(\ell^1(G))$ is rationally injective for all finitely presented discrete groups $G$. This verifies the $\ell^1$-analogue of the Strong Novikov Conjecture for such groups. The same methods show that the Strong Novikov Conjecture for all finitely presented groups can be reduced to proving a certain (conjectural) rigidity of the cyclic homology group $HC_1^t(H^{CM}_m(F))$ where $F$ is a finitely-generated free group and $H^{CM}_m(F)$ is the “maximal” Connes-Moscovici algebra associated to $F$.
Crichton Ogle
MAR
08
2011
3:30PM
Hermitian K-theory of Spaces
A fundamental question (still unanswered) is whether certain “periodic” functors on the category of discrete groups, such as the topological $K$-theory of $C^*(\pi)$ or the Witt theory of $\mathbb Z[\pi]$, can be extended to a functors on the category of basepointed topological spaces which depend (rationally) on more than just the fundamental group. Following earlier work of Burghelea-Fiedorowicz, Fiedorowicz-Vogt, and Vogell, I will propose a model for a functor $X\mapsto AH(X)$, which may be thought of as an Hermitian analogue of Waldhausen’s functor $X\mapsto A(X)$, and occurs as the $\mathbb Z/2$ fixed-point set of an involution defined on a certain model of $A(X)$. I will also explain how a suitable $\mathbb Z/2$-equivariant version of Waldhausen’s splitting $Q(X_+)\to A(X)\to Q(X_+)$ verifies the Novikov conjecture for $\pi_1(X)$.
Frank Connolly
MAR
15
2011
3:30PM
Involutions on Tori and Topological Rigidity
How many involutions on the $n$ torus have an isolated fixed point?
This is a report of joint work with Jim Davis and Qayum Khan.
We prove that there is only one involution on the $n$-torus, $T^n$, up to conjugacy, for which the fixed set contains an isolated point. But here, $n$ must be of the form $4k$ or $4k+1$ (or else, n must be $\leq 3$). In the other dimensions, we classify all such involutions, using surgery theory and the calculation of the groups $UNil_n(Z,Z,Z).$
We also introduce a Topological Rigidity Conjecture and we show that the above result is a consequence of it.
Indira Chatterji
APR
05
2011
3:30PM
Discrete linear groups containing arithmetic groups
We discuss a question by Nori, which is to determine when a discrete Zariski dense subgroup in a semisimple Lie group containing a lattice has to be itself a lattice. This is joint work with Venkataramana.
Michael Davis
APR
19
2011
3:30PM
Random graph products of groups
There is a theory of random graphs due to Erdos and Renyi. Associated to any group and a graph there is a notion of its graph product. So, there also is a notion of a random graph products of groups. For example, by letting the group be Z/2, the graph product can be any right-angled Coxeter group. We compute the cohomological invariants of random graph products. This is joint work with Matt Kahle.
Paul Goerss
APR
26
2011
2:30PM
Picard groups in stable homotopy theory
In any symmetric monoidal category $C$, the Picard group is the group of isomorphism classes of invertible objects. For the usual stable homotopy category, the only invertible objects are the sphere spectra $S^n$, with $n$ an integer. However, if $E_\star$ is a good (i.e., complex-orientable) homology theory, Mike Hopkins noticed that the $E$-local stable homotopy category could have a rich and curious Picard group—and that this group could give information about how homotopy theory of spectra reassembles from localizations. I’ll review this theory, revisit some of the curious examples, and report on recent calculations. This is joint work with Hans-Werner Henn, Mark Mahowald, and Charles Rezk.
Andy Putman
MAY
04
2011
4:30PM
Teichmüller space
Boris Tsygan
MAY
10
2011
1:00PM
Algebraic structures on Hochschild and cyclic complexes
The Hochschild chain and cochain complexes and the cyclic complex of an associative algebra serve as noncommutative analogs of classical geometric objects on a manifold, such as differential forms and multivector fields. These complexes are known to possess a very nontrivial and rich algebraic structure that is analogous to, and goes well beyond, the classical algebraic structures known in geometry. In this talk, I will give a review of the subject and outline an approach that is based on an observation that differential graded categories form a two-category up to homotopy.
Stacy Hoehn
MAY
17
2011
1:00PM
Obstructions to Fibering Maps
Given a fibration p, we can ask when p is fiber homotopy equivalent to a topological fiber bundle with compact manifold fibers; assuming that the fibration p does admit a compact bundle structure, we can also ask to classify all such bundle structures on p. Similarly, given a map f between compact manifolds, we can ask when f is homotopic to a topological fiber bundle with compact manifold fibers, and assuming that the map f does fiber, we can ask to classify all of the different ways to fiber f. In this talk, we will begin by describing the space of all compact bundle structures on a fibration, which is nonempty if and only if p admits a compact bundle structure. We will then show that, as long as we are willing to stabilize by crossing with a disk, the obstructions to stably fibering a map f are related to the space of bundle structures on the fibration p associated to f.
John Klein
MAY
24
2011
3:30PM
Bundle structures and Algebraic $K$-theory
This talk will describe (Waldhausen type) algebraic $K$-theoretic obstructions to lifting fibrations to fiber bundles having compact smooth/topological manifold fibers. The surprise will be that a lift can often be found in the topological case. Examples will be given realizing the obstructions.
Dan Burghelea
MAY
31
2011
3:30PM
Courtney Thatcher
JUN
07
2011
3:30PM
On free $Z/p$ actions on products of spheres
We consider free actions of large prime order cyclic groups on products of spheres. The equivariant homotopy type will be determined and the simple structure set discussed. Similar to lens spaces, the first $k$-invariant generally determines the homotopy type, however for homotopy equivalences between products of an even number of spheres the Whitehead torsion vanishes and the quotients are also simple homotopy equivalent. Unlike lens spaces which are determined by their Reidemeister torsion and $\rho$-invariant, the $\rho$-invariant vanishes for products of an even number of spheres and the Pontrjagin classes become p-localized homeomorphism invariants for a given dimension. The cohomology classes, Pontrjagin classes, and the set of normal invariants will also be discussed.
Bobby Ramsey
JUL
15
2011
2:30PM
Zhixu Su
AUG
25
2011
3:30PM
Non-simply-connected rational surgery
We will generalize Sullivan’s rational surgery realization theorem to the case when the fundamental group is finite; given a finite group action on a rational Poincaré duality algebra, does there exist a closed manifold realizing the algebra as its cohomology ring with the group acting freely on it?
