Seminars and Colloquia by Series

Monday, August 26, 2019 - 14:00 , Location: Skiles 006 , Jasmine Powell , University of Michigan , jtpowell@umich.edu , Organizer: Justin Lanier
Friday, May 31, 2019 - 14:00 , Location: Skiles 006 , David Ayala , Montana State University , Organizer: Kirsten Wickelgren
Roughly, factorization homology pairs an n-category and an n-manifold to produce a vector space.  Factorization homology is to state-sum TQFTs as singular homology is to simplicial homology: the former is manifestly well-defined (ie, independent of auxiliary choices), continuous (ie, beholds a continuous action of diffeomorphisms), and functorial; the latter is easier to compute.  

 

Examples of n-categories to input into this pairing arise, through deformation theory, from perturbative sigma-models.  For such n-categories, this state sum expression agrees with the observables of the sigma-model — this is a form of Poincare’ duality, which yields some surprising dualities among TQFTs.  A host of familiar TQFTs are instances of factorization homology; many others are speculatively so.  

 

The first part of this talk will tour through some essential definitions in what’s described above.  The second part of the talk will focus on familiar manifold invariants, such as the Jones polynomial, as instances of factorization homology, highlighting the Poincare’/Koszul duality result.  The last part of the talk will speculate on more such instances.  

Wednesday, May 29, 2019 - 14:00 , Location: Skiles 006 , Paolo Aceto , University of Oxford , paoloaceto@gmail.com , Organizer: JungHwan Park

We prove that every rational homology cobordism class in the subgroup generated by lens spaces contains a unique connected sum of lens spaces whose first homology embeds in any other element in the same class. As a consequence we show that several natural maps to the rational homology cobordism group have infinite rank cokernels, and obtain a divisibility condition between the determinants of certain 2-bridge knots and other knots in the same concordance class. This is joint work with Daniele Celoria and JungHwan Park.

Wednesday, May 15, 2019 - 14:00 , Location: Skile 005 , Roger Casals , UC Davis , Organizer:

In this talk, I will discuss progress in our understanding of Legendrian surfaces. First, I will present a new construction of Legendrian surfaces and a direct description for their moduli space of microlocal sheaves. This Legendrian invariant will connect to classical incidence problems in algebraic geometry and the study of flag varieties, which we will study in detail. There will be several examples during the talk and, in the end, I will indicate the relation of this theory to the study of framed local systems on a surface. This talk is based on my work with E. Zaslow.

Monday, May 13, 2019 - 14:00 , Location: Skiles 006 , Inna Zakharevich , Cornell , Organizer: Kirsten Wickelgren

One of the classical problems in scissors congruence is
this: given two polytopes in $n$-dimensional Euclidean space, when is
it possible to decompose them into finitely many pieces which are
pairwise congruent via translations?  A complete set of invariants is
provided by the Hadwiger invariants, which measure "how much area is
pointing in each direction."  Proving that these give a complete set
of invariants is relatively straightforward, but determining the
relations between them is much more difficult.  This was done by
Dupont, in a 1982 paper. Unfortunately, this result is difficult to
describe and work with: it uses group homological techniques which
produce a highly opaque formula involving twisted coefficients and
relations in terms of uncountable sums.  In this talk we will discuss
a new perspective on Dupont's proof which, together with more
topological simplicial techniques, simplifies and clarifies the
classical results.  This talk is partially intended to be an
advertisement for simplicial techniques, and will be suitable for
graduate students and others unfamiliar with the approach.

Monday, April 22, 2019 - 15:30 , Location: Skiles 006 , Eli Grigsby , Boston College , Organizer: Caitlin Leverson

One can regard a (trained) feedforward neural network as a particular type of function \mathbb{R}^d \rightarrow (0,1), where \mathbb{R}^d is a (typically high-dimensional) Euclidean space parameterizing some data set, and the value N(x) \in (0,1) of the function on a data point x is the probability that the answer to a particular yes/no question is "yes." It is a classical result in the subject that a sufficiently complex neural network can approximate any function on a bounded set. Last year, J. Johnson proved that universality results of this kind depend on the architecture of the neural network (the number and dimensions of its hidden layers). His argument was novel in that it provided an explicit topological obstruction to representability of a function by a neural network, subject to certain simple constraints on its architecture. I will tell you just enough about neural networks to understand how Johnson's result follows from some very simple ideas in piecewise linear geometry. Time permitting, I will also describe some joint work in progress with K. Lindsey aimed at developing a general theory of how the architecture of a neural network constrains its topological expressiveness.

Monday, April 22, 2019 - 14:00 , Location: Skiles 006 , Adam Levine , Duke University , Organizer: Caitlin Leverson

Given an m-dimensional manifold M that is homotopy equivalent to an n-dimensional manifold N (where n<m), a spine of M is a piecewise-linear embedding of N into M (not necessarily locally flat) realizing the homotopy equivalence. When m-n=2 and m>4, Cappell and Shaneson showed that if M is simply-connected or if m is odd, then it contains a spine. In contrast, I will show that there exist smooth, compact, simply-connected 4-manifolds which are homotopy equivalent to the 2-sphere but do not contain a spine (joint work with Tye Lidman). I will also discuss some related results about PL concordance of knots in homology spheres (joint with Lidman and Jen Hom).

Monday, April 15, 2019 - 14:00 , Location: Skiles 006 , Patrick Orson , Boston College , patrick.orson@bc.edu , Organizer: JungHwan Park

The question of which high-dimensional knots are slice was entirely solved by Kervaire and Levine. Compared to this, the question of which knots are doubly slice in high-dimensions is still a largely open problem. Ruberman proved that in every dimension, some version of the Casson-Gordon invariants can be applied to obtain algebraically doubly slice knots that are not doubly slice. I will show how to use L^2 signatures to recover the result of Ruberman for (4k-3)-dimensional knots, and discuss how the derived series of the knot group might be used to organise the high-dimensional doubly slice problem.

Monday, April 8, 2019 - 14:00 , Location: Skiles 006 , Tye Lidman , NCSU , Organizer: Jennifer Hom

We will use Heegaard Floer homology to analyze maps between a certain family of three-manifolds akin to the Gromov norm/hyperbolic volume.  Along the way, we will study the Heegaard Floer homology of splices.  This is joint work with Cagri Karakurt and Eamonn Tweedy.

Wednesday, April 3, 2019 - 14:00 , Location: Skiles 006 , Peter Feller , ETH Zurich , peter.feller@math.ethz.ch , Organizer: JungHwan Park

Following an idea of Hugelmeyer, we give a knot theory reproof of a theorem of Schnirelman: Every smooth Jordan curve in the Euclidian plane has an inscribed square. We will comment on possible generalizations to more general Jordan curves.

Our main knot theory result is that the torus knot T(2n,1) in S^1xS^2 does not arise as the boundary of a locally-flat Moebius band in S^1xB^3 for square-free integers n>1. For context, we note that for n>2 and the smooth setting, this result follows from a result of Batson about the non-orientable 4-genus of certain torus knots. However, we show that Batson's result does not hold in the locally flat category: the smooth and topological non-orientable 4-genus differ for the T(9,10) torus knot in S^3.

Based on joint work with Marco Golla.

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