Francesco Sala | Master's Degree

In what follows, you may find some examples of possible topics for a Master's degree thesis under my supervision.

The proof of P=W conjecture by Hausel, Mellit, Minets, and Schiffmann

Description: Simpson's non-abelian Hodge correspondence establishes an homeomorphism between two significant moduli spaces in algebraic geometry: the Hitchin moduli space and the character variety. This correspondence implies an identification between their corresponding cohomologies. P=W conjecture, now proven, arose from the investigation of the relationship between non-abelian Hodge correspondence and deeper topological structures, such as the perverse filtration versus the weight filtration. The thesis will focus on the proof P=W conjecture given in arXiv:2209.05429, which employs techniques stemming from the theory of cohomological Hall algebras of zero-dimensional sheaves on a smooth surface.

References: the paper arXiv:2209.05429.

Cohomological Hall algebra of zero-dimensional sheaves on a smooth surface

References: the papers arXiv:1901.07641 and arXiv:2311.13415.

The elliptic Hall algebra

Description: Since its introduction by Burban and Schiffmann, the elliptic Hall algebra has played a prominent role in algebraic geometry, particularly in connection with the study of moduli spaces (such as the K-theory of Hilbert schemes of points on the affine plane), and in representation theory concerning quantum groups. The thesis will introduce the construction of the elliptic Hall algebra and its two presentations: the original one stemming from the theory of Hall algebras, and the second resembling Drinfeld's new realization of quantum groups. Prospective students should have a background in the theory of abelian categories and Lie theory, as well as an interest in geometric realizations of associative algebras.

References: the first two chapters of Schiffmann's lecture notes on Hall algebras, arXiv:math/0611617, Burban and Schiffmann's paper, arXiv:math/0505148, and Schiffmann's paper, arXiv:1004.2575.

Shuffle algebra realization of Hall algebras of curves

Description: Shuffle algebras are algebras of symmetric functions in infinitely many variables, with multiplication defined by a twisted version of symmetrization, where the twist is determined by a fixed function. The classical Hall algebra of a fixed abelian category (satisfying certain finite conditions) is an associative algebra. Its underlying vector space is generated by isomorphism classes of objects of the category, and the multiplication depends on the space of extensions between objects. The thesis will delve into the classical Hall algebra of the category of coherent sheaves on a smooth projective curve over a finite field, examining its combinatorial realization as a shuffle algebra. In cases where the curve has genus one, a more detailed characterization of the corresponding shuffle algebra will be analyzed. Prospective students should possess a background in the theory of abelian categories, Lie theory, and complex analysis, as well as an interest in the combinatorial realizations of associative algebras.

References: chapters 1, 2, and 4 of Schiffmann's lecture notes on Hall algebras, arXiv:math/0611617, Section 1 of Schiffmann's and Vasserot's paper, arXiv:1009.0678, and section 2 of Negut's paper, arXiv:1209.3349.

The elliptic Hall algebra and the K-theory and bounded derived category of coherent sheaves of Hilbert schemes of points on a smooth surface

Description: Nakajima and Grojnowski independently demonstrated that the cohomology group of the Hilbert scheme of points on a smooth quasi-projective complex surface S is isomorphic to the Fock space of a Heisenberg-Clifford algebra, modeled on the cohomology of S. The extension of this result to K-theory was initially explored by Schiffmann and Vasserot for the affine plane. Here, the elliptic Hall algebra takes the role of the Heisenberg-Clifford algebra. This result was later generalized to any smooth projective complex surface by Negut, who also categorized such a construction within the bounded derived categories. The thesis will focus on Negut's construction within the bounded derived categories of coherent sheaves, which provides a "weak" categorification of the elliptic Hall algebra. Prospective students should have a background in algebraic geometry and Lie theory, as well as an interest in the geometric realizations of associative algebras and the study of the geometry of moduli spaces through representation theory techniques.

References: sections 1, 2, and 3 of Schiffmann and Vasserot's paper, arXiv:0905.2555, Negut's paper, arXiv:1804.03645.

Two-dimensional cohomological Hall algebras of quivers

Description: The 2-dimensional cohomological Hall algebra (COHA) associated with a fixed quiver is an associative algebra. Its underlying vector space is the (Borel-Moore) homology of the cotangent of the stack of finite-dimensional representations of the quiver. Geometrically, it realizes half of a quantum group associated with the same quiver, known as the Yangian. Additionally, it acts on the cohomology of the corresponding Nakajima quiver varieties, offering a new perspective on the representation theory connected to these quiver varieties. The thesis will delve into the theory of COHAs of quivers, focusing on the characterization of their generators and the study of their (graded) dimensions. Prospective students should possess a background in algebraic geometry and Lie theory, as well as an interest in the algebraic structures which arise from the study of the topology of moduli stacks.

References: Schiffmann and Vasserot's paper, arXiv:1705.07488, and Bozec, Schiffmann, and Vasserot's paper, arXiv:1701.01797.