Stefan Haller
SEP
21
2011
3:30PM
The cohomology of symplectic fiber bundles
The deRham cohomology of a Poisson manifold comes equipped with a canonical filtration. For symplectic manifolds this filtration is well understood and can be computed from the cup product action of the cohomology class represented by the symplectic form. In this talk we will discuss said filtration for the total space of sympletic fiber bundles. The latter constitute a class of Poisson manifolds closely related to the topology of the sympletic group of the typical fiber.
Dan Burghelea
OCT
07
2011
4:30PM
New topological invariants (bar codes and Jordan cell) at work (part I)
Bar codes and Jordan cells provide a new type of linear algebra invariants which can be used in topology. In joint work with Tamal Dey we have associated with any angle valued generic map $f : X \to S^1$, $X$ a compact nice space (ANR), $\kappa$ any field and any integer $r$, $0 \leq r \leq \dim X$, a collection of such bar codes and Jordan cells. They can be effectively computed in case $\kappa = \mathbb{C}$ or $\mathbb{Z}_2$, $X$ is a simplicial complex and f a simplicial map by algorithms implementable by familiar software (Matematica, Mapple or Matlab). In this lecture I will describe some joint work with S Haller.
1. We prove that the Jordan cells defined using $f$ are homotopy invariants of the pair $(X,\xi)$, $\xi ∈ H^1(X;\mathbb{Z})$ representing $f$.
2. We calculate the homology $H_∗(\tilde{X};\kappa)$ as a $\kappa[t^{−1},t]$ module, $\tilde{X}$ the infinite cyclic cover of $X$ induced by $\xi$, as well as and the Novikov homology and Milnor-Turaev torsion of $(X;\xi)$ in terms of bar codes and Jordan cells.
3. As a consequence we introduce Lefchetz zeta function of a pair $(X;\xi)$ which generalizes the familiar Lefschetz zeta function of a self map of a compact manifold and the Alexander polynomial of a knot, and relate this function to dynamics.
Dan Burghelea
OCT
14
2011
4:30PM
New topological invariants (bar codes and Jordan cell) at work (part II)
Bar codes and Jordan cells provide a new type of linear algebra invariants which can be used in topology. In joint work with Tamal Dey we have associated with any angle valued generic map $f : X \to S^1$, $X$ a compact nice space (ANR), $\kappa$ any field and any integer $r$, $0 \leq r \leq \dim X$, a collection of such bar codes and Jordan cells. They can be effectively computed in case $\kappa = \mathbb{C}$ or $\mathbb{Z}_2$, $X$ is a simplicial complex and f a simplicial map by algorithms implementable by familiar software (Matematica, Mapple or Matlab). In this lecture I will describe some joint work with S Haller.
1. We prove that the Jordan cells defined using $f$ are homotopy invariants of the pair $(X,\xi)$, $\xi ∈ H^1(X;\mathbb{Z})$ representing $f$.
2. We calculate the homology $H_∗(\tilde{X};\kappa)$ as a $\kappa[t^{−1},t]$ module, $\tilde{X}$ the infinite cyclic cover of $X$ induced by $\xi$, as well as and the Novikov homology and Milnor-Turaev torsion of $(X;\xi)$ in terms of bar codes and Jordan cells.
3. As a consequence we introduce Lefchetz zeta function of a pair $(X;\xi)$ which generalizes the familiar Lefschetz zeta function of a self map of a compact manifold and the Alexander polynomial of a knot, and relate this function to dynamics.
Matt Sequin
OCT
21
2011
4:30PM
An Algebraic Proof of the Equivalence of Two Quantum 3-Manifold Invariants
We will compare two different quantum 3-manifold invariants, both of which are given using a finite dimensional Hopf Algebra $H$. One is the Hennings invariant, given by an algorithm involving the link surgery presentation of a 3-manifold and the Drinfeld double $D(H)$; the other is the Kuperberg invariant, which is computed using a Heegaard diagram of the 3-manifold and the same $H$. We have shown that when $H$ has the property of being involutory, these two invariants are actually equivalent. The proof is totally algebraic and does not rely on general results involving categorical invariants. We will also briefly discuss some results in the case where $H$ is not involutory.
John Oprea
OCT
28
2011
4:30PM
LS category, the fundamental group and Bochner-type estimates
The LS category of a space X is a numerical invariant that measures the complexity of a space. While it is usually very hard to compute explicitly, there are estimates and approximating invariants that help us to understand category better. A big problem is to understand the effect of the fundamental group on category. Recently, extending work of Dranishnikov, Jeff Strom and the speaker have given an upper bound for category using Ralph Fox’s 1-category (and another approximating invariant). Using an interpretation of this 1-category given by Svarc, we have also been able to refine Bochner’s bound on the first Betti number in the presence of non-negative Ricci curvature. Finally, the 1-category forms a bridge between the theorems of Yamaguchi and Kapovitch-Petrunin-Tuschmann on manifolds with almost non-negative sectional curvature.
Ron Fintushel
NOV
01
2011
10:30AM
Surgery on nullhomologous tori and smooth structures on 4-manifolds
By studying the example of smooth structures on $CP^2 \# 3(-CP^2)$, I will illustrate how surgery on a single embedded nullhomologous torus can be utilized to change the symplectic structure, the Seiberg-Witten invariant, and hence the smooth structure on a 4-manifold.
Grigori Avramidi
NOV
09
2011
4:30PM
Isometries of aspherical manifolds
I will describe some recent results on isometry groups of aspherical Riemannian manifolds and their universal covers. The general theme is that topological properties of an aspherical manifold often restrict the isometries of an arbitrary complete Riemannian metric on that manifold. These topological properties tend to be established by using a specific "nice" metric on the manifold.
I will illustrate this by explaining why on an irreducible locally symmetric manifold, no metric has more symmetry than the locally symmetric metric. I will also discuss why moduli space is a minimal orbifold and relate this phenomenon to symmetries of arbitrary metrics on moduli space.
Max Forester
NOV
18
2011
4:30PM
Higher Dehn functions of some abelian-by-cyclic groups
I will discuss the geometry of certain abelian-by-cyclic groups and show how to establish the optimal top-dimensional isoperimetric inequality that holds in these groups. This is joint work with Noel Brady.
Steve Ferry
NOV
22
2011
3:10PM
Volume Growth, DeRham Cohomology, and the Higson Compactification
We construct a variant of DeRham cohomology and use it to prove that the Higson compactification of $R^n$ has uncountably generated $n^{\mbox{th}}$ integral cohmology. We also explain that there is, nevertheless, a way of using the Higson compactification to prove the Novikov conjecture for a large class of groups.
