This is a reading seminar organised by members of the Algebra and Geometry group in the Mathematics Department at Sapienza Università di Roma on topics in differential geometry.
This semester (Fall 2026) we meet on Monday, 14–16, Aula L, Dipartimento di Matematica G. Castelnuovo, Sapienza, and the focus of the seminar is on very recent existence and classification results for 4-dimensional Einstein metrics with a 2-torus symmetry.
09/11/2026
Yifan Chen
Hyperkähler gravitational instantons
02/11/2026
Simon Jubert
On Toric Hermitian ALF gravitational instantons
We will discuss Theorem A of the paper by Biquard–Gauduchon (arXiv:2112.12711), which provides a classification of smooth toric Hermitian ALF gravitational instantons. We will outline the main steps of the proof and illustrate the strategy using the Taub–NUT metric as a guiding example.
26/10/2026
Lorenzo Foscolo
Introduction to Einstein 4-manifolds and the recent results by Broder–Lafuente, Chu–Stern, Li–Sun, Ozuch
19/10/2026
Federico Giusti
Conifold transitions and their geometry
12/10/2026
Gayana Jayasinghe
Witten deformation and Morse inequalities
Witten deformation is a technique used to construct instanton complexes that categorify Morse polynomials and Lefschetz numbers. The better known among these is for the de Rham complex, but it extends to Dolbeault complexes and other Dirac operators in certain settings. Versions are also used in the study of gauge theory, pioneered by Taubes. The proofs involve semiclassical analysis and supersymmetry, and the methods are flexible enough to be used in different settings including singular spaces, foliated settings. I'll present a general framework, and specialize to one or two complexes for intuition, and present the proofs. If there's enough time, some applications in the study of gravitational instantons.