Mircea Mustaţă


I am a Professor in the Department of Mathematics at University of Michigan. Here is my contact information.

My work is in algebraic geometry. Over the past few years I have been involved in a long-term project with Mihnea Popa, studying certain invariants of hypersurface singularities (Hodge ideals) that arise naturally from Saito's theory of mixed Hodge modules. I am interested in general in invariants of singularities of algebraic varieties, such as minimal log discrepancies, log canonical thresholds, multiplier ideals, Bernstein-Sato polynomials, and F-thresholds. Various points of view and techniques come in the picture when studying these invariants: resolutions of singularities, jet schemes, D-modules or positive characteristic methods.

Here is my CV, including a list of publications.


Papers

You can find here my papers on the archive. Here are a few recent ones:

1. The minimal exponent of cones over smooth complete intersection projective varieties (with Qianyu Chen and Bradley Dirks), available at arXiv2404.18289.
2. Erratum to the paper: Asymptotic Invariants of Base Loci (with Lawrence Ein, Robert Lazarsfeld, Michael Nakamaye, and Mihnea Popa), available at arXiv:2309.16722.
3. The minimal exponent and k-rationality for locally complete intersections (with Qianyu Chen and Bradley Dirks), available at arXiv:2212.01898.
4. An estimate for F-jumping numbers via the roots of the Bernstein-Sato polynomial, available at arXiv:2209.00753.
5. V-filtrations and minimal exponents for locally complete intersection singularities (with Qianyu Chen, Bradley Dirks, and Sebastián Olano), available at arXiv:2208.03277.
6. On k-rational and k-Du Bois local complete intersections (with Mihnea Popa), available at arXiv:2207.08743.
7. On a conjecture of Bitoun and Schedler (with Sebastián Olano), available at arXiv:2207.02047.
8. The log canonical threshold and rational singularities (with Raf Cluckers and János Kollár), available at arXiv:2202.08425.
9. Hodge filtration on local cohomology, Du Bois complex, and local cohomological dimension (with Mihnea Popa), available at arXiv:2108.05192.
10. The Du Bois complex of a hypersurface and the minimal exponent (with Sebastián Olano, Mihnea Popa, and Jakub Witaszek), available at arXiv:2105.01245.
11. Minimal exponents of hyperplane sections: a conjecture of Teissier (with Bradley Dirks), available at arXiv:2008.10345.
12. On a conjecture of Teissier: the case of log canonical thresholds (with Eva Elduque), available at arXiv:2005.03803.
13. The Hilbert series of Hodge ideals of hyperplane arrangements (with Bradley Dirks), available at arXiv:2003.11681.
14. Upper bounds for roots of B-functions, following Kashiwara and Lichtin (with Bradley Dirks), available at arXiv:2003.03842.
15. Hodge ideals and minimal exponents of ideals (with Mihnea Popa), available at arXiv:1912.08072.
16. Bernstein-Sato polynomials for general ideals vs. principal ideals, availableat arXiv:1906.03086.
17. Hodge filtration, minimal exponent, and local vanishing (with Mihnea Popa), available at arXiv:1901.05780.