Dave Constantine
DEC
02
2011
2:30PM
Group actions and compact Clifford-Klein forms of homogeneous spaces
A compact Clifford-Klein form of the homogeneous space $J\backslash H$ is a compact manifold $J\backslash H/\Gamma$ constructed using a discrete subgroup $\Gamma$ of $H$. I will survey the existence problem for compact forms, with particular attention to the case when there is an action by a large group on $J\backslash H/\Gamma$. I will also make some remarks on a conjecture of Kobayashi on the scarcity of compact forms.
Pablo Su\'arez-Serrato
JAN
10
2012
3:30PM
Using 4-manifolds to describe groups
We will describe decompositions of finitely presented groups, using descriptions of smooth and of symplectic four-manifolds. Every finitely presented group admits a decomposition into a triple consisting of the fundamental groups of two compact complex Kähler surfaces with boundary and the fundamental group of a three manifold. We will exhibit various ways of obtaining similar decompositions of finitely presented groups into graphs, via descriptions of smooth 4-manifolds into Lefschetz fibrations. We distill this data into invariants, by considering the minimal number of edges these graphs may have. These ideas are related to the minimal euler characteristic of symplectic four-manifolds and the minimal genus of a Lefschetz fibration, seen as group invariants.
Eric Babson
JAN
12
2012
3:30PM
Random simplicial complexes
The study of fundamental groups of random two dimensional simplicial complexes calls attention to the small subcomplexes of such objects. Such subcomplexes have fewer triangles than some multiple of the number of their vertices. One gets that this condition with constant less than two on a connected complex (and all of its subcomplexes) implies that it is homotopy equivalent to a wedge of circles, spheres and projective planes. This analysis yields parameter regimes for vanishing, hyperbolicity and Kazhdanness of these groups. For clique complexes of random graphs there is a similar problem involving complexes with fewer edges than thrice the number of their vertices resulting in similar results on the fundamental groups of their clique complexes. This is based on joint work with Hoffman and Kahle.
Graham Denham
JAN
17
2012
3:30PM
Duality properties for abelian covers
In parallel with a classical definition due to Bieri and Eckmann, say an FP group G is an abelian duality group if H^p(G,Z[G^{ab}]) is zero except for a single integer p=n, in which case the cohomology group is torsion-free. We make an analogous definition for spaces. In contrast to the classical notion, the abelian duality property imposes some obvious constraints on the Betti numbers of abelian covers.
While related, the two notions are inequivalent: for example, surface groups of genus at least 2 are (Poincaré) duality groups, yet they are not abelian duality groups. On the other hand, using a result of Brady and Meier, we find that right-angled Artin groups are abelian duality groups if and only if they are duality groups: both properties are equivalent to the Cohen-Macaulay property for the presentation graph. Building on work of Davis, Januszkiewicz, Leary and Okun, hyperplane arrangement complements are both duality and abelian duality spaces. These results follow from a slightly more general, cohomological vanishing theorem, part of work in progress with Alex Suciu and Sergey Yuzvinsky.
Jean Lafont
JAN
24
2012
3:30PM
Riemannian vs. metric non-positive curvature on 4-manifolds
I’ll outline the construction of smooth 4-manifolds which support locally CAT(0)-metrics (the metric version of non-positive curvature), but do not support any Riemannian metric of non-positive sectional curvature. This is joint work with Mike Davis and Tadeusz Januszkiewicz.
Thomas Kerler
JAN
31
2012
3:30PM
Faithful Representations of the Braid Groups via Quantum Groups
In the last couple of decades the study of representations of braid groups $B_n$ attracted attention from two rather different motivations. One deals with the linearity of the braid groups, that is, whether the $B_n$ can be faithfully represented.
This question was answered in the positive for the Lawrence-Krammer-Bigelow (LKB) representation independently by Krammer and Bigelow around 2001. The LKB representation is given by the natural action of $B_n$ in the second homology of a rank two free abelian cover of the two-point configuration space on the $n$-punctured disc. It is thus naturally a module over the ring of Laurent polynomials in two variables.
The other development is the construction of braid representations from quantum groups. One such class of $B_n$-representations of is constructed using quantum-$sl_2$, which is a one-parameter Hopf algebra deformation of the universal enveloping algebra of $sl_2$. The algebra admits a quasi-triangular R-matrix which can be used to represent $B_n$ on the $n$-fold tensor product of a Verma module with generic highest weight.
We prove that the latter representation, with some refinement of the ground ring, is isomorphic to the LKB representation where the two parameters corresponding to the generators of the Deck transformation group are identified with the deformation parameter of quantum-$sl_2$ and the generic highest weight of the Verma module respectively. We also show irreducibility of this representation over the fraction field of the ring of Laurent polynomials.
Time permitting we will discuss relations to other types of braid group representations that may shed light on this curious connection, as well as reducibility issues at certain choices of parameters.
Joint work with Craig Jackson.
Niles Johnson
FEB
07
2012
3:30PM
Obstruction theory for homotopical algebra maps
We take an obstruction-theoretic approach to the question of algebraic structure in homotopical settings. At its heart, this is an application of the Bousfield-Kan spectral sequence adapted for the action of a monad T on a topological model category.
This talk will focus on the special case where T is a monad encoding E_infty structure in spectra and H_infty structure in the derived category of spectra. We will present examples from rational homotopy theory illustrating the obstructions to rigidifying homotopy algebra maps to strict algebra maps, and explain in a precise way how the edge homomorphism of this obstruction spectral sequence measures the difference between up-to-homotopy and on-the-nose T-algebra maps.
Dan Burghelea
FEB
14
2012
2:30PM
New topological invariants for a continuous nonzero complex valued function
Given a compact ANR $X$ and $f : X \to \mathbb{C} \setminus 0$ a continuous map, for any $0 \leq r \leq \dim X$, one proposes three monic complex valued polynomials $P_{r,s}(z)$, $P_{r,a}(z)$ and $P_{r,m}(z)$, with $\deg(P_{r,s}(z) = \beta_r(X)$ where $\beta_r(X)$ is the r−th Betti number, $\deg(P_{r,a}(z) = \beta_r^N(X,f)$, where $\beta_r^{N}(X,f)$ is the $r$−th Novikov Betti number, $P_{r,m}(z)$ a homotopy invariant of $f$. The first two are continuous assignments with respect to compact open topology, the last is locally constant (on the space of continuous functions with compact open topology).
