TBA

• Dan Freed (University of Austin)

Surfaces in 4-manifolds

• Andras Stipsicz

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.

Surfaces in 4-manifolds

• Arunima Ray (Max Planck Institute for Mathematics)

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.

TBA

• Thomas Nikolaus (University of Münster)

TQFT and Surgery

• David Reutter

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.

Stable equivalence relations of 4-manifolds

• Daniel Kasprowski (University of Southampton)

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.

Homology manifolds and their Spivak normal fibration

• Markus Land

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.

Chern-Simons invariants, concordance, and homology cobordism

• Matt Hedden

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.

On non-isotopic Seifert surfaces in the 4-ball

• Peter Feller

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.

How to get rid of pointings for constructiong TFTs

• Claudia Scheimbauer (University of Munich)

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.

The 0-1 part of extended 2d QFTs

• André Henriques

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.

Entanglement of sections

• Michael Freedman

TBA

Machine learning and hard problems in topology

• Sergei Gukov

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.

Strings in (low dimensional?) manifolds

• Nathalie Wahl

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.

Towards a (3+1)-dimensional TQFT from TMF

• Slava Krushkal

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.

The Geometry of Light

• James Conant (Gemological Institute of America)

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.

Division, group rings, and negative curvature

• Grigori Avramidi

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.

TBA

• Luciana Basualdo Bonatto (MPIM)