Popa received his bachelor's degree in 1996 from the University of Bucharest. He studied mathematics at the University of California, Los Angeles from 1996 to 1997 and then in 2001, he received his Ph.D. from the University of Michigan under the supervision of Robert Lazarsfeld. His thesis was titled Linear Series on Moduli Spaces of Vector Bundles on Curves. From 2001 to 2005, Popa was a Benjamin Peirce Assistant Professor at Harvard University and from 2005 to 2007 an assistant professor at the University of Chicago. He joined the University of Illinois at Chicago as an associate professor in 2007 and became a full professor in 2011. In 2014 he became a professor at Northwestern University.
Since 1996 Popa is a member of the Institute of Mathematics of the Romanian Academy. He was an AMS Centennial Fellow in 2005–2007, a Sloan Research Fellow in 2007–2009, and a Simons Fellow in 2015–2016. In 2015 he became a fellow of the American Mathematical Society. In 2018 he was an Invited Speaker of the International Congress of Mathematicians in Rio de Janeiro.
- Pareschi, Giuseppe; Popa, Mihnea (2003). "Regularity on abelian varieties I". Journal of the American Mathematical Society. 16: 285–302. arXiv:math/0110003. doi:10.1090/S0894-0347-02-00414-9. MR 1949161.
- with Gavril Farkas: "Effective divisors on , curves on K3 surfaces, and the Slope Conjecture", Journal of Algebraic Geometry, Vol. 14, 2005, pp. 241–267. Arxiv
- with Lawrence Ein, Robert Lazarsfeld, Mircea Mustaţă, Michael Nakamaye: "Asymptotic invariants of base loci", Ann. Inst. Fourier, Volume 56, 2006, pp. 1701–1734. Arxiv
- with Robert Lazarsfeld: Derivative complex, BGG correspondence, and numerical inequalities for compact Kähler manifolds, Inventiones Mathematicae, Vol. 182, 2010, pp. 605–633. Arxiv
- with Christian Schnell: Generic vanishing theory via mixed Hodge modules, Forum of Mathematics, Sigma 1, 2013, pp. 1–60. Arxiv
- with Christian Schnell: Kodaira dimension and zeros of holomorphic one-forms, Ann. of Math., Vol. 179, 2014, pp. 1–12. Arxiv
- Kodaira-Saito vanishing and applications, L'Enseignement Mathémathique, Vol. 62, 2016, pp. 49–89. Arxiv
- Positivity for Hodge modules and geometric applications, in Proceedings of Symposia in Pure Mathematics, Vol. 97, Part I, Algebraic Geometry: Salt Lake City 2015, pp. 555–584. Arxiv
- with Mircea Mustaţă: Hodge Ideals, Memoirs AMS 2016, Arxiv
- with C. Schnell: Viehweg's hyperbolicity conjecture for families with maximum variation, Invent. Math., Vol. 208, 2017, pp. 677–713, Arxiv
- with Giuseppe Pareschi, Christian Schnell: Hodge modules on complex tori and generic vanishing for compact Kähler manifolds, Geom. Topol., Volume 21, 2017, pp. 2419–2460 Arxiv
- with M. Mustaţă: Hodge ideals for -Divisors, Parts I,II, Arxiv 2018, part I, Arxiv, Part II