Ian Hambleton
FEB
21
2012
3:30PM
Co-compact discrete group actions and the assembly map
A discrete group $\Gamma$ can act freely and properly on $S^n \times R^m$, for some $n, m >0$ if and only if $\Gamma$ is a countable group with periodic Farrell cohomology: Connolly-Prassidis (1989) assuming $vcd(\Gamma)$ finite, Adem-Smith (2001). For free co-compact actions there are additional restrictions, but no general sufficient conditions are known. The talk will survey this problem and its connection to the Farrell-Jones assembly maps in K-theory and L-theory.
James Davis
FEB
28
2012
3:30PM
Smith Theory
P.A. Smith, in the first half of the 20th century, developed homological tools to study actions of finite p-groups on topological spaces. The standard applications of Smith theory are that if a p-group acts on a {disk, sphere, manifold, finite-dimensional space} then the fixed set of the action is a {mod p homology disk, mod p homology sphere, mod p homology manifold, finite dimensional space}.
This talk will review the classical theory, give applications to actions on aspherical manifolds, and extend the theory to give restrictions on periodic knots.
Bobby Ramsey
MAR
06
2012
3:30PM
Amenability and Property A
We discuss Yu’s property A as a generalization of amenability for countable groups. A few characterizations of amenability and property A are given, including Johnson’s cohomological characterization of amenability and the recent work of Brodzki, Nowak, Niblo, and Wright which characterizes property A in a similar manner. These characterizations play a major role in the relative versions of these properties.
Bobby Ramsey
MAR
13
2012
3:30PM
Relative Property A and Relative Amenability
We define the notion of a group having relative property A with respect to a finite family of subgroups. Many characterizations for relative property A are given. In particular a cohomological characterization shows that if $G$ has property A relative to a family of subgroups $\mathcal{H}$, and if each $H \in \mathcal{H}$ has property A, then $G$ has property A. This result leads to new classes of groups that have property A. Specializing the definition of relative property A, an analogue definition of relative amenability for discrete groups are introduced and similar results are obtained.
Randy McCarthy
APR
03
2012
3:30PM
On the Algebraic K-theory of Brave New Tensor Algebras
Waldhausen’s A-theory of a space $X$ is sometimes described as the “universal Euler class”. Along these lines, the “universal generalized Lefschetz class” of an endomorphism would be the reduced algebraic K-theory of an associated “brave new” tensor algebra. Recent joint work with Ayelet Lindenstrauss describing this spectrum, when one is working in an analytic range (in the sense of Goodwillie’s calculus of functors) will be discussed.
For $\pi_0$, these results go back to Almkvist, Rincki and L\"uck. More generally these results are related to the theses of Lydakis and Iwachita which built upon the computation of the $A$-theory of the suspension of a space by Carlsson, Cohen, Goodwillie and Hsiang.
Matt Kahle
APR
10
2012
3:30PM
Sharp vanishing thresholds for cohomology of random flag complexes
The random flag complex is a natural combinatorial model of random topological space. In this talk I will survey some results about the expected topology of these objects, focusing on recent work which gives a sharp vanishing threshold for kth cohomology with rational coefficients.
This recent work provides a generalization of the Erdos-Renyi theorem which characterizes how many random edges one must add to an empty set of n vertices before it becomes connected. As a corollary, almost all d-dimensional flag complexes have rational homology only in middle degree (d/2).
This is topology seminar, so I will assume that people know what homology and cohomology are, but I will strive to make the talk self contained and define all the necessary probability as we go.
Ruben Sanchez-Garcia
APR
24
2012
3:30PM
Classifying spaces and the Isomorphism Conjectures
For each discrete group G one can find a universal G-space with stabilizers in a prescribed family of subgroups of G. These spaces play a prominent role in the so-called Isomorphism Conjectures, namely the Baum-Connes and the Farrell-Jones conjectures. We will discuss the former conjecture in more detail and describe its topological side: the equivariant K-homology of the universal space for proper actions. Finally, we will report on joint work with Jean-François Lafont and Ivonne Ortiz on the rationalized topological side for some low dimensional groups.
Ivonne Ortiz
APR
26
2012
3:30PM
The lower algebraic $K$-theory of three-dimensional crystallographic groups
In this joint work with Daniel Farley, we compute the lower algebraic $K$-groups of all split three-dimensional crystallographic groups $G$. These groups account for 73 isomorphism types of three-dimensional crystallographic groups, out of 219 types in all. Alves and Ontaneda in 2006, gave a simple formula for the Whitehead group of a 3-dimensional crystallographic group $G$ in terms of the Whitehead groups of the virtually infinite cyclic subgroups of $G$. The main goal in this work in progress is to obtain explicit computations for $K_0(ZG)$ and $K_{-1}(ZG)$ for these groups.
Dan Isaksen
MAY
15
2012
3:30PM
Sums-of-squares formulas and motivic cohomology
A sums-of-squares formula over a field $k$ is a polynomial identity of the form $\left( x_1^2 + \cdots + x_r^2 \right) \left( y_1^2 + \cdots + y_s^2 \right) = z_1^2 + \cdots z_t^2,$ where the $z$’s are bilinear in the $x$’s and $y$’s over $k$. If a sums-of-squares formula exists over $\mathbb{R}$, then a theorem of Hopf from 1940 gives numerical restrictions on $r$, $s$, and $t$. This result was one of the earliest uses of the cup product in singular cohomology.
I will describe some joint work with D. Dugger on generalizing Hopf’s result to arbitrary fields of characteristic not $2$. The basic idea is to use motivic cohomology instead of singular cohomology.
This leads into the broader subject of computations in motivic homotopy theory.
This is a joint talk with algebraic geometry seminar.
Andrew Salch
MAY
22
2012
3:30PM
Algebraic G-theory via twisted deformation theory
We review some old problems in algebraic topology--namely, the classification of finite-dimensional modules over subalgebras of the Steenrod algebra, and related classification problems in representation theory and finite CW complexes--and some old techniques in deformation theory--namely, the use of Hochschild 1- and 2-cocycles with appropriate coefficients to classify first-order deformations of modules and algebras, respectively. Then we work out how one has to adapt these old methods to solve these old problems, ultimately using some modern technology: a deformation-theoretic interpretation of twisted nonabelian higher-order Hochschild cohomology.
