About

Xianzhe Dai is a Professor in the Department of Mathematics at the University of California, Santa Barbara. His research interests focus on Differential Geometry and Geometric Analysis. Throughout his career, Professor Dai has been actively involved in the academic community, participating in various seminars and international conferences related to geometry and analysis on manifolds. These include the Differential Geometry Seminar, the International Conference on Geometry and Analysis on Manifolds, and the International Conference on Metric and Differential Geometry. His engagement with the Mathematics Colloquium and topics intersecting Geometry, Topology, and Physics further highlight his commitment to advancing research in these interconnected fields.

Research topics

• Artificial Intelligence
• Computer Science
• Mathematics
• Quantum mechanics
• Algorithm
• Mathematical analysis
• Statistical physics
• Geology
• Physics

Selected publications

• Epigenetically poised chromatin states regulate PRR and NLR genes in soybean

  aBIOTECH · 2025-08-08

  articleOpen access

  In the plant innate immune system, pattern recognition receptor (PRR) and nucleotide-binding domain leucine-rich repeat (NLR) proteins recognize pathogens and activate defenses. To prevent excessive immune responses that could affect growth, plants regulate PRRs and NLRs at the transcriptional and post-transcriptional levels. Poised or bivalent chromatin states, marked by the simultaneous presence of active and repressive epigenetic modifications, maintain genes in a transcriptionally primed state, keeping their expression low while enabling their rapid activation in response to stress. Here, we investigated how poised chromatin states regulate PRR and NLR genes in soybean (Glycine max). Our integrative epigenomic and transcriptomic analysis revealed that although NLR and PRR genes both harbor abundant active and repressive histone modifications and exhibit high chromatin accessibility, their basal expression levels remain relatively low. Moreover, clustered NLR and PRR genes residing within the same topologically associating domains shared similar chromatin states and expression dynamics, suggesting coordinated control. These gene families had distinct epigenetic features: NLR genes displayed narrow H3K27me3 peaks together with strong pausing of RNA Polymerase II at their 5′ ends, whereas PRR genes were characterized by broader H3K27me3 peaks. Together, our results shed light on the role of poised chromatin states in coordinating growth and defense responses in soybean.

  PublisherOA PDFDOI

• Degeneration of Riemann surfaces and small eigenvalues of the Laplacian

  ArXiv.org · 2025-09-07

  preprintOpen access1st authorCorresponding

  For a one-parameter degeneration of compact Riemann surfaces endowed with the Kähler metric induced from the Kähler metric on the total space of the family, we determine the exact magnitude of the small eigenvalues of the Laplacian as a function on the parameter space, under the assumption that the singular fiber is reduced. The novelty in our approach is that we compute the asymptotic behavior of certain difference of (logarithm of) analytic torsions in the degeneration in two ways. On the one hand, via heat kernel estimates, it is shown that the leading asymptotic is determined by the product of the small eigenvalues. On the other hand, using Quillen metrics, the leading asymptotic is connected with the period integrals, which we explicitly evaluate.

  PublisherOA PDFDOI

• Positive mass theorem for asymptotically flat spin manifolds with isolated conical singularities

  Transactions of the American Mathematical Society · 2024-10-04 · 1 citations

  preprint1st authorCorresponding

  There has been a lot of interest in positive mass theorems for singular metrics on smooth manifolds. We prove a positive mass theorem for asymptotically flat (AF) spin manifolds with isolated conical singularities or more generally horn singularities. In particular, we allow topological singularities in the space as we do not require the cross sections of the conical singularities to be spherical. Note that the negative mass Schwarzschild metric is AF with a horn singularity.

  PublisherDOI

• Perelman’s functionals on manifolds with non-isolated conical singularities

  Calculus of Variations and Partial Differential Equations · 2024-11-07

  article1st authorCorresponding
  PublisherDOI

• Positive scalar curvature and isolated conical singularity

  arXiv (Cornell University) · 2024-12-04

  preprintOpen access1st authorCorresponding

  We prove a Geroch type result for isolated conical singularity. Namely, we show that there is no Riemannian metric $g$ on $ X \# T^n $ with an isolated conical singularity which has nonnegative scalar curvature on the regular part, and is positive at some point. In particular, this implies that there is no metric on tori with an isolated conical singularity and positive scalar curvature. We also prove that a scalar flat Riemannian metric $g$ on $X \# T^n$ with finitely many isolated conical singularities must be flat, and extend smoothly across the singular points. We do not a priori assume that a conically singular point on $X$ is a manifold point; i.e., the cross section of the conical singularity may not be spherical.