Teaching


Summer 2024. Course on D-modules, Hodge modules, and singularities at the University of Tokyo. Here are the lecture notes from the course (version of July 17).
Winter 2024. Math 594. Algebra II.
Fall 2023. Math 593. Algebra I.
Winter 2023. Math 732. D-modules and singularities. Here are the lecture notes from the course.
Fall 2022. Math 614. Commutative algebra. Here are the lecture notes from the class.
Winter 2022. Math 732. Introduction to singularities. Here are the course lecture notes.
Fall 2021. Math 412. Introduction to modern algebra (Section 001).
Fall 2020. Math 217. Linear algebra (Section 014) and Math 565. Combinatorics and graph theory.
Fall 2019. Math 731. Introduction to Hodge theory and Math 420. Advanced linear algebra.
Winter 2018. Math 632. Algebraic geometry II.
Fall 2017. Math 631. Algebraic geometry I.
Winter 2017. Math 732. Rationality of algebraic varieties. Here are some notes that Takumi Murayama Live TeX-ed during lectures. For most of the course I also posted lecture notes here.
Fall 2016. Math 593. Algebra I.
Fall 2015. Math 565. Combinatorics and graph theory and Math. 412. Introduction to modern algebra.
Winter 2015. Math 412. Introduction to modern algebra.
Fall 2014. Math 711. Topics in birational geometry.
Winter 2014. Math 537. Introduction to differential topology.
Fall 2013. Math 731. Spaces of arcs and singularities in birational geometry.
Winter 2013. Math 732. Introduction to birational geometry.
Winter 2012. Math 732. Introduction to diophantine approximation on abelian varieties.
Winter 2011. Math 732. Zeta functions in algebraic geometry. Here are the lecture notes from the course.
Winter 2010. Math 632. Algebraic geometry II.
Fall 2009. Math 631. Algebraic geometry I.
Fall 2007. Math 631. Algebraic geometry I.
Winter 2006. Math 632. Algebraic geometry II (Schemes and cohomology). Here are the homework assignments and the problems covered in the discussion session.
Fall 2004, Winter 2005. Math 731 and 732. Topics in Algebraic Geometry I and II (Toric varieties). Here are the lecture notes, though some chapters are still missing. The first chapters are just expanded versions of the corresponding chapters in Bill Fulton's book "Introduction to toric varieties", using also Bill's lecture notes for a course he taught a few years ago.

Other activities

1. Qianyu Chen and I organized the Spring school Singularities in Ann Arbor, between May 13-17, 2024.
2. Paolo Aluffi, Dave Anderson, Milena Hering, Sam Payne, and I organized the conference Facets of algebraic geometry in Ann Arbor, over the week-end of October 18-20, 2019.
3. I was one of the organizers of the Spring 2019 MSRI program Algebraic geometry and moduli spaces. At the same time, there was a program in Derived algebraic geometry.
4. Daniel Erman, Claudiu Raicu, Greg Smith, and I organized the conference “A view towards algebraic geometry“, on the occasion of David Eisenbud's 70th birthday. This was held at the Harbor View Hotel on Martha’s Vineyard, between May 1-5, 2017.
5. Tommaso de Fernex, Karl Schwede, and I organized the Summer school and conference Higher dimensional algebraic geometry, held at the University of Utah, July 18-26, 2016.
6. Tommaso de Fernex, Brendan Hassett, Martin Olsson, Mihnea Popa, Richard Thomas, and I organized a successor to Seattle 2005 (and Santa Cruz 1995, Bowdoin 1985, Arcata 1974, Woods Hole 1964). This took place between July 13-31, 2015, at the University of Utah.

Some events in Ann Arbor


1. During the week of November 27-December 1, 2023, Junliang Shen gave the Spring Lectures in Algebraic Geometry.


Future and past conferences

1. École d'Été: Singularités en Géométrie Algébrique, Centre Henri Lebesgue, Rennes, May 26-June 13, 2025.
2. Higher Du Bois and higher rational singularities, AIM, Pasadena, October 28-November 1, 2024.
3. D-modules, local systems, and applications, CRM, Montreal, September 16-20, 2024.
4. The first algebraic geometry Gero Symposium, Gero (Japan), June 17-20, 2024.
5. Winter School on New Applications of Mixed Hodge Modules, Simons Center, Stony Brook, January 15-26, 2024.
6. Higher Dimensional Algebraic Geometry, celebrating James McKernan's 60th birthday, UCSD, January 10-14, 2024.
7. Mirror Symmetry and Hodge Ideals, Granada, November 6-10, 2023.

Photos

Here are some pictures taken in 2008, while hiking in Utah. The photo at the top of the page was taken by Maya Mustata (2023).

mmustata@umich.edu