Grigori Avramidi
AUG
28
2012
3:00PM
Flat tori in the homology of some locally symmetric spaces
I’ll show that many finite covers of $\mathrm{SL}(m,Z)\backslash \mathrm{SL}(m,R)/\mathrm{SO}(m)$ have non-trivial homology classes generated by totally geodesic flat $(m-1)$-tori. This is joint work with Tam Nguyen Phan.
Stefan Haller
SEP
04
2012
3:00PM
Commutators of diffeomorphisms
Suppose $M$ is a smooth manifold and let $G$ denote the connected component of the identity in the group of all compactly supported diffeomorphisms of $M$. It has been known for quite some time that the group $G$ is simple, i.e. has no non-trivial normal subgroups. Consequently, $G$ is a perfect group, i.e. each element $g$ of $G$ can be written as a product of commutators, $$g=[h_1,k_1]\circ\cdots\circ[h_N,k_N].$$ Actually, all available proofs (Herman, Mather, Epstein, Thurston) for the simplicity of $G$ first establish perfectness; it is then rather easy to conclude that $G$ has to be simple.
In the talk I will discuss a new, more elementary, proof for the perfectness of the group $G$. This approach also shows that the factors $h_i$ and $k_i$ in the presentation above can be chosen to depend smoothly on $g$. Moreover, it leads to new estimates for the number of commutators necessary. If $g$ is sufficiently close to the identity, then $N=4$ commutators are sufficient; for certain manifolds (e.g. mapping tori) even $N=3$ will do.
This talk is based on joint work with T. Rybicki and J. Teichmann.
Christopher Davis
SEP
18
2012
3:00PM
Computing Abelian rho-invariants of links via the Cimasoni-Florens signature
The solvable filtration of the knot concordance group has been studied closely since its definition by Cochran, Orr and Teichner in 2003. Recently Cochran, Harvey and Leidy have shown that the successive quotients in this filtration contain infinite rank free abelian groups and even exhibit a kind of primary decomposition. Unfortunately, their construction relies on an assumption of non-vanishing of certain rho-invariants. By relating these rho-invariants to the signature function defined by Cimasoni and Florens in 2007, we remove this ambiguity from the construction of Cochran-Harvey-Leidy.
Duane Randall
OCT
02
2012
3:00PM
On Homotopy Spheres
We present results concerning the existence of nontrivial homotopy spheres and also discuss the determination of the smallest dimensional Euclidean spaces in which they smoothly embed.
Niles Johnson
OCT
09
2012
3:00PM
Ecological Niche Topology
The ecological niche of a species is the set of environmental conditions under which a population of that species persists. This is often thought of as a subset of "environment space" -- a Euclidean space with axes labeled by environmental parameters. This talk will explore mathematical models for the niche concept, focusing on the relationship between topological and ecological ideas. We also describe applications of machine learning to develop empirical models from data in the field. These lead to novel questions in computational topology, and we will discuss recent progress in that direction. This is joint with John Drake in ecology and Edward Azoff in mathematics.
Paul Arne Østvær
OCT
16
2012
3:00PM
Motivic slices and the graded Witt ring
We compute the motivic slices of hermitian K-theory and higher Witt-theory. The corresponding slice spectral sequences relate motivic cohomology to Hermitian K-groups and Witt groups, respectively. Using this we compute the Hermitian K-groups of number fields, and (re)prove Milnor’s conjecture on quadratic forms for fields of characteristic different from 2. Joint work with Oliver Röndigs.
Pierre-Emmanuel Caprace
OCT
30
2012
3:00PM
Rank one elements in Coxeter groups and CAT(0) cube complexes
This talk centers around the Rank Rigidity Conjecture for groups acting properly and cocompactly on CAT(0) spaces. After discussing some generalities on the conjecture and some of its consequences, I will focus on the two special cases alluded to in the title.
Jean Lafont
NOV
06
2012
3:00PM
Comparing semi-norms on 3rd homology
It follows from work of Crowley-Loeh (d>3) and Barge-Ghys (d=2) that in all degrees distinct from d=3, the l^1-seminorm and the manifold semi-norm coincide on homology of degree d. We show that when d=3, the two semi-norms are bi-Lipschitz to each other, with an explicitly computable constant. This was joint work with Christophe Pittet (Univ. Marseille).
Jim Fowler
NOV
13
2012
3:00PM
Numeric methods in topology
Often the answer to a topological question is, by its own admission, nonconstructive, but even when the answer is constructive, serious difficulties can arise in carrying out that construction. We will consider a couple cases like this. As an approachable, low-dimensional example, we decompose surfaces as a square complex with a fixed number of squares meeting at a vertex. As a high-dimensional example, we consider the possible Pontrjagin numbers of highly connected 32-manifolds. To address this latter case, we will be confronted with needing to compute the coefficients of the Hirzebruch $L$-polynomial; some topology provides a recursive method faster than naive symmetric reduction.
Neal Stoltzfus
NOV
27
2012
3:00PM
Knots with Cyclic Symmetries and Recursion in Knot Polynomial of Link families
For knots invariant under a finite order cyclic symmetry, Seifert, Murasugi and others developed relations constraining the Alexander polynomial of such knots.
We develop similar constraints using the transfer method of generating functions is applied to the ribbon graph rank polynomial. This polynomial, denoted $R(D;X,Y,Z)$, is due to Bollobás, Riordan, Whitney and Tutte. Given a sequence of ribbon graphs, $D_n$, constructed by successive amalgamation of a fixed pattern ribbon graph, we prove by the transfer method that the associated sequence of rank polynomials is recursive: that is, the polynomials $R(D_n;X,Y,Z)$ satisfy a linear recurrence relation with coefficients in $Z[X,Y,Z]$.
We develop conditions for the Jones polynomial of links which admit a periodic homeomorphism, by applying the above result and the work of Dasbach et al showing that the Jones polynomial is a specialization of the ribbon graph rank polynomial.
This is joint work with Jordan Keller and Murphy-Kate Montee)
Jenny George
DEC
04
2012
3:00PM
TQFTs from Quasi-Hopf Algebras and Group Cocycles
The original Hennings TQFT is defined for quasitriangular Hopf algebras satisfying various nondegeneracy requirements. We extend this construction to quasitriangular quasi-Hopf algebras with related nondegeneracy conditions and prove that this new “quasi-Hennings” algorithm is well-defined and gives rise to TQFTs. The ultimate goal is to apply this construction to the Dijkgraaf-Pasquier-Roche twisted double of the group algebra, and then show that the resulting TQFT is equivalent to a more geometric one, described by Freed and Quinn.
