# 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.

My research is partially supported by NSF grant DMS-2200690.### 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 |

### Publications/Preprints

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. To appear in Forum Math. Pi.

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), Geom. Topol..

**26**(2022), 2507-2564.

Openness of K-semistability for Fano varieties (joint with Yuchen Liu and Chenyang Xu), Duke Math. J..

**171**(2022), 2753-2797.

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), Ann. Sci. Éc. Norm. Supér.

**55**(2022), 1-41

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.### Teaching

This semester I am teaching Math 5320 - Intro to Modern Algebra II. The course website is on Canvas.Last semester I taught Math 7800 - Topics in Algebraic Geometry. The goal of the course was to give an introduction to the algebraic theory of K-stability. Partial lecture notes from the course can be found here. Note that these notes are very much under construction and various sections are missing. Any comments/suggestions are appreciated.

A list of my past teaching can be found here.