Welcome to My Website

I am Hsuan-Yi Liao (廖軒毅), a faculty member in the Department of Mathematics at National Tsing Hua University.

Email:
hyliao@math.nthu.edu.tw

Office:
Room 607, GEN III Building¹

Weekly Seminar: Student Geometry Seminar
Current Teaching (Fall 2025):
Calculus (I)
Geometric Models for Noncommutative Algebras

Experience

• Associate Professor, Department of Mathematics, National Tsing Hua University, Taiwan (Aug 2024 - now)
• Assistant Professor, Department of Mathematics, National Tsing Hua University, Taiwan (Feb 2021 - Jul 2024)
  • CARMIN visiting position, Institut des Hautes Études Scientifiques, France (Jan 8 - Feb 22, 2024 & Mar 1 - Jul 14, 2023)
  • Shapiro visitor, Department of Mathematics, Pennsylvania State University, USA (Jan 1 - Feb 28, 2023)
• Research Fellow, School of Mathematics, Korea Institute for Advanced Study, South Korea (Oct, 2018 - Dec, 2020)
• Graduate Assistant (Ph.D program), Department of Mathematics, Pennsylvania State University, USA (Aug, 2012 - Aug 2018)

Learn more: CV

Research

Research Interests

My research primarily focuses on differential graded manifolds, with broader interests spanning several interconnected fields of mathematics and physics:

• Differential Geometry: Exploring the properties and applications of geometric structures on differentiable manifolds.
• Poisson Geometry: Studying the geometric aspects of Poisson brackets and their role in mechanics and beyond.
• Homotopy Algebra: Investigating algebraic structures that arise from homotopy theory, particularly in the context of higher structures.
• Noncommutative Geometry: Delving into spaces where the coordinate rings are noncommutative, with implications for quantum theory.
• Mathematical Physics: Applying mathematical techniques to solve problems in theoretical physics, particularly those related to the geometry and algebra of physical systems.

Publications/Preprints

• (with Zachary McGuirk, Dang Khoa Nguyen and Byungdo Park) Brown functors of directed graphs; arXiv:2503.14865, 2025.
• (with Kai Behrend and Ping Xu) Differential graded manifolds of finite positive amplitude; Int. Math. Res. Not. IMRN, 2024. (arXiv:2307.10242)
• (with Hua-Shin Chang) Vertical isomorphisms of Fedosov dg manifolds associated with a Lie pair; J. Geom. Phys., 2024. (arXiv:2309.02826)
• Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs; Comm. Math. Phys., 2023. (arXiv:2211.03273)
• (with Kai Behrend and Ping Xu) On the structure of étale fibrations of L-infinity-bundles; arXiv:2307.08179, 2023.
• (with Seokbong Seol) Keller admissible triples and Duflo theorem; J. Math. Pures Appl., 2023. (arXiv:2109.06413)
• (with Kai Behrend and Ping Xu) Derived differentiable manifolds; arXiv:2006.01376, 2020.
• (with Mathieu Stiénon and Ping Xu) Formality and Kontsevich-Duflo type theorems for Lie pairs; Adv. Math., 2019. (arXiv:1605.09722)
• (with Mathieu Stiénon) Formal exponential map for graded manifolds; Int. Math. Res. Not. IMRN, 2019. (arXiv:1508.02780)
• (with Mathieu Stiénon and Ping Xu) Formality theorem for differential graded manifolds; C. R. Math. Acad. Sci. Paris, 2018.
• (with Mathieu Stiénon and Ping Xu) Formality theorem for g-manifolds; C. R. Math. Acad. Sci. Paris, 2017. (arXiv:1701.04872)
• (with Jyh-Haur Teh) Semi-topological Galois theory; Algebra Colloq., 2015. (arXiv:1006.1166)

Lists from Other Websites:

• arXiv
• Google Scholar
• MathSciNet
• Research Gate

Activities (Co-)Organizer

Conference/Workshop

• GAP (Geometry and Physics — Séminaire itinérant)
  • GAP 2024 — Rome (September 9-13, 2024)
  • GAP 2023 — Paris (May 22-26, 2023)
  • GAP 2022 — Vancouver (May 16-20, 2022)
• NCTS workshop on groupoids and quantization (Hsinchu, January 21-23, 2025)
• 2024 Taipei conference on geometry and quantization (Taipei, December 17-20, 2024)
• Higher Structures on Saturday (Paris, May 27, 2023)
• 2022 NCTS Workshop on Geometry and Physics (Hsinchu, December 30, 2022)
• Workshop on Atiyah classes and related topics (Seoul, January 6-9, 2020)

Seminar/Colloquium

• Student Geometry Seminar (2022-now)
• Colloquium (2021-2022)
• KIAS Topology Seminar (co-organizer in 2019-2020)
• Student Geometric Functional Analysis Seminar at Penn State University (organizer in 2013-2014)

Teaching

Students and Resources

• Students under my guidance

Lecture notes:

• Calculus
• Advanced Linear Algebra
• Algebraic Topology
• Homological Algebra

Other resources:

• NTHU support for student seminar
• NCTS URP²
• NCTS USRP³
• NSTC URP⁴
• Writing Mathematical Papers in English

Special Programs

• Student Geometry Seminar
• Undergraduate research programs:
  • Maslov Index, Floer Homology and Fukaya Category, 2023-2024 (NCTS URP)
  • $L_\infty$ Algebra and Its Applications, 2022-2023 (NCTS URP)
  • Star products and PBW maps⁵, 2022-2023
  • Comparison of Various Methods in Deformation Quantization, Summer 2022 (NCTS USRP)
  • PBW Approach to Fedosov’s Quantization, 2021-2022 (NCTS URP)

Courses

At National Tsing Hua University

• Geometric Models for Noncommutative Algebras, Fall 2025 (11410MATH660000)
• Calculus (I), Fall 2025 (11410MATH101001)
• Homological Algebra, Spring 2025 (11320MATH559000)
• Calculus (II), Spring 2025 (11320MATH102007)
• Calculus (I), Fall 2024 (11310MATH101007)
• Algebraic Topology, Spring 2024 (11220MATH552500)
• Calculus II, Spring 2024 (11220MATH104006)
• Calculus I, Fall 2023 (11210MATH103006)
• Advanced Linear Algebra, Fall 2022 (11110MATH331000)
• Calculus I, Fall 2022 (11110MATH103008)
• Complex Analysis, Spring 2022 (11020MATH305000)
• Disquisition, Spring 2022 (11020MATH717200)
• Algebraic Topology, Fall 2021 (11010MATH552500)
• Disquisition, Fall 2021 (11010MATH717200)
• Calculus (I), Spring 2021 (10920MATH101000)

At Penn State University

• MATH 220: MATRICES (2013 - 2018)

  Topics: Linear systems, vectors, linear transformations, invertibility of matrices, dimension, basis, determinants, eigenvalues/vectors, diagonalization, inner product, and orthogonality.

• MATH 251: ODE AND PDE (Spring 2015)

  Topics: First- and second-order ODEs, ODEs with constant coefficients, Laplace transform solutions, systems of ODEs, and 1-dimensional second-order heat equations.

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1. Room 607, GEN III Building,
   Department of Mathematics, National Tsing Hua University,
   101, Section 2, Kuang-Fu Road,
   Hsinchu 300, Taiwan ↩︎

2. 理論中心大學生研究計畫 ↩︎

3. 理論中心暑期大學生研究計畫 ↩︎

4. 國科會大專生研究計畫 ↩︎

5. 此為該年的國科會大專生研究計畫之一 ↩︎