Tam Nguyen Phan
JAN
08
2013
4:30PM
Aspherical manifolds obtained by gluing locally symmetric manifolds
Aspherical manifolds are manifolds that have contractible universal covers. I will explain how to construct closed aspherical manifolds by gluing the Borel-Serre compactifications of locally symmetric spaces using the reflection group trick. I will also discuss rigidity aspects of these manifolds, such as whether a homotopy equivalence of such a manifold is homotopic to a homeomorphism.
Anh T. Tran
JAN
15
2013
3:00PM
On the AJ conjecture for knots
We consider the AJ conjecture that relates the A-polynomial and the colored Jones polynomial of a knot. Using skein theory, we show that the conjecture holds true for some classes of two-bridge knots and pretzel knots.
Moshe Cohen
JAN
22
2013
3:00PM
Kauffman’s clock lattice as a graph of perfect matchings: a formula for its height.
Kauffman gives a state sum formula for the Alexander polynomial of a knot using states in a lattice that are connected by his clock moves. We show that this lattice is more familiarly the graph of perfect matchings of a bipartite graph obtained from the knot diagram by overlaying the two dual Tait graphs of the knot diagram.
Using a partition of the vertices of the bipartite graph, we give a direct computation for the height of Kauffman’s clock lattice obtained from a knot diagram with two adjacent regions starred and without crossing information specified.
We prove structural properties of the bipartite graph in general and mention applications to Chebyshev or harmonic knots (obtaining the popular grid graph) and to discrete Morse functions.
This talk is accessible to those without a background in knot theory. Basic graph theory is assumed.
John Harper
FEB
12
2013
3:00PM
Completions in topology and homotopy theory
I will give a historical overview of completions in topology and homotopy theory starting with the work of D. Sullivan, together with motivation and applications of these constructions, including H.R. Miller’s proof of the Sullivan conjecture and Mandell’s "homotopical double dual" result for algebraically characterizing p-adic homotopy types. I will then describe a variation of these completion ideas for the enriched algebraic-topological context of homotopy theoretic commutative rings that arises naturally in algebraic K-theory, derived algebraic geometry, and algebraic topology. I will finish by describing some recent results on completion in this new context, which are joint with M. Ching.
Nathan Dunfield
FEB
26
2013
3:00PM
Integer homology 3-spheres with large injectivity radius
Conjecturally, the amount of torsion in the first homology group of a hyperbolic 3-manifold must grow rapidly in any exhaustive tower of covers (see Bergeron-Venkatesh and F. Calegari-Venkatesh). In contrast, the first betti number can stay constant (and zero) in such covers. Here "exhaustive" means that the injectivity radius of the covers goes to infinity. In this talk, I will explain how to construct hyperbolic 3-manifolds with trivial first homology where the injectivity radius is big almost everywhere by using ideas from Kleinian groups. I will then relate this to the recent work of Abert, Bergeron, Biringer, et. al. In particular, these examples show a differing approximation behavior for L^2 torsion as compared to L^2 betti numbers. This is joint work with Jeff Brock.
Martin Frankland
MAR
05
2013
3:00PM
The homotopy of p-complete K-algebras
Morava E-theory is an important cohomology theory in chromatic homotopy theory. Rezk described the algebraic structure found in the homotopy of $K(n)$-local commutative E-algebras, via a monad on $E_\ast$-modules that encodes all power operations. However, the construction does not see that the homotopy of a $K(n)$-local spectrum is L-complete (in the sense of Greenlees-May and Hovey-Strickland). We show that the construction can be improved to a monad on $L$-complete $E_\ast$-modules, and discuss some applications. Joint with Tobias Barthel.
Nobuyuki Oda
MAR
19
2013
3:00PM
Brown-Booth-Tillotson products and exponentiable spaces
To study the exponential law for function spaces with the compact-open topology, R. Brown introduced a topology for product set, which is finer than the product topology, and showed the exponential law for any Hausdorff spaces. The method was improved by P. Booth and J. Tillotson, making use of test maps, and they removed the Hausdorff condition for spaces. The product space they used is called the BBT-product. If we use any class of exponentiable spaces, then we can define a topology for function spaces which enables us to prove the exponential law with the BBT-product for any spaces. We can apply the result to based spaces and we get various good results for homotopy theory. For example, we can prove a theorem of pairings of function spaces without imposing conditions on spaces and base points. If we look at the techniques carefully, we find that the results can also be applied to study group actions on function spaces.
The BBT-product is asymmetric in general and we can define the ‘centralizer’ of the BBT-product, which contains the class of k-spaces defined by the class. The centralizer of the BBT-product has good properties for homotopy theory.
This talk is based on joint work with Yasumasa Hirashima.
Charles Estill
MAR
26
2013
3:00PM
Matroid Connection: Matroids for Algebraic Topology
In our paper "Polynomial Invariants of Graphs on Surfaces" we found a relationship between two polynomials cellularly embedded in a surface, the Krushkal polynomial, based on the Tutte polynomial of a graph and using data from the algebraic topology of the graph and the surface, and the Las Vergnas polynomial for the matroid perspective from the bond matroid of the dual graph to the circuit matroid of the graph, $\mathcal{B}(G^\ast) \to \mathcal{C}(G)$.
With Vyacheslav Krushkal having (with D. Renardy) expanded his polynomial to the $n$th dimension of a simplicial or CW decomposition of a $2n$-dimensional manifold, a matroid perspective was found whose Las Vergnas polynomial would play a similar role to that in the 2-dimensional case.
We hope that these matroids and the perspective will prove useful in the study of complexes.
Ian Leary
APR
02
2013
3:00PM
Platonic triangle complexes
I will discuss work arising from Raciel Valle’s thesis concerning complexes built from triangles that are highly symmetrical and have vertex links the join of $n$ pairs of points (equivalently the $1$-skeleton of the $n$-dimensional analogue of the octahedron).
Michael A. Mandell
APR
09
2013
3:00PM
The homotopy theory of cyclotomic spectra
In joint work with Andrew Blumberg, we construct a category of cyclotomic spectra that is (something like) a closed model category and which has well-behaved mapping spectra. We show that topological cyclic homology (TC) is the corepresentable functor on this category given by maps out of the sphere spectrum, verifying a conjecture of Kaledin.
Mark Meilstrup
APR
16
2013
3:00PM
Reduced forms for one-dimensional Peano continua
We will discuss a few reduced forms for homotopy types of 1-dim Peano continua. "Deforested" continua contain no attached strongly contractile subsets (dendrites). For 1-dim continua this always gives a minimal deformation retract, or core. In a core 1-dim continuum, the points which are not homotopically fixed form a graph. Furthermore, this can be homotoped to an "arc reduced" continuum, where the non-homotopically fixed points are in fact a union of arcs.
