Radu Laza

My research lies at the intersection of algebra and geometry. I study algebraic varieties, which are geometric spaces defined by polynomial equations, and seek to understand how they can be classified and how they vary in families. Such families are often organized by another geometric space, called a moduli space, whose points represent the different varieties being studied. I am particularly interested in degenerations, which are situations in which a smoothly varying family develops singularities. These limiting objects often reveal the underlying principles governing the entire family. A major focus of my research is on Calabi–Yau threefolds, remarkable geometric spaces that are important both in mathematics and in theoretical physics, especially string theory.

My current research focuses on the geometry of moduli spaces of Calabi–Yau threefolds, including both their local and global theory; the classification and boundedness of hyper-Kähler manifolds; and the theory of higher Du Bois and higher rational singularities.

Previous research highlights include several aspects of the geometry of hyper-Kähler manifolds, such as the LLV decomposition, the LSV construction, and degenerations, as well as degenerations of Hodge structures and their applications to moduli. I have also worked on the birational geometry of moduli spaces of K3 surfaces; the moduli and periods of cubic fourfolds and threefolds; and the birational geometry of moduli spaces, particularly in connection with the Hassett–Keel program and its interplay with singularities.

• Deformations of Fibered Calabi–Yau Varieties (with B. Bakker, K. DeVleming, S. Filipazzi, J. Li, R. Svaldi, C. Wang, and J. Zhao), preprint 2026, submitted.
• Hodge Theory of Degenerations, (III): A Vanishing-Cycle Calculus for Non-Isolated Singularities (with M. Kerr), to appear in J. Singul.
• Isotrivial Lagrangian Fibrations of Compact Hyper-Kähler Manifolds (with Y.-J. Kim and O. Martin), J. Math. Pures Appl. (9) 205 (2026), Paper No. 103810, 49 pp.
• Hodge Theory of Degenerations, (II): Vanishing Cohomology and Geometric Applications (with M. Kerr), in Current Developments in Hodge Theory, Simons Symposia, Springer, Cham, 2025, 299–348.
• Deformations of Singular Fano and Calabi–Yau Varieties (with R. Friedman), J. Differential Geom. 131 (2025), no. 1, 65–131.
• Deformations of Calabi–Yau Varieties with Isolated Log Canonical Singularities (with R. Friedman), Int. Math. Res. Not. IMRN (2025), no. 10, Paper No. rnaf112, 36 pp.
• Higher Du Bois and Higher Rational Singularities (with R. Friedman; appendix by M. Saito), Duke Math. J. 173 (2024), no. 10, 1839–1881.
• Deformations of Calabi–Yau Varieties with k-Liminal Singularities (with R. Friedman), Forum Math. Sigma 12 (2024), Paper No. e59, 25 pp.
• Non-Isomorphic Smooth Compactifications of the Moduli Space of Cubic Surfaces (with S. Casalaina-Martin, S. Grushevsky, and K. Hulek), Nagoya Math. J. 254 (2024), 315–365.
• The Higher Du Bois and Higher Rational Properties for Isolated Singularities (with R. Friedman), J. Algebraic Geom. 33 (2024), no. 3, 493–520.
• Deformations of Some Local Calabi–Yau Manifolds (with R. Friedman), Épijournal Géom. Algébrique, Special Volume in Honor of Claire Voisin (2024), Article 18, 35 pp.
• Deformation of Rational Singularities and Hodge Structure (with M. Kerr and M. Saito), Algebr. Geom. 9 (2022), no. 4, 476–501.

• Cohomology of the Moduli Space of Cubic Threefolds and Its Smooth Models (with S. Casalaina-Martin, S. Grushevsky, and K. Hulek), Mem. Amer. Math. Soc. 282 (2023), no. 1395, v+100 pp.
• The LLV Decomposition of Hyper-Kähler Cohomology (with M. Green, Y.-J. Kim, and C. Robles), Math. Ann. 382 (2022), nos. 3–4, 1517–1590.
• Automorphisms and Periods of Cubic Fourfolds (with Z. Zheng), Math. Z. 300 (2022), no. 2, 1455–1507.
• Hodge Theory of Degenerations, (I): Consequences of the Decomposition Theorem (with M. Kerr; appendix by M. Saito), Selecta Math. (N.S.) 27 (2021), no. 4, Paper No. 71.
• Birational Geometry of the Moduli Space of Quartic K3 Surfaces (with K. O’Grady), Compos. Math. 155 (2019), no. 9, 1655–1710.
• On the Moduli Space of Pairs Consisting of a Cubic Threefold and a Hyperplane (with G. Pearlstein and Z. Zhang), Adv. Math. 340 (2018), 684–722.
• Remarks on Degenerations of Hyper-Kähler Manifolds (with J. Kollár, G. Saccà, and C. Voisin), Ann. Inst. Fourier (Grenoble) 68 (2018), no. 7, 2837–2882.
• A Hyper-Kähler Compactification of the Intermediate Jacobian Fibration Associated to a Cubic Fourfold (with G. Saccà and C. Voisin), Acta Math. 218 (2017), no. 1, 55–135.
• The KSBA Compactification for the Moduli Space of Degree Two K3 Pairs, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 2, 225–279.
• Log Canonical Models and Variation of GIT for Genus Four Canonical Curves (with S. Casalaina-Martin and D. Jensen), J. Algebraic Geom. 23 (2014), 727–764.
• Semi-Algebraic Horizontal Subvarieties of Calabi–Yau Type (with R. Friedman), Duke Math. J. 162 (2013), no. 12, 2077–2148.
• Moduli Space of Cubic Fourfolds via the Period Map, Ann. of Math. (2) 172 (2010), no. 1, 673–711.
• Moduli Space of Cubic Fourfolds (the GIT compactification), J. Algebraic Geom. 18 (2009), 511–545.
• The Moduli Space of Cubic Threefolds via Degenerations of the Intermediate Jacobian (with S. Casalaina-Martin), J. Reine Angew. Math. 633 (2009), 29–65.
• Deformations of Singularities and Variation of GIT Quotients, Trans. Amer. Math. Soc. 361 (2009), no. 4, 2109–2161 (published version of my Columbia thesis).

