Homepage of Bingxiao LIU

Dr. Bingxiao Liu

Photo from Archives of the Mathematisches Forschungsinstitut Oberwolfach

Professional address:

Department Mathematik/Informatik, Abteilung Mathematik,

Universität zu Köln,

Weyertal 86 - 90,

D-50931 Köln,

Deutschland

Email address: bingxiao.liu AT uni-koeln.de

I am currently a research assistant at the Mathematics Institute of the University of Cologne from October 2020. I work with Prof. Drewitz and Prof. Marinescu on the SPP project Random polynomials and random Kähler geometry.

I was a Postdoctoral Fellow at Max Planck Institute for Mathematics in Bonn from September 2018 to September 2020. Before Bonn, I was a Ph.D. student in Université Paris-Sud (Orsay, France).

During the university year 2017/2018, I had an A.T.E.R. position in Université Paris-Sud(Orsay), with a teaching load of 192 hours for undergraduates in Université Paris-Sud and Polytech Paris-Sud.

My Ph.D. supervisor: Professor Jean-Michel Bismut (Université Paris-Sud, Orsay).

My thesis title: Hypoelliptic Laplacian and twisted trace formula.

Research interests:

Local index theory, analytic torsion, and microlocal analysis;
Complex and Kähler geometry, random holomorphic sections;
Symmetric spaces, semisimple Lie groups, and their representations;
Loop spaces and mathematical physics;

Publications & Preprints

with George Marinescu and Huan Wang (Henan Academy of Sciences, China), Nakano-Griffiths inequality, holomorphic Morse inequalities, and extension theorems for q-concave domains (49 pages). arXiv:2506.00879.
with Turgay Bayraktar (Sabanci University, Turkey), Dan Coman (Syracuse University, USA) and George Marinescu, Tian's theorem for Grassmannian embeddings and degeneracy sets of random sections (32 pages). arXiv:2504.19731.
with Dominik Zielinski, Semipositive line bundles on punctured Riemann surfaces: Bergman kernels and random zeros . Analysis and Mathematical Physics, 15(2), article number 38, 61 pages, 2025. arXiv:2407.15106.
with Alexander Drewitz and George Marinescu, Toeplitz operators and zeros of square-integrable random holomorphic sections (66 pages). arXiv:2404.15983.
with Hisatoshi Kodani (Kyushu University, Japan), Integral invariants for framed 3-manifolds associated to trivalent graphs possibly with self-loops (52 pages). arXiv:2311.02682.
with Alexander Drewitz and George Marinescu, Gaussian holomorphic sections on noncompact complex manifolds . Journal of the Institute of Mathematics of Jussieu. Volume 24 Issue 4 , pp. 1197 - 1262, 2025. arXiv:2302.08426.
with Alexander Drewitz and George Marinescu, Large deviations for zeros of holomorphic sections on punctured Riemann surfaces . Michigan Math. J. 75 (2) 381 - 421, 2025. DOI: 10.1307/mmj/20226253. arXiv:2109.09156.
On full asymptotics of analytic torsions for compact locally symmetric orbifolds . Anal. PDE, 17(4):1261–1329, 2024. arXiv:2005.02055.
Hypoelliptic Laplacian and twisted trace formula . Annales de l'Institut Fourier, Tome 73 (2023) no. 5, pp. 1909-1985. arXiv:1910.11382.
Asymptotic equivariant real analytic torsions for compact locally symmetric spaces . J. Funct. Anal., 281(7):Paper No. 109117, 54 pp., 2021. Journal.
Hypoelliptic Laplacian and twisted trace formula . Comptes Rendus Mathématique, Volume 357, Issue 1, 2019, Pages 74-83. DOI:10.1016/j.crma.2018.11.010.

Current activities

Hong Kong Laureate Forum 2025, 5-8 November 2025.
QuaSiDy@Pisa, October 1st-3rd, 2025.
Seminar semiklassische Analysis und Darstellungstheorie, Universität zu Köln.
AG Random Geometry, Universität zu Köln.

Teachings

Master's Lecture: Introduction to the Atiyah-Singer index theory, Summer semester 2025, Universität zu Köln. Please visit the lecture wepage for more detials.
Teaching assistant for the lecture Random Kähler Geometry, Winter semester 2023/2024, Universität zu Köln.

Organization

Conference Geometric and topological properties of random algebraic varieties, 4th - 6th Oct. 2023, Universität zu Köln.

