I am working on Riemann & Kähler & Metric Geometry.
Email address: Jialongdeng@gmail.com
I am currently an Oberwolfach Leibinz fellow, MFO. Previously, I was a post-doc researcher with Prof. Akito Futaki at Yau Mathematical Sciences Center, Tsinghua University. Before that, I completed my Ph.D. study at the University of Göttingen (w/ Prof. Thomas Schick) .
Papers and Preprints:
6. Quasiconformal mappings and curvatures on metric measure spaces. Proceedings of the International Geometry Center Vol. 15, no. 3-4 (Mar. 2023) pp. 203-218.
In an attempt to develop higher-dimensional quasiconformal mappings on metric measure spaces with curvature conditions, i.e. from Ahlfors to Alexandrov, I showed that a quasiconformal homeomorphism f of a non-collapsed RCD(0, n) (n>1) space with Euclidean volume growth is quasisymmetric, if it maps bounded sets to bounded sets.
5.Sphere theorems with and without Smoothing. Journal of Geometry.113, 33 (2022).
Using harmonic maps, I demonstrated rigidity results for scalar curvature without relying on the spin condition. By extending the concept of mean distance from graph theory, I defined analogous metric invariants on metric measure spaces. These invariants were utilized to establish rigidity and sphere theorems on compact non-collapsed RCD(n-1,n) spaces.
4.Curvature dimension condition meets Gromov's n-volumic scalar curvature. (Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday, 2020.) Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 17 (2021), 013.
One of my main results was defining the weighted scalar curvature on a weighted Riemannian manifold and show its properties. This weighted scalar curvature includes Perelman's scalar curvature, Chang-Gursky-Yang's conformally invariant scalar curvature, and Case's scalar curvature (for the weighted Yamabe constants).
3. Enlargeable Length-structures and Scalar Curvatures. Annals of Global Analysis and Geometry, 60.2 (2021) pp. 217–230.
One of my main results was defining enlargeable length-structures. I used this concept to show that certain closed locally CAT(0) manifolds, including closed non-positively curved manifolds, cannot carry a Riemannian metric with positive scalar curvature. This result holds when these manifolds are connected summed with a closed manifold.
2. Metric topology on the moduli space. Applied General Topology 22(1):11–15, 2021.
The smooth Lipschitz topology on the moduli space of Riemannian metrics was defined and we showed that each conformal class is dense in the moduli space endowed with Gromov-Hausdorff topology.
1. A note about scalar curvature on the total space of a vector bundle. arXiv:1907.02723.
Doctoral Thesis: Foliated positive scalar curvature
General Relativity on metric measure spaces was discussed in Section 5.2.
Professional Service:
Reviewer for zbMATH. (43)
• Referee for The Journal of Geometric Analysis. (1)
• The first International Congress of Basic Science (ICBS 2023), Frontiers of Science Award (Differential Geometry), Subcommittee Coordinator.
Presentations:
2025
Geometry and Topology Seminar, University of Wisconsin-Madison, Jan. 24, 2025.
Postdoc workshop I 2025 of Yau Mathematical Sciences Center-Beijing Institute of Mathematical Sciences and Applications (YMSC-BIMSA), Beijing, China, April 4, 2025.
Postdoc workshop II 2025 of Yau Mathematical Sciences Center-Beijing Institute of Mathematical Sciences and Applications (YMSC-BIMSA), Beijing, China, May 10, 2025.
2024
1. Research Teaching Group 2491 Autumn School: T-Duality, K-Theory, and Representation Theory (Sept. 30-Oct. 4, 2024), poster, University of Göttingen, Germany, Oct. 1, 2024.
2. The 184th Session of Math Forum, Southwestern University of Finance and Economics, Chengdu, China, Sept. 28, 2024.
3. Geometry seminar, Yau Mathematical Sciences Center (YMSC), Beijing, China, Sept. 14, 2024.
4. Postdoc workshop II of Yau Mathematical Sciences Center-Beijing Institute of Mathematical Sciences and Applications (YMSC-BIMSA), Beijing, China, June 23, 2024.
5. 2024 Geometry and Analysis Seminar (Mar. 30-31, 2024), Tsinghua University, Beijing, China, Mar. 30, 2024.
2023
1. Differential Geometry Seminar, Beijing Institute of Mathematical Sciences and Applications (BIMSA), Beijing, China, Nov. 28, 2023.
2. Geometry Seminar, Capital Normal University, Beijing, China, Nov. 14, 2023.
3. Series of academic activities for the 70th anniversary of Capital Normal University, Capital Normal University, Beijing, China, Nov. 8, 2023.
4. Differential Geometry Seminar (online), University of Houston, Houston, USA, Oct. 16, 2023.
5. International Talent Forum，Nankai University, Tianjin, China, Sept. 26, 2023.
6. Korea Institute for Advanced Study, Geometric Analysis Seminar (online), Sept. 26, 2023.
7. Workshop on Curvature and Global Shape, poster, Muenster, Germany, Aug. 1, 2023.
8. Workshop on Metric Geometry and Complex Analysis (online), Foshan, China, June 11, 2023.
9. Geometry seminar, Yau Mathematical Sciences Center (YMSC), Beijing, China, June 10, 2023.
10. World Conference on Physics and Mathematics, Berlin, Germany, invited speaker (online), May 23, 2023.
11. Postdoc workshop of Yau Mathematical Sciences Center-Beijing Institute of Mathematical Sciences and Applications (YMSC-BIMSA), Beijing, China, Mar. 25, 2023.
12. Geometry Seminar of Yau Mathematical Sciences Center (YMSC), Beijing, China, Mar. 4, 2023.
2022
1. Geometry seminar of Yau Mathematical Sciences Center (YMSC), Beijing, China, Nov. 27, 2022.
2. Postdoc workshop B of Yau Mathematical Sciences Center-Beijing Institute of Mathematical Sciences and Applications (YMSC-BIMSA), Beijing, China, Nov. 20, 2022.
3. Topology seminar of Yau Mathematical Sciences Center (YMSC), Beijing, China, Oct. 25, 2022.
4. Yau Mathematical Sciences Center (YMSC), one-day seminar, Beijing, China, Sept. 18, 2022.
5. Sun Yat-sen University, Guangzhou, China, Sept. 18, 2022 (online).
6. Sun Yat-sen University, Guangzhou, China, Sept. 11, 2022 (online).
7. Sun Yat-sen University, Guangzhou, China, Jan. 18, 2022 (online).
8. South China University of Technology, Guangzhou, China, Jan. 13, 2022.
9. Jinan University, Guangzhou, China, Jan. 12, 2022.
10. Huazhong University of Science and Technology, Wuhan, China, Jan. 5, 2022.
2021
1. Graduate Student Topology and Geometry Conference (GSTGC), Indiana University, April 9-11, 2021, online poster.
2. Young Geometers Meeting, University of Copenhagen, April 19-23, 2021, online poster.
3. International Conference of Young Mathematicians (online), Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine, June 3-5, 2021, 15-minute talk.
4. The 38th Annual (online) Workshop in Geometric Topology, Texas Christian University, June 15-17, 2021, 20-minute talk.
5. TopFlavours 2021 (online), University of Warwick, June 17-18, 25-minute talk and 7-minute Gongshow.
6. The 3rd Geometric Analysis Festival, July 25-28, 2021, 60-minute talk.