Christopher Davis
APR
23
2013
3:00PM
Satellite operators as a group action
Let $P$ be a knot in a solid torus, $K$ be a knot in $3$-space and $P(K)$ be the satellite knot of $K$ with pattern $P$. This correspondence defines an operator, the satellite operator, on the set of knot types and induces a satellite operator $P:C\to C$ on the set of smooth concordance classes of knots. In a recent paper with Tim Cochran and Arunima Ray, we show that for many patterns this map is injective. I will approach this result from a different perspective, namely by showing that satellite operators really come from a group action. In 2001, Levine studied homology cylinders over a surface modulo the relation of homology cobordism as a group containing the mapping class group. We show that this group also contains satellite operators and acts on an enlargement of knot concordance. In doing so we recover the injectivity result. I will also present some preliminary results on the surjectivity of satellite operators on knot concordance. This is joint work with Arunima Ray of Rice University.
Andrew Salch
APR
30
2013
3:00PM
Chromatic convergence and its discontents
The chromatic convergence theorem of Ravenel and Hopkins asserts that, if $X$ is a $p$-local finite spectrum, then the homotopy limit $\text{holim}_n L_{E(n)}(X)$ of the localizations of $X$ at each of the Johnson-Wilson $E$-theories $E(n)$ is homotopy-equivalent to $X$ itself. One way of seeing the chromatic convergence theorem is by regarding the functor sending a spectrum $X$ to $\text{holim}_n L_{E(n)}(X)$ as a kind of completion, "chromatic completion," which has the agreeable property that $p$-local finite spectra are all already chromatic complete. Then there are two natural questions:
1. Given a (not necessarily finite) spectrum $X$, is there a criterion that lets us decide easily whether $X$ is chromatically complete or not?
2. Given a nonclassical setting for homotopy theory, such as equivariant spectra or motivic spectra, what analogue of the chromatic convergence theorem might hold?
We give an answers to each of these two questions. For a symmetric monoidal stable model category $C$ satisfying some reasonable hypotheses, we produce a natural notion of "chromatic completion," as well as the notion of a "chromatic cover," a commutative monoid object which shares important properties with the complex cobordism spectrum $MU$ from classical stable homotopy theory. We show that, if a chromatic cover exists in $C$, then any object $X$ satisfying Serre’s condition $S_n$ for any $n$ is chromatically complete if and only if the microlocal cohomology of $X$ vanishes. (Of course we have to define Serre’s condition $S_n$ as well as microlocal cohomology in this context!)
We get two important corollaries: first, by computing some microlocal cohomology groups, we find that large classes of non-finite classical spectra are not chromatically complete, such as the connective spectra $ku$ and $BP\langle n \rangle$ for all finite $n$. We also get some non-chromatic-completeness results for $\text{ko}$, $\text{tmf}$, and $\text{taf}$. Second, we get conditions under which a chromatic completion theorem can hold for motivic and equivariant spectra: one needs a chromatic cover to exist in those categories of spectra. We identify a candidate for such a chromatic cover for motivic spectra over $\text{Spec}\, C$, assuming the Dugger-Isaksen nilpotence conjecture.
Andrew Salch
MAY
01
2013
3:00PM
Explicit class field theory and stable homotopy groups of spheres
One knows from Artin reciprocity that, for any abelian Galois extension $L/K$ of $p$-adic number fields, there is an isomorphism $K^{\times} / N_{L/K} L^{\times} \to Gal(L/K)$ from the units in $K$ modulo the norms of units in $L$ to the Galois group of $L/K$; this isomorphism is called the "norm residue symbol." Computing the norm residue symbol explicitly on specific elements of its domain is quite difficult and is an open area of research in algebraic number theory.
Given an abelian Galois extension $L$ of $Q_p$ and a finite CW-complex $X$, we use Lubin-Tate theory and the Goerss-Hopkins-Miller theorem to produce a particular subgroup of the $K(1)$-local stable homotopy groups of $X$. We show that this construction provides a filtration, indexed by the abelian extensions of $Q_p$, of the $K(1)$-local stable homotopy groups of finite CW-complexes, and we use Dwork’s computation of the norm residue symbol on the maximal abelian extension of $Q_p$ to compute this filtration explicitly on some interesting finite CW-complexes, such as mod $p$ Moore spaces. We then use the nilpotence and localization theorems of Ravenel-Devinatz-Hopkins-Smith to produce a "dictionary" that lets us pass between norm residue symbols computations from explicit class field theory, and families of nilpotent elements in the $K(1)$-local stable homotopy groups of finite ring spectra.
Time allowing, we will discuss what versions of a (so far only conjectural) $p$-adic Langlands correspondence would permit these methods to be extended to higher heights, i.e., $K(n)$-local stable homotopy groups and nonabelian Galois extensions of $Q_p$.
Dan Burghelea
SEP
10
2013
3:00PM
A (computer friendly) alternative to Morse-Novikov theory
We present an alternative to Morse-Novikov theory which works for a considerably larger class of spaces and maps rather than smooth manifolds and Morse maps. One explains what Morse-Novikov theory does for dynamics and topology and indicates how our theory does almost the same for a considerably larger class of situations as well as its additional features.
Taehee Kim
SEP
24
2013
3:00PM
Concordance of knots and Seifert forms
Two knots in the 3-sphere are said to be concordant if they cobound a locally flat, properly embedded annulus in the product of the 3-sphere and the unit interval. The notion of concordance originates from Fox and Milnor, and it is related with other 3- and 4-dimensional topological properties such as homology cobordism and topological surgery theory. In this talk, I will discuss various relationships between concordance and Seifert forms (or the Alexander polynomial) of knots. In particular, I will explain Cha-Orr’s extension of Cochran-Orr-Teichner’s concordance invariants, which are von Neumann rho-invariants, and show its application to this subject.
Kun Wang
OCT
03
2013
1:50PM
On group actions on $\mathrm{CAT}(0)$-spaces and the Farrell-Jones Isomorphism Conjecture.