Mini-lectures

Mini-lecture (jointly with A. Drewitz and G. Marinescu) Random holomorphic sections and their zeros, at Summer School PRobabilistic mEthoDs in Complex geomeTry, Como Lake, June 16 - 20, 2025. Program at Lake Como School of Advanced Studies. Link for (partial) Notes
Mini-course on superconnection and complex immersions, August 5th - 9th 2024, the mini-course was given during the Workshop on Global Analysis on Manifolds 2024 at Wuhan University, consisting of five lectures.
On-line lectures: Introduction to random zeros of holomorphic sections, 19, 21, 26, 28/03/2024, the lectures were given on-line in Chinese with English notes, see abstracts .
Lecture Note 1 Lecture Note 2 Lecture Note 3 Lecture Note 4 Latexified notes with more details and references still under construction.
On-line mini-lecture: Introduction to Hypoelliptic Laplacian of Bismut on Symmetric Spaces, Nov 17th, 24th, and 26th, 2021. School of Mathematical Sciences, Tongji University, Shanghai.
Mini-course: An introduction to symmetric spaces, July 29th - Aug 2nd, 2019, in Workshop on Global Analysis on Manifolds, USTC, Hefei.

Talks in Seminars & Conferences

July 16th, 2025, Institutskolloquium: From Kac Polynomials to Complex Geometry: Study of Random Holomorphic Zeros, 16th July 2025, 16:30 at Departmenthörsaal (Raum 203), Abteilung Mathematik.
June 25th, 2025, Asymptotic analytic torsions for compact locally symmetric orbifolds. Bernoulli External Program 2025: New Trends and Applications Around Generalized Fokker-Planck Operators (Week 2 of the 4-week programme). June 23rd - 27th, 2025. Bernoulli Center for Fundamental Studies, Lausanne, Switzerland.
May 12th, 2025, Tian's theorem for Grassmannian embeddings and degeneracy sets of random sections. Conference: From Complex Analysis to Quantization -- in honour of Professor George Marinescu's 60th Birthday. May 12th - 16th, 2025. Chern Institute of Mathematics (CIM), Nankai University, Tianjin, China
Mar 25th, 2025, Théorème de Tian et valeurs propres d’opérateur de Toeplitz. Séminaire Analyse et Dynamique, Université Côte d'Azur, Nice, France.
Mar 13th, 2025, L’espace de Wiener abstrait et sections holomorphes gaussiennes de carré intégrable. Séminaire équipe EMA, Université du Littoral Côte d'Opale, Calais, France.
Mar 11th, 2025, Tian's approximation theorem and eigenvalues of Toeplitz operators. Séminaire Analyse Harmonique, Université Paris-Saclay, Orsay, France.
Dec 19th, 2024, Toeplitz-Fubini-Study forms and lowest eigenvalues of Toeplitz operators. Conference: Around local index theory, Dec 16th - 20th 2024, École polytechnique, CMLS, Palaiseau, France.
Sept 27th, 2024, Toeplitz-Fubini-Study forms and lowest eigenvalues of Toeplitz operators. Seminar on several complex variables and complex geometry, Academy of Mathematics and Systems Science, CAS, Beijing, China.
Sept 26th, 2024, Toeplitz operators and zeros of square-integrable Gaussian holomorphic sections. Seminar on several complex variables and complex geometry, Academy of Mathematics and Systems Science, CAS, Beijing, China. slides
Sept 10th, 2024, Toeplitz operators and zeros of square-integrable Gaussian holomorphic sections. Conference: Randomness in Complex Analysis and Complex Geometry, Sept 9th - 13th 2024, Nesin Mathematics Village (Şirince), Izmir, Türkiye (Turkey).
Sept 3rd, 2024, Asymptotic analytic torsions for compact locally symmetric orbifolds. Conference: Analytic torsion and interactions, Sept 2nd - 6th 2024, Universität Münster, Germany.
August 28th, 2024, Toeplitz-Fubini-Study forms and lowest eigenvalues of Toeplitz operators (on-line talk). Conference on Complex analysis and Geometry, August 26th - 29th 2024, Vietnam Institute for Advanced Study in Mathematics (VIASM), Hanoi, Vietnam.
July 12th, 2024, Toeplitz-Fubini-Study forms and lowest eigenvalues of Toeplitz operators. Joint Seminar on Complex Algebraic Geometry and Complex Analysis (Bochum - Essen - Köln - Münster - Wuppertal) . Ruhr-Universität Bochum, Bochum, Germany.
April 29th, 2024, Toeplitz Operators and Zeros of Square-integrable Gaussian Holomorphic Sections. Probability Seminar Series, NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, NYU Shanghai, Shanghai, China.
April 4th, 2024, Opérateur de Toeplitz et zéros de sections holomorphes gaussiennes de carré intégrable. Séminaire d'algèbre, topologie et géométrie, Laboratoire J.A. Dieudonné, Université Côte d'Azur, Nice, France.
Oct 26th, 2023, Random holomorphic sections on noncompact complex manifolds. Workshop on Pluripotential Theory and Holomorphic foliations, Oct 25th - 27th 2023, Université Polytechnique Hauts-de-France, Valenciennes, France.
