Geometry, Topology, and Physics Seminar, Winter 2020

Organizers: Dave Morrison and Zhenghan Wang.
Meets 4:00 - 5:30 p.m. on selected Fridays in South Hall 6635.

January 10

### Embeddability of simplicial complexes is undecidable

We consider the following decision problem EMBED(k,d) in computational topology (where k \leq d are fixed positive integers): Given a finite simplicial complex K of dimension k, does there exist a (piecewise-linear) embedding of K into R^d?

The special case EMBED(1,2) is graph planarity, which is decidable in linear time, by the well-known algorithm of Hopcroft and Tarjan.

In higher dimensions, EMBED(2,3) and EMBED(3,3) are known to be decidable (as well as NP-hard), and recent results of Čadek et al., in combination with a classical theorem of Haefliger and Weber, imply that EMBED(k,d) can be solved in polynomial time for any fixed pair (k,d) of dimensions in the so-called \emph{metastable range} $d\geq (3(k+1))/2$.

Here, by contrast, we prove that EMBED(k,d) is algorithmically undecidable for almost all pairs of dimensions outside the metastable range, namely for $8\leq d<\floor{(3(k+1))/2}$. This almost completely resolves the decidability vs.\ undecidability of EMBED(k,d) in higher dimensions and establishes a sharp dichotomy between polynomial-time solvability and undecidability.

Our proof builds on work by Čadek et al., who showed how to encode an arbitrary system of Diophantine equations into a homotopy-theoretic extension problem. We turn their construction into an embeddability problem, using techniques from piecewise-linear (PL) topology due to Zeeman, Irwin, and others.

January 24

### A deformation invariant of 2D SQFTs

The elliptic genus is a powerful deformation invariant of 2D SQFTs: if it is nonzero, then it protects the SQFT from admitting a deformation to one with spontaneous supersymmetry breaking. I will describe a "secondary" invariant, defined in terms of mock modularity, that goes beyond the elliptic genus, protecting SQFTs with vanishing elliptic genus. The existence of this invariant supports the hypothesis that the space of minimally supersymmetric 2D SQFTs provides a geometric model for universal elliptic cohomology. Based on joint works with D. Gaiotto and E. Witten.

February 24

### BCOV axioms of Gromov-Witten theory of Calabi-Yau 3-fold

One of biggest and most difficult problems in the subject of Gromov-Witten theory is to compute higher genus Gromov-Witten invariants of compact Calabi-Yau 3-fold such as the quintic 3-folds. There have been a collection of remarkable conjectures from physics (BCOV B-model) regarding the universal structure or axioms of higher genus Gromov-Witten theory of Calabi-Yau 3-folds. In the talk, I will first explain 4 BCOV axioms explicitly for the quintic 3-folds. Then, I will outline a solution for 3+1/2 of them. This talk is based on the joint works with Q. Chen, F. Janda and S. Guo.

March 6

### 3-Dimensional Topological QFTs from Non-Semisimple Modular Categories

The famous Reshetikhin-Turaev construction of 3d TQFTs has as an input data a modular tensor category that is assumed to be semi-simple. In middle of 90's Lyubashenko has proposed a reasonable non-semisimple version of modular categories and it was later shown that they produce mapping class group representations with new features not present in the RT construction. Examples of such categories are given by small quantum groups. However, a proper TQFT construction for Lyubashenko's theory was missing. In this talk, I will present our recent construction that provides such TQFTs with input data given by any not necessarily semi-simple modular category over a field.