ALF gravitational instantons, of which the Taub-NUT and Atiyah-Hitchin metrics are prototypes, are the complete non-compact hyperkähler 4-manifolds with cubic volume growth. Examples have been known since the 1970's, but a complete classification was only given around 10 years ago. In this talk, I will present joint work with Haskins and Nordström where we extend some of these results to complete non-compact 7-manifolds with holonomy G2 and an asymptotic geometry, called ALC (asymptotically locally conical), that generalises to higher dimension the asymptotic geometry of ALF spaces.
I will describe a generalisation of the Seiberg-Witten equations to a Spin-c manifold of any dimension. The equations are for a U(1) connection A and spinor \phi and also an odd-degree differential form b (of inhomogeneous degree). Clifford action of the form is used to perturb the Dirac operator D_A. The first equation says that (D_A+b)(\phi)=0. The second equation involves the Weitzenböck remainder for D_A+b, setting it equal to q(\phi), where q(\phi) is the same quadratic term which appears in the usual Seiberg-Witten equations. This system is elliptic modulo gauge in dimensions congruent to 0,1 or 3 mod 4. In dimensions congruent to 2 mod 4 one needs to take two copies of the system, coupled via b. I will also describe a variant of these equations which make sense on manifolds with a Spin(7) structure. The most important difference with the familiar 3 and 4 dimensional stories is that compactness of the space of solutions is, for now at least, unclear. This is joint work with Partha Ghosh and, in the Spin(7) setting, Ragini Singhal.
In this talk, we will make an explicit link between self-dual Yang-Mills instantons on the Taub-NUT space, and G2-instantons on the BGGG space, by displaying the latter space as a fibration by the former. In doing so, we will discuss analysis on non-compact manifolds, circle symmetries, and a new method of constructing solutions to quadratically singular ODE systems. This talk is based on joint work with Matt Turner: https://arxiv.org/pdf/2409.03886.
I’ll tell you about some of my favorite algebraic varieties, which are beautiful in their own right, and also have some dramatic applications to algebraic combinatorics. These include the top-heavy conjecture (one of the results for which June Huh was awarded the Fields Medal), as well as non-negativity of Kazhdan—Lusztig polynomials of matroids.
Namikawa has shown that the functor of flat graded Poisson deformations of a conic symplectic singularity is unobstructed and pro-representable. In a subsequent work, Losev showed that the universal Poisson deformation admits, a quantization which enjoys a rather remarkable universal property. In a recent work, we have repackaged the latter theorem as an expression of the representability of a new functor: the functor of quantizations. I will describe how this theorem leads to an easy proof of the existence of a universal equivariant quantizations, and outline a work in progress in which we describe a presentation of a rather complicated quantum Hamiltonian reduction: the finite W-algebra associated to a nilpotent element in a classical Lie algebra. The latter result hinges on new presentations of twisted Yangians.
Given a singularity with a crepant resolution, a symmetry of the derived
category of coherent sheaves on the resolution may often be constructed
using the formalism of spherical functors. I will introduce this, and
new work (arXiv:2409.19555) on general constructions of such symmetries
for hypersurface singularities. This builds on previous results with
Segal, and is inspired by work of Bodzenta-Bondal.
Joint work with Paul Hacking (U Mass Amherst). We first explain how to
prove homological mirror symmetry for a maximal normal crossing
Calabi-Yau surface Y with split mixed Hodge structure. This includes the
case when Y is a type III K3 surface, in which case this is used to
prove a conjecture of Lekili-Ueda. We then explain how to build on this
to prove an HMS statement for K3 surfaces. On the symplectic side, we
have any K3 surface (X, ω) with ω integral Kaehler; on the algebraic
side, we get a K3 surface Y with Picard rank 19. The talk will aim to be
accessible to audience members with a wide range of mirror symmetric
backgrounds.
A number of moduli problems are, via Hodge theory, closely related to
ball quotients. In this situation there is often a choice of possible
compactifications such as the GIT compactification´and its Kirwan
blow-up or the Baily-Borel compactification and the toroidal
compactificatikon. The relationship between these compactifications is
subtle and often geometrically interesting. In this talk I will discuss
several cases, including cubic surfaces and threefolds and
Deligne-Mostow varieties. This discussion links several areas such as
birational geometry, moduli spaces of pointed curves, modular forms and
derived geometry. This talk is based on joint work with S.
Casalaina-Martin, S. Grushevsky, S. Kondo, R. Laza and Y. Maeda.
A well-known problem in algebraic geometry is to construct smooth projective Calabi--Yau varieties $Y$. In the smoothing approach, we construct first a degenerate (reducible) Calabi--Yau scheme $V$ by gluing pieces. Then we aim to find a family $f\colon X \to C$ with special fiber $X_0 = f^{-1}(0) \cong V$ and smooth general fiber $X_t = f^{-1}(t)$. In this talk, we see how infinitesimal logarithmic deformation theory solves the second step of this approach: the construction of a family out of a degenerate fiber $V$. This is achieved via the logarithmic Bogomolov--Tian--Todorov theorem as well as its variant for pairs of a log Calabi--Yau space $f_0\colon X_0 \to S_0$ and a line bundle $\mathcal{L}_0$ on $X_0$.