Research

Research interests, publications, and seminar notes of He Wang.

Research Overview

My research interests are in algebraic topology and group theory, and their relations with algebra, geometry, combinatorics, and data science. I am also interested in applications of algebraic and topological ideas to machine learning, deep learning, and topological data analysis.

Research Areas

Algebraic Topology

Rational homotopy theory, formality properties, Massey products, spectral sequences, twisted cohomology, and stable homotopy computations.

Group Theory and Lie Algebras

Finitely generated groups, Malcev Lie algebras, holonomy Lie algebras, Chen Lie algebras, lower central series ranks, and filtered-formality.

Cohomology Jumping Loci

Resonance varieties, characteristic varieties, Chen ranks, Alexander-type invariants, and their relation to the structure of groups.

Braid-Type Groups

Pure braid groups, virtual braid groups, welded braid groups, McCool groups, upper McCool groups, and their algebraic and topological invariants.

Topological Data Analysis

Persistent homology, simplicial complexes, clique complexes, shape of data, and algebraic-topological methods in data science.

Mathematics of AI

Linear algebra, geometry, probability, optimization, and topology as mathematical foundations for machine learning and artificial intelligence.

Publications

10. Taylor expansions of groups and filtered-formality

He Wang and Alexander I. Suciu
European Journal of Mathematics, 6, 1073–1096, 2020.
DOI · arXiv:1905.10355

Abstract

Let \(G\) be a finitely generated group, and let \(\Bbbk G\) be its group algebra over a field of characteristic \(0\). A Taylor expansion is a certain type of map from \(G\) to the degree completion of the associated graded algebra of \(\Bbbk G\) which generalizes the Magnus expansion of a free group. The group \(G\) is said to be filtered-formal if its Malcev Lie algebra is isomorphic to the degree completion of its associated graded Lie algebra. We show that \(G\) is filtered-formal if and only if it admits a Taylor expansion, and derive some consequences.

9. Chen ranks and resonance varieties of the upper McCool groups

He Wang and Alexander I. Suciu
Advances in Applied Mathematics, 110, 197–234, 2019.
MR3983125 · arXiv:1804.06006

Abstract

The group of basis-conjugating automorphisms of the free group of rank \(n\), also known as the McCool group or the welded braid group \(P\Sigma_n\), contains a much-studied subgroup called the upper McCool group \(P\Sigma_n^+\). Starting from the cohomology ring of \(P\Sigma_n^+\), we use a Gröbner basis computation to find a simple presentation for the infinitesimal Alexander invariant of this group, from which we determine the resonance varieties and the Chen ranks of the upper McCool groups. These computations show that the Chen ranks conjecture does not hold for \(P\Sigma_n^+\) for any \(n \ge 4\). We also determine the scheme structure of the resonance varieties \(\mathcal{R}_1(P\Sigma_n^+)\) and show that these schemes are not reduced for \(n \ge 4\).

8. Formality properties of finitely generated groups and Lie algebras

He Wang and Alexander I. Suciu
Forum Mathematicum, 31 no. 4, 867–905, 2019.
DOI · MR3975666 · arXiv:1504.08294

Abstract

We explore the graded and filtered formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic \(0\) attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how these notions behave with respect to split injections, coproducts, direct products, and field extensions, and how they are inherited by solvable and nilpotent quotients. A key tool is the \(1\)-minimal model of the group and its relation to these Lie algebras. Another approach to formality is provided by Taylor expansions from the group to the completion of the associated graded algebra of the group ring.

7. Cup products, lower central series, and holonomy Lie algebras

He Wang and Alexander I. Suciu
Journal of Pure and Applied Algebra, 223 no. 8, 3359–3385, 2019.
DOI · MR3926216 · arXiv:1701.07768

Abstract

We generalize basic results relating the associated graded Lie algebra and the holonomy Lie algebra from finitely presented commutator-relators groups to arbitrary finitely presented groups. In the process, we give an explicit formula for the cup product in the cohomology of a finite \(2\)-complex, and an algorithm for computing the corresponding holonomy Lie algebra using a Magnus expansion method. Examples include link groups, one-relator groups, and fundamental groups of Seifert fibered manifolds.

6. Pure virtual braids, resonance, and formality

He Wang and Alexander I. Suciu
Mathematische Zeitschrift, 286 no. 3–4, 1495–1524, 2017.
DOI · MR3671586 · arXiv:1602.04273

Abstract

We investigate the resonance varieties, lower central series ranks, and Chen ranks of the pure virtual braid groups and their upper-triangular subgroups. As an application, we give a complete answer to the \(1\)-formality question for this class of groups. We also explore connections between Alexander-type invariants of finitely generated groups and several graded Lie algebras associated to them.

5. The pure braid groups and their relatives

He Wang and Alexander I. Suciu
In Perspectives in Lie Theory, 403–426, Springer INdAM Series vol. 19, 2017.
DOI · MR3751136 · arXiv:1602.05291

Abstract

This survey investigates the resonance varieties, lower central series ranks, Chen ranks, and formality properties of several families of braid-like groups: the pure braid groups \(P_n\), the welded pure braid groups \(wP_n\), the virtual pure braid groups \(vP_n\), and their upper variants \(wP_n^+\) and \(vP_n^+\). We also discuss natural homomorphisms between these groups and methods for distinguishing among pure braid groups and their relatives.

4. On a spectral sequence for twisted cohomologies

He Wang, Weiping Li, and Xiugui Liu
Chinese Annals of Mathematics, Series B, 35 no. 4, 633–658, 2014.
DOI · MR3227750 · arXiv:0911.1417

Abstract

Let \((\Omega^{\ast}(M), d)\) be the de Rham cochain complex for a smooth compact closed manifold \(M\). For an odd-degree closed form \(H\), there is a twisted de Rham cochain complex \((\Omega^{\ast}(M), d + H_\wedge)\) and its associated twisted de Rham cohomology \(H^*(M,H)\). We construct a spectral sequence derived from the filtration of \(\Omega^{\ast}(M)\) that converges to the twisted de Rham cohomology, and describe the differentials in terms of cup products and Massey products.

3. Some products involving the fourth Greek letter family element in the Adams spectral sequence

He Wang and Xiugui Liu
Turkish Journal of Mathematics, 35 no. 2, 311–321, 2011.
DOI · MR2839725

Abstract

This paper studies products in the cohomology of the mod \(p\) Steenrod algebra using the classical Adams spectral sequence. For an odd prime \(p\), we analyze the nontriviality of certain products involving elements in the fourth Greek letter family and provide explicit computations in the \(E_2\)-term of the Adams spectral sequence.

2. On the cohomology of the mod \(p\) Steenrod algebra

He Wang and Xiugui Liu
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 85 no. 9, 143–148, 2009.
DOI · MR2573964

Abstract

For an odd prime \(p > 7\), this paper proves nontriviality and triviality results for certain products in the cohomology of the mod \(p\) Steenrod algebra. The proof uses explicit combinatorial analysis of the modified May spectral sequence.

1. Ph.D. Thesis

Resonance varieties, Chen ranks and formality properties of finitely generated groups
Northeastern University, 2016.
MathSciNet: MR3517822

Description

The thesis studies resonance varieties, Chen ranks, and formality properties of finitely generated groups, with emphasis on the interaction between group theory, algebraic topology, and Lie-theoretic invariants.

Seminar and Workshop Notes

Notes on spectral sequences Seminar notes

Higher Massey products and their applications Seminar notes

Some functors from topological spaces to Lie algebras and varieties Seminar notes

Rational homotopy theory Seminar notes