My research focuses on existence of constant scalar curvature Kähler (cscK) metrics on compact Kähler manifolds, and related notions of stability. For my PhD thesis my work was centered around developing a theory of K-stability for arbitrary (possibly non-projective) Kähler manifolds, that extends the classical theory for polarized manifolds. More recently, I have been studying the connections between algebraic stability notions (e.g. K-stability, J-stability) and analytic stability notions (e.g. geodesic stability, properness of the Mabuchi K-energy and Donaldson's J-functional). Recent projects include proving existence of cscK metrics on smooth minimal models, work on openness of uniform K-stability for general polarizations, and the study of effective conditions to test existence of solutions to several geometric PDE on surfaces, including the deformed Hermitian Yang Mills and Z-critical equations. 

Keywords: Uniform K-stability, cscK metrics, cohomological test configurations, Monge-Ampère equations, geodesic stability, J-stability, Donaldson's J-equation, deformed Hermitian Yang Mills equation, Z-critical equation, properness conjectures, Yau-Tian-Donaldson conjecture.

Publications and preprints: 

1. The set of destabilizing curves for deformed Hermitian Yang-Mills and Z-critical equations on surfaces (with S. Khalid). Preprint [arXiv:2210.06374] (2022). International Mathematics Research Notices, Volume 2024, Issue 7, April 2024, p. 5773-5814. 

2. Openness of uniform K-stability in the Kähler cone. 

Preprint [arXiv:2011.14806] (2020). 

3. Existence of cscK metrics on smooth minimal models. 

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 23 (1) (2022), 223 - 232. 

See [arXiv:2004.02832] or the journal version.

4. Optimal lower bounds for Donaldson's J-functional. 

Advances in Mathematics, Volume 374 (2020), to appear. DOI: 10.1016/j.aim.2020.107271. 

See [arXiv:1907:01486] or the journal version.

5. A partial comparison of stability notions in Kähler geometry. 

Moduli of K-stable varieties, Springer INdAM series, Volume 31 (2019), Springer International Publishing, ISBN 978-3-030-13157-9. See [arXiv:1812.10958] or the journal version. 

6. On K-polystability of cscK manifolds with transcendental cohomology class (with an appendix by R. Dervan). International Mathematics Research Notices (IMRN), Volume 2020, Issue 9 (2020), 2769 - 2817. 

See [arXiv:1711.11482] or the journal version.

7. K-semistability of cscK manifolds with transcendental cohomology class. 

Journal of Geometric Analysis, Volume 28, Issue 4 (2018), 2927-2960. 

See [arXiv:1601.07659] or the journal version. 

PhD thesis "K-stabilité et variétés kähleriennes avec classe transcendante” (2017), under the supervision of S. Boucksom and V. Guedj. Available in English here. 

Master’s thesis ”Bergman kernels for high powers of positive line bundles” (2014), under the supervision of H. Samuelsson Kalm and R. Berman. 

Invited conference talks: 

Gothenburg Complex Geometry 2024, Gothenburg, August 2024 (upcoming).

Nordic complex analysis meeting (NORDAN) 25th anniversary edition, Östanskär, Sweden, May 2024.

New equivariant methods in algebraic and differential geometry, University of Cambridge, UK, February 2024.

Stability in Kähler geometry (Geometry & TACoS synchronous online conference), May 2023. 

Cohomology of Complex Manifolds and Special Structures - III, Trento, Italy, January 2023.  

Nordic Congress of Mathematicians XXVIII, Complex Analysis and Geometry session, Aalto University, Helsinki, Finland. August 2022. 

Recent advances in Complex and Symplectic geometry, Parma, Italy, January 2022. 

Cohomology of Complex Manifolds and Special Structures - II, Centro Internazionale per la Ricerca Matematica (CIRM), Levico Terme, Italy, July 2021. 

The 6th Workshop of Complex Geometry and Lie Groups, Toki Messe, Niigata, Japan. May 2020, postponed to February 2021.

Trends in Modern Geometry, Tsinghua University, Beijing, China. June 2019.

Newcastle Complex Geometry Workshop, University of Newcastle, England. July 2018.

Second Symposium of Geometry and Differential Equations, Shanghai Tech University, Shanghai, China. June 2018. 

Constant scalar curvature metrics in Kahler and Sasaki geometry, Centre International de Rencontres Mathématiques (CIRM), Marseille, France, January 2018.

Moduli of K-stable varieties, Università di Roma “La Sapienza”, Italy,  July 2017.

Recent Advances in Kähler Geometry, Notre Dame, USA.  June 2017.

Complex Analysis and Geometry - XXIII, Centro Internazionale per la Ricerca Matematica (CIRM), Levico Terme, Italy, June 2017.

Rencontre EMARKS "Research in Pairs", Université Claude Bernard Lyon 1, France. May 2017.

Limits and Flows in Kähler Geometry, Université de Nantes, France.  April 2017.

Some selected seminar talks: 

Jussieu "Séminare d'Analyse et Géometrie", Paris, France. May 2022.

Newcastle Algebraic Geometry Seminar (online), Newcastle, England. April 2022.

IMAR Geometry Seminar (online), Bucharest, Romania. April 2022.

Jagiellonian University Geometry Seminar (online), Krakow, Poland. November 2021.

Aarhus University Geometry Seminar, Aarhus, Denmark. May 2021.

Ludwig-Maximilians-Universität Geometry Seminar (online), Munich, Germany. November 2020. 

Mathematics at Shanghai (online), Shanghai, China. August 2020.

Università degli Studi di Firenze Geometry Seminar, Italy. December 2019.

IGAP Kähler Geometry Seminar, Trieste, Italy. November 2019.

University of Ljubljana Geometry Seminar, Slovenia. April 2019. 

ICTP/SISSA Kähler Geometry Seminar, Trieste, Italy. July 2019.

ICTP/SISSA Kähler Geometry Seminar, Trieste, Italy. January 2019.

ICTP Mathematics Seminar, Trieste, Italy. November 2018.

KASS Complex Analysis Seminar, Chalmers University of Technology, Göteborg, Sweden. September 2018. 

BICMR Peking University Mathematics Seminar, China. Part I. March 2018.

BICMR Peking University Mathematics Seminar, China. Part II. March 2018.

LMBA Geometry and Topology Seminar, Brest, France. March 2017. 

Université Libre de Bruxelles Geometry Seminar, Belgium. October 2016. 

École Polytechnique Mathematics Seminar, Paris, France. April 2016.

CMLS/CMAP PhD Student Seminar, École Polytechnique, Paris. April 2016.

Picard PhD Student Seminar, Université Paul Sabatier, Toulouse, France. March 2016.

Université Paul Sabatier Complex Geometry Seminar, Toulouse, France. February 2016.