Talk_id  Date  Title 
1018

Thursday 1/12 2:00 PM

Projective coordinates for the analysis of data
 Projective coordinates for the analysis of data
 01/12/2017
 2:00 PM  2:50 PM
 C304 Wells Hall
 Jose Perea, MSU
Barcodes  the persistent homology of data  have been shown to be effective quantifiers of multiscale structure in finite metric spaces. Moreover, the universal coefficient theorem implies that (for a fixed field of coefficients) the barcodes obtained with persistent homology are identical to those obtained with persistent cohomology. Persistent cohomology, on the other hand, is better behaved computationally and allows one to use convenient interpretations such as the Brown representability theorem. We will show in this talk how one can use persistent cohomology to produce maps from data to (real and complex) projective space, and conversely, how to use these projective coordinates to interpret persistent cohomology computations.

231

Thursday 1/19 2:00 PM

TuraevViro invariants of links and the colored Jones polynomial
 TuraevViro invariants of links and the colored Jones polynomial
 01/19/2017
 2:00 PM  2:50 PM
 C304 Wells Hall
 Renaud Detcherry, MSU
In a recent work by Tian Yang and Qingtao Chen, it has been observed that one can recover the hyperbolic volume from the asymptotic of TuraevViro invariants of 3manifolds at a specific root of unity. This is reminiscent of the volume conjecture for the colored Jones polynomial.
In the case of link complements, we derive a formula to express TuraevViro invariants as a sum of values of colored Jones polynomial, and get a proof of Yang and Chen's conjecture for a few link complements. We also discuss the link between this conjecture and the volume conjecture. This is joint work with Effie Kalfagianni and Tian Yang.

1019

Thursday 1/26 2:00 PM

Orders from $\widetilde{PSL_2(\mathbb{R})}$ Representations and Nonexamples
 Orders from $\widetilde{PSL_2(\mathbb{R})}$ Representations and Nonexamples
 01/26/2017
 2:00 PM  2:50 PM
 C304 Wells Hall
 Xinghua Gao, UIUC
It is still unknown for the case of rational homology 3sphere whether the leftorderability of its fundamental group and it not be a HeegaardFloer Lspace are equivalent. Let $M$ be an integer homology 3sphere. One way to study leftorderability of $\pi_1(M)$ is to construct a nontrivial representation from $\pi_1(M)$ to $\widetilde{PSL_2(\mathbb{R})}$. However this method does not always work. In this talk, I will give examples of non Lspace irreducible integer homology 3spheres whose fundamental groups do not have nontrivial $\widetilde{PSL_2(\mathbb{R})}$ representations.

230

Thursday 2/2 2:00 PM

Quantum cluster algebras from geometry
 Quantum cluster algebras from geometry
 02/02/2017
 2:00 PM  2:50 PM
 C304 Wells Hall
 Leonid Chekhov, Steklov Mathematical Institute
We identify the Teichmuller space $T_{g,s,n}$ of (decorated) Riemann
surfaces $\Sigma_{g,s,n}$ of genus $g$, with $s>0$ holes and $n>0$
bordered cusps located on boundaries of holes uniformized by Poincare with
the character variety of $SL(2,R)$monodromy problem. The effective
combinatorial description uses the fat graph technique; observables are
geodesic functions of closed curves and $\lambda$lengths of paths
starting and terminating at bordered cusps decorated by horocycles. Such
geometry stems from special 'chewing gum' moves corresponding to colliding
holes (or sides of the same hole) in a Riemann surface with holes. We
derive Poisson and quantum structures on sets of observables relating them
to quantum cluster algebras of Berenstein and Zelevinsky. A seed of the
corresponding quantum cluster algebra corresponds to the partition of
$\Sigma_{g,s,n}$ into ideal triangles, $\lambda$lengths of their sides
are cluster variables constituting a seed of the algebra; their number
$6g6+3s+2n$ (and, correspondingly, the seed dimension) coincides with the
dimension of $SL(2,R)$character variety given by
$[SL(2,R)]^{2g+s+n2}/\prod_{i=1}^n B_i$,
where $B_i$ are Borel subgroups associated with bordered cusps. I also discuss the
very recent results enabling constructing monodromy matrices of SL(2)connections out of
the corresponding cluster variables.
The talk is based on the joint papers with with M.Mazzocco and V.Roubtsov

228

Thursday 2/9 2:00 PM

Jones slopes and Murasugi sums of links
 Jones slopes and Murasugi sums of links
 02/09/2017
 2:00 PM  2:50 PM
 C304 Wells Hall
 Christine Lee, University of Texas, Austin
A Jones surface for a knot in the threesphere is an essential surface whose boundary slopes, Euler characteristic, and number of sheets correspond to quantities defined from the asymptotics of the degrees of colored Jones polynomial. The Strong Slope Conjecture by Garoufalidis and KalfagianniTran predicts that there are Jones surfaces for every knot.
A link diagram D is said to be a Murasugi sum of two links D' and D' if a state graph of D has a cut vertex, which separates the graph into two state graphs of D' and D', respectively. We may obtain a state surface in the complement of the link K represented by D by gluing the state surface for D and the state surface for D' along the disk filling the circle represented by the cut vertex in the state graph. The resulting surface is called the Murasugi sum of the two state surfaces.
We consider nearadequate links which are Murasugi sums of certain nonadequate link diagrams with an adequate link diagram along their allA state graphs with an additional graphical constraint. For a nearadequate knot, the Murasugi sum of the corresponding state surface is a Jones surface by the work of Ozawa. We discuss how this proves the Strong Slope Conjecture for this class of knots and raises interesting questions about constraints on the possible Murasugi sumdecompositions of a link diagram.