  PublisherOA PDFDOI

• Volume entropy and rigidity for RCD-spaces

  arXiv (Cornell University) · 2024-11-07 · 1 citations

  preprintOpen access

  We develop the barycenter technique of Besson--Courtois--Gallot so that it can be applied on RCD metric measure spaces. Given a continuous map $f$ from a non-collapsed RCD$(-(N-1),N)$ space $X$ without boundary to a locally symmetric $N$-manifold we show a version of BCG's entropy-volume inequality. The lower bound involves homological and homotopical indices which we introduce. We prove that when equality holds and these indices coincide $X$ is a locally symmetric manifold, and $f$ is homotopic to a Riemannian covering whose degree equals the indices. Moreover, we show a measured Gromov--Hausdorff stability of $X$ and $Y$ involving the homotopical invariant. As a byproduct, we extend a Lipschitz volume rigidity result of Li--Wang to RCD$(K,N)$ spaces without boundary. Finally, we include an application of these methods to the study of Einstein metrics on $4$-orbifolds.

  PublisherOA PDFDOI

• The positive mass theorem for asymptotically flat manifolds with isolated conical singularities

  Science China Mathematics · 2024-11-20 · 2 citations

  article1st authorCorresponding
  PublisherDOI

• Singular metrics with nonnegative scalar curvature and RCD

  arXiv (Cornell University) · 2024-12-12

  preprintOpen access1st authorCorresponding

  We show that a uniformly Euclidean metric with isolated singularity on $M^n = T^n \# M_0$, where $4\leq n\leq 7$ or $n\geq 4$, $M_0$ spin, and nonnegative scalar curvature on the smooth part is Ricci flat and extends smoothly over the singularity. This confirms Schoen's Conjecture in these cases. The key to the proof is to show that the space has nonnegative synthetic Ricci curvature, i.e., an $RCD(0, n)$ space. Our result also holds when the singular set consists of a finite union of submanifolds (of possibly different dimensions) intersecting transversally under additional assumption on the co-dimension and the location of the singular set.

  PublisherOA PDFDOI

• Positive mass theorem for asymptotically flat manifolds with isolated conical singularities

  arXiv (Cornell University) · 2024-01-14

  preprintOpen access1st authorCorresponding

  We prove the positive mass theorem for asymptotical flat (AF for short) manifolds with finitely many isolated conical singularities. We do not impose the spin condition. Instead we use the conformal blow up technique which dates back to Schoen's final resolution of the Yamabe conjecture.

  PublisherOA PDFDOI

• Singular Weyl’s law with Ricci curvature bounded below

  Transactions of the American Mathematical Society Series B · 2023 · 9 citations

  1st authorCorresponding
  • Artificial Intelligence
  • Computer Science
  • Algorithm

  We establish two surprising types of Weyl’s laws for some compact <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R upper C upper D left-parenthesis upper K comma upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>RCD</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {RCD}(K, N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> /Ricci limit spaces. The first type could have power growth of any order (bigger than one). The other one has an order corrected by logarithm similar to some fractals even though the space is 2-dimensional. Moreover the limits in both types can be written in terms of the singular sets of null capacities, instead of the regular sets. These are the first examples with such features for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R upper C upper D left-parenthesis upper K comma upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>RCD</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {RCD}(K,N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spaces. Our results depend crucially on analyzing and developing important properties of the examples constructed in Pan and Wei [Geom. Funct. Anal. 32 (2022), pp. 676–685], showing them isometric to the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Grushin halfplanes. Of independent interest, this also allows us to provide counterexamples to conjectures in Cheeger and Colding [J. Differential Geom. 46 (1997), pp. 406–480] and Kapovitch, Kell, and Ketterer [Math. Z. 301 (2022), pp. 3469–3502].

  PublisherDOI

Recent grants

• Dirac Operator, Eta Invariant and Applications

  NSF · $130k · 2004–2008

• Geometric Applications of Dirac Operator and Atiyah-Singer Index Theory

  NSF · $137k · 2010–2013

• Dirac operator, Atiyah-Singer index theory, and applications

  NSF · $221k · 2007–2012

• Analytic Torsion, Conical Singularity and Geometric Applications

  NSF · $153k · 2016–2019

• EMSW21-RTG: UCSB RTG in Topology and Geometry

  NSF · $803k · 2011–2018

Frequent coauthors

• Guofang Wei

  27 shared

• Changliang Wang

  Tongji University

  12 shared

• Weiping Zhang

  University of Science and Technology of China

  11 shared

• Yukai Sun

  9 shared

• Xiaodong Wang

  5 shared

• Zhenlei Zhang

  China University of Petroleum, Beijing

  5 shared

• Junrong Yan

  4 shared

• Pablo Suárez‐Serrato

  3 shared