The Farrell-Jones isomorphism conjecture (FJIC) plays an important role in manifold topology as well as computations in algebraic $K$- and $L$-theory. It implies, for example, the Borel conjecture of topological rigidity of closed aspherical manifolds and the Novikov conjecture of homotopy invariance of higher signatures. By the work of A. Bartels, W. Lueck and C. Wegner, it’s now known that FJIC holds for $\mathrm{CAT}(0)$-groups, i.e. groups admitting proper, cocompact actions on finite dimensional proper $\mathrm{CAT}(0)$-spaces. This includes for example fundamental groups of nonpositively curved closed Riemannian manifolds. It’s a natural question that if a group admits a "nice" but not necessary proper action on a $\mathrm{CAT}(0)$-space and if the point stabilizers satisfy FJIC, whether the original group satisfies FJIC. In this talk, after outlining the general strategy for proving FJIC, I will talk about the progress that I have made concerning the above question.
Ryan Kowalick
OCT
15
2013
3:00PM
Discrete Systolic Inequalities and Applications
We investigate a discrete analogue of Gromov’s systolic estimate and use it to prove facts about triangulations of surfaces. We also discribe a procedure for obtaining Gromov’s result from the discrete version.
Wouter van Limbeek
OCT
22
2013
3:00PM
Riemannian manifolds with local symmetry
In this talk I will discuss the problem of classifying all closed Riemannian manifolds whose universal cover has nondiscrete isometry group. Farb and Weinberger solved this under the assumption that $M$ is aspherical. Roughly, they proved that any such $M$ is a fiber bundle with locally homogeneous fibers. However, if $M$ is not aspherical, many new examples and phenomena appear. I will exhibit some of these, and discuss progress towards a classification. As an application, I will characterize simply-connected manifolds with both a compact and a noncompact finite volume quotient.
Xiaolei Wu
NOV
12
2013
3:00PM
Farrell-Jones conjecture for Baumslag-Solitar groups
The Baumslag-Solitar groups are a particular class of two-generator one-relation groups which have played a surprisingly useful role in combinatorial and geometric group theory. They have provided examples which mark boundaries between different classes of groups and they often provide a test-cases for theories and techniques. In this talk, I will illustrate the proof of the Farrell-Jones conjecture for them. This is a joint work with my advisor Tom Farrell.
Somnath Basu
NOV
19
2013
1:50PM
The closed geodesic problem for four manifolds
We will explain why a generic metric on a smooth four manifold (with second Betti number at least three) has the exponential growth property, i.e., the number of geometrically distinct periodic geodesics of length at most l grow exponentially as a function of l. Time permitting, we shall explain related topological consequences.
Dan Burghelea
NOV
26
2013
3:00PM
Alexander Polynomial revisited
I will provide alternative definitions and methods of calculations for the Alexander Polynomial of a knot and ultimately a generalization of this invariant to all odd dimensional manifolds with large fundamental group. The generalization is a "rational function" on the variety of complex rank K representations of the fundamental group.
Crichton Ogle
DEC
03
2013
3:00PM
Fundamental Theorems for the $K$-theory of $S$-algebras
We show how recent results of Dundas-Goodwillie-McCarthy can be used to give efficient proofs of i) a Fundamental Theorem for the K-theory of connective S-algebras, ii) an integral localization theorem for the relative K-theory of a 1-connected map of connective S-algebras, iii) a generalized localization theorem for the p-complete relative K-theory of a 1-connected map of connective S-algebras. Following Weibel, we define homotopy K-theory for general S-algebras, and prove that the corresponding NK-groups of the sphere spectrum are non-trivial.
Much of this work arose in an attempt to apply recent results and methods from topological cyclic homology to update Waldhausen’s original program for studying the effect of Ravenel’s chromatic tower on the algebraic K-theory of the sphere spectrum. We will give a brief summary of this program, along with recent results of Blumberg-Mandell and how they fit into some deep conjectures of Rognes. As time permits, we will add some conjectures to the list.
Stratos Prassidis
DEC
10
2013
3:00PM
Equivariant Rigidity of Quasi-toric Manifolds
We show that quasi-toric manifolds are topologically equivariant rigid with the natural torus action. The proof of the rigidity is done in three steps. First we show that for the manifold equivariantly homotopy equivalent to the quasi-toric manifold the action of the torus is locally standard (it resembles the standard action of the torus on the complex space). The second step is that the manifold is equivariantly homeomorphic to the standard model of such actions. The final step is based on the topological rigidity of the quotient space which is a manifold with corners. This is joint work with Vassilis Metaftsis.
Dave Constantine
JAN
14
2014
3:00PM
On volumes of compact anti-de Sitter 3-manifolds
Anti de-Sitter manifolds are Lorentzian manifolds with constant curvature $-1$. In a loose analogy with Teichmuller space, there is a moduli space of AdS 3-manifolds with a given fundamental group. This space is not entirely understood—for instance, we do not know how many connected components it has—but we do know a fair amount. We know much less about how the geometry of the manifolds varies across the moduli space. I’ll present the some preliminary results on how volume varies across the moduli space and state a few questions the results so far raise.
Michael Davis
FEB
11
2014
3:00PM
When are two Coxeter orbifolds diffeomorphic?
One can define what it means for a compact manifold with corners to be a “contractible manifold with contractible faces.” Two combinatorially equivalent, contractible manifolds with contractible faces are diffeomorphic if and only if their $4$-dimensional faces are diffeomorphic. It follows that two simple convex polytopes are combinatorially equivalent if and only if they are diffeomorphic as manifolds with corners. On the other hand, by a result of Akbulut, for each n greater than 3, there are smooth, contractible n-manifolds with contractible faces which are combinatorially equivalent but not diffeomorphic. Applications are given to rigidity questions for reflection groups and smooth torus actions.
Allan Edmonds
FEB
25
2014
3:00PM
Introduction to Haken $n$-manifolds
Haken $n$-manifolds have recently been defined and studied by B. Foozwell and H. Rubinstein in analogy with the classical Haken manifolds of dimension 3, using the the theory of boundary patterns developed by K. Johannson. They can be systematically cut apart along essential codimension-one hypersurfaces until one obtains a system of $n$-cells with a boundary pattern recording some of the information carried by the original manifold and the cutting hypersurfaces. Haken manfolds in all dimensions are aspherical and, in general are amenable to proofs by induction on the length of a hierarchy (and on dimension). As such they provide a a context to explore the classical Euler characteristic conjecture for closed aspherical manifolds, which we are doing in some joint work with M. Davis.
Effie Kalfagianni
MAR
25
2014
3:00PM
Geometric structures and stable coefficients of Jones knot polynomials
We will discuss a way to “re-package" the colored Jones polynomial knot invariants that allows to read some of the geometric properties of knot complements they detect.