232

Thursday 2/23 2:00 PM

Abundant quasifuchsian surfaces in cusped hyperbolic 3manifolds
 Abundant quasifuchsian surfaces in cusped hyperbolic 3manifolds
 02/23/2017
 2:00 PM  2:50 PM
 C304 Wells Hall
 David Futer, Temple University
I will discuss a proof that every finite volume hyperbolic 3manifold M contains an abundant collection of immersed, $\pi_1$injective surfaces. These surfaces are abundant in the sense that their lifts to the universal cover separate any pair of disjoint geodesic planes. The proof relies in a major way on the corresponding theorem of Kahn and Markovic for closed 3manifolds. As a corollary, we recover Wise's theorem that the fundamental group of M is acts properly and cocompactly on a cube complex. This is joint work with Daryl Cooper.

3394

Thursday 3/2 2:00 PM

The genus of a special cube complex and its applications
 The genus of a special cube complex and its applications
 03/02/2017
 2:00 PM  2:50 PM
 C304 Wells Hall
 Corey Bregman, Rice University
Recently, the geometry of nonpositively curved (NPC) cube complexes has featured prominently in lowdimensional topology. We introduce an invariant of NPC special cube complexes called the genus, which generalizes the classical notion of genus for a closed orientable surface. We then show that having genus one characterizes special cube complexes with abelian fundamental group and discuss some applications.

229

Thursday 3/23 2:00 PM

Phillips Lecture Series
 Phillips Lecture Series
 03/23/2017
 2:00 PM  2:50 PM
 C304 Wells Hall
 Peter Ozsvath, Princeton University
No abstract available.

3659

Thursday 4/6 2:00 PM

Concordance in homology spheres
 Concordance in homology spheres
 04/06/2017
 2:00 PM  2:50 PM
 C304 Wells Hall
 Tye Lidman, North Carolina State University
Although not every knot in the threesphere can bound a smooth embedded disk in the threesphere, it must bound a PL disk in the fourball. This is not true for knots in the boundaries of arbitrary smooth contractible manifolds. We give new examples of knots in homology spheres that cannot bound PL disks in any bounding homology ball and thus not concordant to knots in the threesphere. This is joint work with Jen Hom and Adam Levine.

3660

Thursday 4/13 2:00 PM

A Khovanov stable homotopy type
 A Khovanov stable homotopy type
 04/13/2017
 2:00 PM  2:50 PM
 C304 Wells Hall
 Robert Lipshitz, University of Oregon
Khovanov homology is a combinatoriallydefined knot invariant which refines the Jones polynomial. After recalling the definition of Khovanov homology we will sketch a construction of a stable homotopy refinement of Khovanov homology. We will conclude with some modest applications and some work in progress. This is joint work with Tyler Lawson and Sucharit Sarkar. Another construction of the Khovanov stable homotopy type was given by HuKrizKriz.

4074

Thursday 4/20 2:00 PM

Constructing SardSmale Fundamental Classes
 Constructing SardSmale Fundamental Classes
 04/20/2017
 2:00 PM  3:00 PM
 C304 Wells Hall
 Thomas H. Parker, MSU
The moduli spaces in gauge theory usually arise as generic fibers of a universal moduli space, and invariants are constructed using cobordisms between generic fibers.
I will describe a topological setting that, in important cases, produces a fundamental class on all fibers, and gives an alternative perspective on the resulting invariants.
This is joint work with E. Ionel.

4079

Tuesday 4/25 3:00 PM

Bipolar filtration of topologically slice knots
 Bipolar filtration of topologically slice knots
 04/25/2017
 3:00 PM  4:00 PM
 C304 Wells Hall
 Min Hoon Kim, KIAS
We show that the bipolar filtration of the smooth concordance group of topologically slice knots introduced by Cochran, Harvey and Horn has nontrivial graded quotients at every stage. To detect a nontrivial element in the quotient, the proof uses CheegerGromov $L^2$ $\rho$invariants and infinitely many Heegaard Floer correction term invariants simultaneously.

2246

Thursday 4/27 2:00 PM

Constructing equivariant cohomology theories
 Constructing equivariant cohomology theories
 04/27/2017
 2:00 PM  2:50 PM
 C304 Wells Hall
 Anna Marie Bohmann, Vanderbilt University
Equivariant cohomology theories are cohomology theories incorporate a group action on spaces. These types of cohomology theories are increasingly important in algebraic topology but can be difficult to understand or construct. In recent work, Angelica Osorno and I have developed a construction for building them out of purely algebraic data by controlling pieces with different isotropy types under the group action. Our method is philosophically similar to classical work of Segal on building nonequivariant cohomology theories.