July 26th, 2023, Random holomorphic sections on noncompact complex manifolds. Conference: Quantization in Geometry, July 24th - 28th 2023, Universität zu Köln, Köln, Germany. slides
May 25th, 2023, Hypoelliptic Laplacian of Bismut on symmetric spaces. Oberwolfach Workshop: Hypoelliptic Operators in Geometry (Workshop ID 2321) , May 21st - 26th 2023, MFO, Oberwolfach, Germany. MFO Report (Reporter: Bingxiao Liu)
Apr 28th, 2023, Hypoelliptic Laplacian of Bismut on symmetric spaces and applications. ECNU Differential Geometry Seminar, East China Normal University, Shanghai.
Mar 27th, 2023, Random polynomials and random Kähler geometry. 2nd Annual Conference of Random Geometric Systems (SPP2265), Mar 27th - 30th 2023, DLR site in Cologne-Porz.
Mar 7th, 2023, Zeros of random holomorphic sections on complex manifolds. Session: Limit theorems, large deviations and extremes, German Probability and Statistics Days (GPSD 2023), Mar 7th - 10th 2023, Essen.
Jan 17th, 2023, Zeros of random holomorphic sections on noncompact complex manifolds. Séminaire Analyse Harmonique, Orsay, Université Paris-Saclay.
Dec 15th, 2022, Zeros of random holomorphic sections on noncompact complex manifolds. ECNU Differential Geometry Seminar, East China Normal University, Shanghai.
Oct 15th, 2022, Zeros of random holomorphic sections on noncompact complex manifolds. Workshop Cologne-Nice Complex and CR Geometries, Dynamics, Nov 14th - 15th 2022, Nice.
Mar 8th, 2022, Large deviation estimates for zeros of random holomorphic sections on punctured Riemann surfaces. Séminaire d'Analyse et Géométrie, Jussieu, Paris.
Oct 26th, 2021, Large deviation estimates for zeros of random holomorphic sections for punctured Riemann surfaces. Conference ACQUA: Quantitative Aspects in Complex analysis, Geometry and Dynamics, October 25th - 29th, Lille, France
July 13th, 2021, Asymptotic real analytic torsions for compact locally symmetric orbifold (on-line). Seminar semiklassische Analysis und Darstellungstheorie, University of Cologne.
Feb 2nd, 2021, joint talk with Prof. Drewitz, Random polynomials and random Kähler geometry (on-line). Probability / SPP 2265 Research Seminar Ruhr.
May 15th, 2020, Asymptotic analytic torsion for compact locally symmetric orbifold. Online Seminar on the Hypoelliptic Laplacian and Noncommutative Geometry.
July 15th, 2019, Asymptotics of equivariant analytic torsion on compact locally symmetric spaces. Conference 'Quantization in symplectic geometry', July 15th - 19th, 2019, Cologne.
International Micro-Workshop on the Hypoelliptic Laplacian, May 7th - 10th, 2019, Department of Mathematics, Penn State University, State College.
Apr 30th, 2019, Hypoelliptic Laplacians on symmetric spaces and twisted orbital integrals. Noncommutative Geometry Festival 2019, Apr 29th - May 3rd, 2019, Washington University in St. Louis.
Apr 12th, 2019, Hypoelliptic Laplacians on symmetric spaces and twisted trace formula. Mini-workshop 'Recent Progress in Path Integration on Graphs and Manifolds', Apr 7th - 13th 2019, MFO, Oberwolfach, Germany. MFO Report
Dec 18th, 2018, Introductions to hypoelliptic Laplacian on symmetric spaces. Oberseminar Global Analysis and Operator Algebras, University of Bonn.
Nov 11th, 2018, Asymptotics of equivariant analytic torsion on compact locally symmetric spaces. Seminar semiklassische Analysis und Darstellungstheorie, University of Cologne.
Oct 9th, 2018, Selberg's twisted trace formula and asymptotics of equivariant analytic torsion. International Workshop on Geometric Quantization and Applications, Oct 8th - 12th, 2018, at Luminy - CIRM. slides
Aug 1st, 2018, Hypoelliptic Laplacian and twisted orbital integrals. Workshop on Global Analysis on Manifolds, USTC, Hefei.
Jun 30th, 2018, Hypoelliptic Laplacian and semisimple orbital integrals. Séminaire Mathjeunes, Université Paris-Sud, Orsay.
May 5th, 2018, Semisimple orbital integrals associated with real reductive groups. Séminaire Mathjeunes, Université Paris-Sud, Orsay.

Events & Miscellaneous

The 2021 Shaw Prize in Mathematical Sciences for Prof. Jean-Michel Bismut. An overview of Bismut's work (in French) by Prof. Xiaonan Ma.
Results of the dissertation of my Bachelor degree (2013) at USTC are cited in the survey articles (English, Chinese) of Prof. Xiao Zhang (AMSS, CAS), he was my supervisor of Bachelor thesis.
Some videos of talks given by Prof. Jean-Michel Bismut.
Conference "Index theory, Duality and Related Fields" at the Chern Institute, June 17th - 20th 2019, Tianjin.
Workshop in Chern Institute of Mathematics (Tianjin): Loop spaces, Supersymmetry and Index theory, July 17th - 21st 2017.