About me        Research        Notes        Teaching        

About me.

I started my undergraduate study in 2013 at School of Mathematical Science, Nankai University, Tianjin, China.

I ontained my Ph. D. degree at University of California, Santa Barbara in 2022. My advisor is Xianzhe Dai.

I am a postdoctoral fellow at Beijing International Center for Mathematical Research (BICMR), Peking University, Beijing, China. My mentor is Xiaobo Liu.

Beginning in the fall of 2024, I will start my postdoctoral position at Northeastern University under the supervision of Maxim Braverman.

Email Address

j_yan@math.ucsb.edu

Curriculum Vitae

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Research Statement

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Besides mathematics, I enjoy playing the card game Yu-Gi-Oh , cycling and swimming.

Research

My main interests are geometric analysis and PDEs. More concretely, my current research focuses on the geometry and topology of the Landau-Ginzburg (LG) models, gluing formula of global spectral invariants and comparison geometry of hypersurfaces. See my Research Statement for more details.

Publications and Preprints

Accepted or published

1. Xianzhe Dai and Junrong Yan. Witten Deformation for Noncompact Manifolds with Bounded Geometry, Journal of the Institute of Mathematics of Jussieu，38 pages
2. Xianzhe Dai and Junrong Yan. Witten Deformation on Non-compact Manifold: Heat Kernel Expansion and Local Index Theorem, Math. Z., 31 pages

Preprints

1. Xinxing Tang and Junrong Yan. Calabi-Yau/Landau-Ginzburg Correspondence for Weil-Peterson Metrics and tt^∗ Structures, ArXiv:2205.05791, 34 pages
2. Junrong Yan. Witten deformation for non-Morse Functions and gluing formula for analytic torsions, ArXiv:2301.01990, 39 pages
3. Junrong Yan. A new analytic proof of gluing formula for analytic torsion forms, ArXiv:2301.02591, 38 pages
4. Fagui Li and Junrong Yan. A first eigenvalue estimate for embedded hypersurfaces in positive Ricci curvature manifolds. ArXiv:2308.02803, 10 pages

In preparation

1. Xianzhe Dai and Junrong Yan. Analytic Torsion for Witten Deformation on Non-compact Manifold
2. Xianzhe Dai and Junrong Yan. The Non-semiclassical Weyl’s Law for Schrodinger Operators on Non-compact Manifolds

My Thesis

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Here are the projects that I am currently working on

Higher Cheeger-Mueller/Bismut-Zhang Theorem

This is a joint project with Martin Puchol and Yeping Zhang.

Gluing formula for Global invariants

Weyl’s law on metric geometry

This is a joint project with Xianzhe Dai.

BCOV torsion for LG models

This is in various parts joint with with Xianzhe Dai and Xinxing Tang.

Notes

1. Notes on Connes Fiberation
2. Prof. Dai’s lecture on Determinants, Analytic Torsion, and Mirror Symmetry
3. Notes on String and String field theory(To be continued)
4. My Guest lecture on Index Theorem, Positive Scalar Curvature and Enlargibility in Prof. Dai’s Math 241C, 2022 Spring.
5. Notes on Heat kernel on Vector Bundles
6. Notes on Unbounded operators in Hilbert Space, min-max principle and EPDEs

Teaching

Instructor

1. Math 3B: Calculus with Applications, Summer 2021

Teaching Assistant

1. Math 8: Transitions to Higher Mathematics, Spring 2018
2. Math 8: Transitions to Higher Mathematics, Fall 2018
3. Math 6B: Vector Calculus, Winter 2019
4. Math 34B: Calculus for Social and Life Sciences, Spring 2019
5. Math 6A: Vector Calculus, Fall 2019
6. Math 6A: Vector Calculus, Winter 2020
7. Math 6A: Vector Calculus, Summer 2020
8. Math 8: Transitions to Higher Mathematics, Fall 2020
9. Math 6A: Vector Calculus, Winter 2021
10. Math 6A: Vector Calculus, Spring 2021

Reader

1. Math 117: Methods of Analysis, Fall 2017
2. Math 108B: Introduction to Linear Algebra, Winter 2018
3. Math 117: Methods of Analysis, Fall 2019

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