Expository Articles

• Classical Period Domains (with Z. Zhang), in Recent Advances in Hodge Theory, London Mathematical Society Lecture Note Series, Cambridge University Press, 2016.
• Perspectives on the Construction and Compactification of Moduli Spaces, in Compactifying Moduli Spaces, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser, 2016, 1–35.
• GIT and Moduli with a Twist, in Handbook of Moduli, vol. 2, Adv. Lect. Math. 25, International Press, 2013, 259–297.

Books Edited

• Calabi–Yau Varieties: Arithmetic, Geometry and Physics (with M. Schütt and N. Yui), lecture notes from the concentrated graduate courses held during the 2013 thematic program on Calabi–Yau varieties at the Fields Institute, Fields Institute Monographs, Springer, 2015.
• Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds (with M. Schütt and N. Yui), Fields Institute Communications, vol. 67, Springer, 2013; proceedings of the 2011 Fields Institute workshop on Calabi–Yau varieties and K3 surfaces.

• Higher Du Bois and Higher Rational Singularities, AIM workshop (with B. Dirks), Pasadena, California, October 28–November 1, 2024.
• School and Workshop on Moduli, K-Trivial Varieties, and Related Topics (with J.-M. Hwang and Y. Lee), Daejeon, South Korea, February 21–29, 2024.
• Discrete Groups and Moduli (with S. Kondo and S. Mukai), Nagoya University, June 17–20, 2019.
• Hodge Theory, Moduli, and Representation Theory, final conference of the Hodge Theory FRG, Stony Brook University, August 14–18, 2017.
• Positivity in Arithmetic and Geometry, spring school, Orsay, France, May 29–June 2, 2017.
• Hyper-Kähler Manifolds, Hodge Theory, and Chow Groups, Sanya, China, December 18–23, 2016.
• Calabi–Yau Varieties: Arithmetic, Geometry, and Physics, Herstmonceux Castle, East Sussex, United Kingdom, June 20–25, 2016.
• Algebraic Cycles and Moduli, Centre de recherches mathématiques, Montréal, June 2–8, 2016.
• Complex, p-Adic, and Logarithmic Hodge Theory and Their Applications, semester program, Simons Center for Geometry and Physics, March–April 2016.
• Moduli Spaces and Singularities in Algebraic and Riemannian Geometry, semester program, Simons Center for Geometry and Physics, August–November 2015.
• Invariants of Singularities in Zero and Positive Characteristics, mini-school, Stony Brook University, December 4, 2015.
• Collapsing Calabi–Yau Manifolds, Simons Center for Geometry and Physics, August 31–September 4, 2015.
• Topology of Algebraic Varieties, Institute for Advanced Study, 2014–2015.
• Perspectives on Complex Algebraic Geometry, Columbia University, May 22–25, 2015.
• New Techniques in Birational Geometry, Stony Brook University, April 6–10, 2015.
• K3, Enriques Surfaces, and Related Topics, Nagoya University, November 10–14, 2014.
• Thematic Program on Calabi–Yau Varieties, Fields Institute, Fall 2013.

• Lisa Marquand (Ph.D. 2023; birational automorphisms of hyper-Kähler manifolds), now Assistant Professor at Rutgers University.
• Olivier Martin (postdoctoral associate), now Assistant Professor at IMPA.
• Sasha Viktorova (Ph.D. 2022; singularities of cubic threefolds), now a postdoctoral fellow at the National Center for Theoretical Sciences in Taipei.
• Yoon-Joo Kim (Ph.D. 2022; Lagrangian fibrations on hyper-Kähler manifolds), now a Ritt Assistant Professor at Columbia University.
• François Greer (RTG postdoctoral associate), now Assistant Professor at Michigan State University.
• Adrian Brunyate (NSF postdoctoral fellow), now at the University of Georgia.
• Giulia Saccà (postdoctoral associate), now Associate Professor at Columbia University.
• Zheng Zhang (Ph.D.; geometric and motivic realizations of variations of Hodge structure), now Assistant Professor at ShanghaiTech University.
• Patricio Gallardo (Ph.D.; moduli of surfaces of general type, especially quintic surfaces), now Associate Professor at the University of California, Riverside.
• Kenneth Ascher (undergraduate honors thesis), now Associate Professor and Chancellor’s Fellow at the University of California, Irvine.
• Dave Jensen (postdoctoral associate), now Professor at the University of Kentucky.

Department of Mathematics
Stony Brook University
Stony Brook, NY 11794-3651