About

Erkao Bao is an assistant professor in the differential geometry group at the School of Mathematics, University of Minnesota. His research interests include Symplectic Geometry and Contact Geometry.

Research topics

• Mathematical physics
• Mathematics
• Pure mathematics
• Mathematical analysis
• Mathematical optimization

Selected publications

• Equivariant Morse-Bott cohomology through stabilization

  ArXiv.org · 2026-01-22

  articleOpen access1st authorCorresponding

  For closed manifolds with compact Lie group actions, we study Austin-Braam's Morse-theoretic construction of Borel equivariant cohomology using the technique of stabilization. We show that a $C^1$-small equivariant perturbation produces stable invariant Morse-Bott functions. This allows us to realize the equivariant transversality and orientability assumptions in Austin-Braam's framework by choosing generic invariant Riemannian metrics.

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• Equivariant Morse-Bott cohomology through stabilization

  arXiv (Cornell University) · 2026-01-22

  preprintOpen access1st authorCorresponding

  For closed manifolds with compact Lie group actions, we study Austin-Braam's Morse-theoretic construction of Borel equivariant cohomology using the technique of stabilization. We show that a $C^1$-small equivariant perturbation produces stable invariant Morse-Bott functions. This allows us to realize the equivariant transversality and orientability assumptions in Austin-Braam's framework by choosing generic invariant Riemannian metrics.

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• A UCB Bandit Algorithm for General ML-Based Estimators

  arXiv (Cornell University) · 2026-01-03

  preprintOpen access

  We present ML-UCB, a generalized upper confidence bound algorithm that integrates arbitrary machine learning models into multi-armed bandit frameworks. A fundamental challenge in deploying sophisticated ML models for sequential decision-making is the lack of tractable concentration inequalities required for principled exploration. We overcome this limitation by directly modeling the learning curve behavior of the underlying estimator. Specifically, assuming the Mean Squared Error decreases as a power law in the number of training samples, we derive a generalized concentration inequality and prove that ML-UCB achieves sublinear regret. This framework enables the principled integration of any ML model whose learning curve can be empirically characterized, eliminating the need for model-specific theoretical analysis. We validate our approach through experiments on a collaborative filtering recommendation system using online matrix factorization with synthetic data designed to simulate a simplified two-tower model, demonstrating substantial improvements over LinUCB

  PublisherDOI

• A UCB Bandit Algorithm for General ML-Based Estimators

  ArXiv.org · 2026-01-03

  articleOpen access

  We present ML-UCB, a generalized upper confidence bound algorithm that integrates arbitrary machine learning models into multi-armed bandit frameworks. A fundamental challenge in deploying sophisticated ML models for sequential decision-making is the lack of tractable concentration inequalities required for principled exploration. We overcome this limitation by directly modeling the learning curve behavior of the underlying estimator. Specifically, assuming the Mean Squared Error decreases as a power law in the number of training samples, we derive a generalized concentration inequality and prove that ML-UCB achieves sublinear regret. This framework enables the principled integration of any ML model whose learning curve can be empirically characterized, eliminating the need for model-specific theoretical analysis. We validate our approach through experiments on a collaborative filtering recommendation system using online matrix factorization with synthetic data designed to simulate a simplified two-tower model, demonstrating substantial improvements over LinUCB

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• Equivariant Morse Homology for Reflection Actions via Broken Trajectories

