Harold Blum

University of Utah - Department of Mathematics

About Me

I am an Assistant Professor in the Department of Mathematics at the University of Utah University of Utah. From 2018 to 20201, I was an NSF postdoc at Stony Brook University and the University of Utah. In 2018, I completed my Ph.D. at the University of Michigan under the supervision of Mircea Mustaţă.

Here is my CV.

Research interests

Algebration geometry: birational geometry, singularities, Fano varieties, K-stability, and moduli.

Contact Info

Email: blum [at] math.utah.edu
Office: LCB 203
Mail: University of Utah
Department of Mathematics
Salt Lake City, UT 84112


Convexity of multiplicities of filtrations on local rings, joint with Yuchen Liu and Lu Qi, arXiv:2208.04902.

The existence of the Kähler-Ricci soliton degeneration, joint with Yuchen Liu, Chenyang Xu, and Ziquan Zhuang, arXiv:2103.15278.

On properness of K-moduli spaces and optimal degenerations of Fano varieties, joint with Daniel Halpern-Leistner, Yuchen Liu and Chenyang Xu, Selecta Math. 27 (2021).

Optimal destablization of K-unstable Fano varieties via stability thresholds, joint with Yuchen Liu and Chuyu Zhou, arXiv:1907.05399. To appear in Geom. Topol.

Openness of K-semistability for Fano varieties, joint with Yuchen Liu and Chenyang Xu, arXiv:1907.02408. To appear in Duke Math. J.

Reductivity of the automorphism group of K-polystable Fano varieties, joint with Jarod Alper, Daniel Halpern-Leistner, and Chenyang Xu, Invent. Math. 222 (2020), 995-1032.

Uniqueness of K-polystable degenerations of Fano varieties, joint with Chenyang Xu, Ann. of Math. 190 (2019), 609-656.

Openness of uniform K-stability in families of Q-Fano varieties, joint with Yuchen Liu, arXiv:1808.09070. To appear in Ann. Sci. Éc. Norm. Supér.

The normalized volume of a singularity is lower semicontinuous, joint with Yuchen Liu, J. Eur. Math. Soc. 23 (2021), 1225-1256.

Thresholds, valuations, and K-stability, joint with Mattias Jonsson, Adv. Math. 365 (2020).

Existence of valuations with smallest normalized volume, Compos. Math. 154 (2018), 820-849.

On divisors computing mld's and lcts's, Bull. Korean Math. Soc. 58 (2021), 113-132.

Other Writing

Singularities and K-stability, Ph.D. Thesis, University of Michigan, link.


This semester I am teaching Math 7800 - Topics in Algebraic Geometry. The purpose of the course is to give an introduction to the algebraic theory of K-stability. My lecture notes, which are very much a work in progress, can be found here. A more up to date version can be found on the Canvas page for the course.

A list of my past teaching can be found here.