Interactions of Low-dimensional Topology and Quantum Field Theory

21 May 2023, 10:00 → 26 May 2023, 12:20 Europe/Zurich

SRS

Christopher Schommer-Pries (University of Notre Dame), Danica Kosanović, Peter Teichner (Max Planck Institute for Mathematics, Bonn, Germany), Robert Schneiderman (Lehman College City University of New York), Stephan Stolz (University of Notre Dame)

This conference will bring together students and close collaborators of Peter Teichner on the occasion of his sixtieth birthday. With Teichner’s contributions to both Low-dimensional Topology and Quantum Field Theory, this will be an opportunity to share ways of thinking between experts of these two areas. The accent will be on recent results which will inspire future collaborations and encourage interactions between these fields.

PeteFest Schedule (v20.05.2023).pdf

Monday 22 May

09:30 → 10:15 TBA

Speaker: Dan Freed (University of Austin)

10:45 → 11:30 Surfaces in 4-manifolds

We will review the genus function (associating to a second homology class the minimal genus of an embedded surface representing it), and show how to extend it. The function turns out to be useful in obstructing smooth sliceness, while the extension might become handy in contemplating
about smooth structures on the four-sphere.

Speaker: Andras Stipsicz

11:35 → 12:20 Surfaces in 4-manifolds

I will describe a general procedure to homotope an immersed positive genus surface in a simply connected 4-manifold to a locally flat embedding. This is a special case of a surface embedding theorem, joint with Daniel Kasprowski, Mark Powell, and Peter Teichner.

Speaker: Arunima Ray (Max Planck Institute for Mathematics)

21:00 → 21:45 TBA

Speaker: Thomas Nikolaus (University of Münster)

Tuesday 23 May

09:30 → 10:15 TQFT and Surgery

How much manifold topology can a given topological quantum field theory see? In this talk, I will answer this question for "semisimple" TQFTs in even dimensions, a certain class of field theories which includes all "once-extended" even-dimensional field theories, i.e. those which also assign linear categories to corners of codimension 2.

These results suggest to think of TQFTs as appropriately "dual" to manifolds, and lead to classification schemes for TQFTs "dual" to surgery theoretic classifications of manifolds. If time permits, I will explain such a classification of linear once-extended 4-dimensional TQFT in terms of certain group theoretical data and bordism invariants, and comment on higher-dimensional variants.

The first part of this talk is based on joint work with Christopher Schommer-Pries, the second part on ongoing joint work with Christopher Schommer-Pries and Noah Snyder, and with Theo Johnson-Freyd.

Speaker: David Reutter

10:45 → 11:30 Stable equivalence relations of 4-manifolds

I will introduce and study relations of 4-manifolds up to connected sum with copies of $S^2\times S^2$ and their relations. This includes stable diffeomorphism and homotopy equivalence. The talk is based on joint work with Johnny Nicholson and Simona Veselá.

Speaker: Daniel Kasprowski (University of Southampton)

11:35 → 12:20 Homology manifolds and their Spivak normal fibration

I propose to present a construction of a Poincare duality space with the two properties: 1) its Spivak normal fibration does not admit a Top-reduction (equivalently, there is no degree one normal map from a topological closed manifold) and 2) its (periodic) total surgery obstruction vanishes. This contradicts the validity of two theorems in the literature, the one stating that PD complexes with trivial periodic tso are homotopy equivalent to homology manifolds, and the other saying that the Spivak normal fibration of a homology manifold admits a Top-reduction. Joint with Hebestreit, Winges, and Weiss.

Speaker: Markus Land

21:00 → 21:45 Chern-Simons invariants, concordance, and homology cobordism

Chern-Simons invariants of homology spheres, analyzed in conjunction with moduli spaces of solutions to the ASD Yang-Mills equations on 4-manifolds, provide a powerful tool for studying the homology cobordism groups of 3-manifolds and the closely related smooth concordance group of knots. I'll give an overview of this technique and discuss some of its applications.

Speaker: Matt Hedden

Wednesday 24 May

09:30 → 10:15 On non-isotopic Seifert surfaces in the 4-ball

Pete (in an informal seminar) and before him Livingston wondered, whether there exist non-isotopic oriented surfaces in the 3-sphere with boundary a fixed knot that remain non-isotopic when pushed into the 4-ball. It turns out that such examples exist, as recently observed by Hayden-Kim-Miller-Park-Sundberg, and, surprisingly, they can be distinguished by a classical invariant: the intersection form on the second homology of the double branched cover.

