Abstract: The Lefschetz (1, 1) theorem is a classical result that tells us that for any smoothprojective variety, a rational (1, 1) Hodge class comes from an algebraic cycle of codimension1.⁠ ⁠Counter-examples to the obvious generalization for singular varieties exist. Inspired byTotaro, in this talk, I will give a modification of the statement of Lefschetz (1, 1) and show thatit is satisfied by several singular…

Abstract: This talk concerns the birational geometry of cAn singularities and their curve-counting invariants through noncommutative methods. I introduce generalised GV invariants for crepant partial resolutions, verify Toda’s formula in this setting, and show these invariants are determined by the isomorphism class of the contraction algebra. Turning to crepant resolutions, I give intrinsic definitions of Type-A potentials on…

Abstract: I will introduce an equivalence relation on Lagrangians in a symplectic manifold called algebraic Lagrangian cobordism. Its purpose is to mirror algebraic equivalence of cycles, in the same way Sheridan—Smith observed that cylindrical Lagrangian cobordisms are “mirror” to rational equivalences of cycles.
I then show that this equivalence relation captures non-trivial symplectic geometry through a Lagrangian…

Abstract: We study semiorthogonal decompositions of derived categories of smooth projective varieties and their behavior under birational transformations. Motivated by Kuznetsov’s conjecture on cubic fourfolds and Kontsevich’s vision of canonical decompositions, we introduce the notion of G-atomic theory: canonical, mutation-equivalence classes of G-invariant semiorthogonal decompositions compatible with derived contractions with respect to a group G-action. We prove…