### Factorization homology: sigma-models as state-sum TQFTs.

- Series
- Geometry Topology Seminar
- Time
- Friday, May 31, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- David Ayala – Montana State University

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- Series
- Geometry Topology Seminar
- Time
- Friday, May 31, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- David Ayala – Montana State University

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.

- Series
- Geometry Topology Seminar
- Time
- Wednesday, May 29, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Paolo Aceto – University of Oxford – paoloaceto@gmail.com

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.

- Series
- Geometry Topology Seminar
- Time
- Wednesday, May 15, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skile 005
- Speaker
- Roger Casals – UC Davis

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.

- Series
- Geometry Topology Seminar
- Time
- Monday, May 13, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Inna Zakharevich – Cornell

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.

- Series
- Geometry Topology Seminar
- Time
- Monday, April 22, 2019 - 15:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Eli Grigsby – Boston College

One can regard a (trained) feedforward neural network as a particular type of function , where is a (typically high-dimensional) Euclidean space parameterizing some data set, and the value of the function on a data point 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.

- Series
- Geometry Topology Seminar
- Time
- Monday, April 22, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Adam Levine – Duke University

Given an m-dimensional manifold M that is homotopy equivalent to an n-dimensional manifold N (where n4, 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).

- Series
- Geometry Topology Seminar
- Time
- Monday, April 15, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Patrick Orson – Boston College – patrick.orson@bc.edu

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.

- Series
- Geometry Topology Seminar
- Time
- Monday, April 8, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Tye Lidman – NCSU

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.

- Series
- Geometry Topology Seminar
- Time
- Wednesday, April 3, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Peter Feller – ETH Zurich – peter.feller@math.ethz.ch

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.

- Series
- Geometry Topology Seminar
- Time
- Monday, April 1, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Ahmad Issa – University of Texas, Austin

Which 3-manifolds smoothly embed in the 4-sphere? This seemingly simple question turns out to be rather subtle. Using Donaldson's theorem, we derive strong restrictions to embedding a Seifert fibered space over an orientable base surface, which in particular gives a complete classification when e > k/2, where k is the number of exceptional fibers and e is the normalized central weight. Our results point towards a couple of interesting conjectures which I'll discuss. This is joint work with Duncan McCoy.

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