We report on work in progress with M. Akka, A. Miller, and A. Wieser, where we use an algebraic approach to finding pairs of such examples. A sample result is that, for all positive integers D with D=3 (mod 4) and (D+1)/4 not a prime nor the square of a prime, there exists a knot with determinant D that has genus one Seifert surfaces that remain non-isotopic when pushed into the 4-ball. The relevant algebra input is that the class group of Q(\sqrt(-D)) has non-trivial elements of order different than two. In fact, this class group corresponds to Gauss's group of equivalence classes of integral binary quadratic forms with discriminant -D, and our main result essentially characterizes (in terms of Gauss's group) when two such quadratic forms can arise as the intersection forms of the double branched cover of pushed-in Seifert surfaces of the same knot.

Speaker: Peter Feller

10:45 → 11:30 How to get rid of pointings for constructiong TFTs

Vector spaces having “duals” are automatically finite dimensional, and this is the case for those appearing as values of TFTs. However, if we assume that the vector spaces are pointed, they are automatically one dimensional (lines). When constructing extended n-dimensional TFTs, a natural family of targets (replaceing Vect) naturally has the feature that pointings are built in. This is due to “factoriazation algebras” having this property. I will explain an approach to fixing this, allowing us to prove a conjecture by Lurie on higher dualizability, which in turn gives fully extended TFTs. This joint work with Eilind Karlsson.

Speaker: Claudia Scheimbauer (University of Munich)

11:35 → 12:20 The 0-1 part of extended 2d QFTs

I will present a conjecture according to which the 0-1 part of an extended 2d QFTs is, up to isomorphism, independent of the QFT.
This conjecture is analogous to the well known fact that there exists a unique separable Hilbert space up to isomorphism (a Hilbert space is the 0 part of a 1d QFT), and has striking consequences about the existence of various kinds of symmetries.

Speaker: André Henriques

21:00 → 21:45 Entanglement of sections

TBA

Speaker: Michael Freedman

Thursday 25 May

09:30 → 10:15 Machine learning and hard problems in topology

Considering the special occasion and to diversify the list of topics, in this talk, intended for a broad audience, I decided to turn to something that hopefully can be fun and entertaining: While the "state-of-the-art" machine learning algorithms already make an impact at the level of high school or undergraduate curriculum, in this talk I want to explore whether they can help us with some of the most difficult mathematical challenges, at the cutting edge of the mathematical research. No prior familiarity with machine learning is required; rather, one of the goals of this talk is to provide a gentle introduction to some of the modern tools in this subject, in part explaining its rapidly increasing role in everyday life and, hopefully, in pure mathematics as well.

Speaker: Sergei Gukov

10:45 → 11:30 Strings in (low dimensional?) manifolds

String topology can be seen as a form of 2d field theory on the homology of the free loop spaces of manifolds. I’ll describe this field theory, and exhibit some of its interesting features.

Speaker: Nathalie Wahl

11:35 → 12:20 Towards a (3+1)-dimensional TQFT from TMF

I will discuss work in progress, joint with Sergei Gukov, Lennart Meier, and Du Pei. It concerns a construction of a 4-manifold invariant using the theory of topological modular forms, and TQFT properties of this invariant. This is a mathematical construction related to a particular instance of the Gukov-Pei-Putrov-Vafa program associating an invariant of 4-manifolds to certain 6-dimensional superconformal field theories.

Speaker: Slava Krushkal

14:00 → 14:45 The Geometry of Light

In the first part of the talk, we give some background on the geometry of diamond cuts, and the special optical properties that make them so captivating to look at. In the second part, we discuss how the classical mathematics of the Maxwell-Cremona correspondence can assist in the enumeration of possible diamond cuts.

Speaker: James Conant (Gemological Institute of America)

Friday 26 May

09:30 → 10:15 Division, group rings, and negative curvature

A basic problem in low dimensional topology is to understand the 2-complexes with a given fundamental group G. I will explain how this can be studied using a division algorithm in the group ring of G, and describe some instances in which such an algorithm is available.

Speaker: Grigori Avramidi

10:45 → 11:30 TBA

Speaker: Luciana Basualdo Bonatto (MPIM)