  International Mathematics Research Notices · 2026-04-06

  preprintOpen access1st authorCorresponding

  Abstract We consider a finite group $G$ acting on a manifold $M$. According to [3, 10], a generic equivariant function on $M$ is Morse. For any equivariant Morse function, there does not always exist an equivariant metric $g$ on $M$ such that the pair $(f,g)$ is Morse–Smale. Here, the pair $(f,g)$ is called Morse–Smale if the descending and ascending manifolds intersect transversely. The best possible metrics $g$ are those that make the pair $(f,g)$ stably Morse–Smale. A diffeomorphism $\phi : M \to M$ is a reflection if $\phi ^{2} = \operatorname{id}$ and the fixed point set of $\phi $ forms a codimension-one submanifold (with $M \setminus M^{\operatorname{fix}}$ not necessarily disconnected). In this note, we focus on the special case where the group $G = \{\operatorname{id}, \phi \}$. We show that the condition of being stably Morse–Smale is generic for metrics $g$. Given a stably Morse–Smale pair, we introduce a canonical equivariant Thom–Smale–Witten complex by counting certain broken trajectories. This has applications to the case when we have a manifold with boundary and when the Morse function has critical points on the boundary. We provide an alternative definition of the Thom–Smale–Witten complexes, which are quasi-isomorphic to those defined by [7]. We also explore the case when $G$ is generated by multiple reflections. As an example, we compute the Thom–Smale–Witten complex of an upright higher-genus surface by counting broken trajectories.

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• Morse homology and equivariance

  arXiv (Cornell University) · 2024-09-07

  preprintOpen access1st authorCorresponding

  In this paper, we develop methods for calculating equivariant homology from equivariant Morse functions on a closed manifold with the action of a finite group. We show how to alter $G$-equivariant Morse functions to a stable one, where the descending manifold from a critical point $p$ has the same stabilizer group as $p$, giving a better-behaved cell structure on $M$. For an equivariant, stable Morse function, we show that a generic equivariant metric satisfies the Morse--Smale condition. In the process, we give a proof that a generic equivariant function is Morse, and that equivariant, stable Morse functions form a dense subset in the $C^0$-topology within the space of all equivariant functions. Finally, we give an expository account of equivariant homology and cohomology theories, as well as their interaction with Morse theory. We show that any equivariant Morse function gives a filtration of $M$ that induces a Morse spectral sequence, computing the equivariant homology of $M$ from information about how the stabilizer group of a critical point acts on its tangent space. In the case of a stable Morse function, we show that this can be further reduced to a Thom-Smale-Witten complex.

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• Computable, obstructed Morse homology for clean intersections

  arXiv (Cornell University) · 2024-09-17

  preprintOpen access1st authorCorresponding

  In this paper, we develop a method to compute the Morse homology of a manifold when descending manifolds and ascending manifolds intersect cleanly, but not necessarily transversely. While obstruction bundle gluing defined by Hutchings and Taubes is a computable tool to handle non-transverse intersections, it has only been developed for specific cases. In contrast, most virtual techniques apply to general cases but lack computational efficiency. To address this, we construct minimal semi-global Kuranishi structures for the moduli spaces of Morse trajectories, which generalize obstruction bundle gluing while maintaining its computability feature. Through this construction, we obtain iterated gluing equals simultaneous gluing.

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• Coherent orientations in symplectic field theory revisited

  Mathematische Zeitschrift · 2023-09-28

  article1st authorCorresponding
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• Semi-global Kuranishi charts and the definition of contact homology

  Advances in Mathematics · 2023 · 11 citations

  1st authorCorresponding
  • Mathematics
  • Pure mathematics
  • Mathematical physics
  PublisherDOI

• Coherent orientations in symplectic field theory revisited

  arXiv (Cornell University) · 2022-06-16

  preprintOpen access1st authorCorresponding

  In symplectic field theory (SFT), the moduli spaces of $J$-holomorphic curves can be oriented coherently (compatible with gluing). In this note, we correct the signs involved in the generating function $\mathbf H$ in SFT so that the master equation $\mathbf H \cdot \mathbf H = 0$ holds assuming transversality. The orientation convention that we use is consistent with that of Hutchings-Taubes from [HT09].

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Frequent coauthors

• Garrett Alston

  4 shared

• Ko Honda

  University of California, Los Angeles

  4 shared

• Tyler Lawson

  2 shared

• Lina Liu

  Harbin Engineering University

  1 shared

• Ke Zhu

  1 shared

• Linqi Song

  City University of Hong Kong

  1 shared

Awards & honors

• Outstanding Achievement
• Distinguished Leadership