Cambridge

The question of profinite rigidity asks whether the isomorphism type of a group Γ can be recovered entirely from its finite quotients. In this talk, I will introduce the study of profinite rigidity in a different setting: the category of modules over a Noetherian domain Λ. I will explore properties of Λ-modules that can be detected in finite quotients and present two profinite rigidity theorems: one for free Λ-modules under a weak homological assumption on Λ, and another for all Λ-modules in the case when Λ is a Dedekind domain. Returning to groups, I will explain how these algebraic results yield new answers to profinite rigidity for certain classes of solvable groups. Time permitting, I will conclude with a sketch of future directions and ongoing collaborations that push these ideas further.

Due to connections to flat space holography, Carrollian geometry, physics and fluid dynamics have received an explosion of interest over the last two decades. In the Carrollian limit of vanishing speed of light c, relativistic fluids reduce to a set of PDEs called the Carrollian fluid equations. Although in general these equations are not well understood, and their PDE theory does not appear to have been studied, in dimensions 1+1 it turns out that there is a duality with the Galilean compressible Euler equations in 1+1 dimensions inherited from the isomorphism of the Carrollian (c to 0) and Galilean (c to infinity) contractions of the Poincar\'e algebra. Under this duality time and space are interchanged, leading to different dynamics in evolution. I will discuss recent work with N. Athanasiou (Thessaloniki), M. Petropoulos (Paris) and S. Schulz (Pisa) in which we establish the first rigorous PDE results for these equations by introducing a notion of Carrollian isentropy and studying the equations using Lax’s method and compensated compactness. In particular, I will explain that there is global existence in rough norms but finite-time blow-up in smoother norms.

I am interested in studying moduli spaces and associated enumerative invariants via degeneration techniques. Logarithmic geometry is a natural language for constructing and studying relevant moduli spaces. In this talk I  will explain the logarithmic Hilbert (or more generally Quot) scheme and outline how the construction helps study enumerative invariants associated to Hilbert/Quot schemes- a story we now understand well. Time permitting, I will discuss some challenges and key insights for studying moduli of stable vector bundles/ sheaves via similar techniques - a theory whose details are still being worked out. 

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.

I will discuss recent and ongoing work with Davesh Maulik that explains how Gromov-Witten invariants behave under simple normal crossings degenerations. The main outcome of the study is that if a projective manifold $X$ undergoes a simple normal crossings degeneration, the Gromov-Witten theory of $X$ is determined, via universal formulas, by the Gromov-Witten theory of the strata of the degeneration. Although the proof proceeds via logarithmic geometry, the statement involves only traditional Gromov-Witten cycles. Indeed, one consequence is a folklore conjecture of Abramovich-Wise, that logarithmic Gromov-Witten theory “does not contain new invariants”. I will also discuss applications of this to a conjecture of Levine and Pandharipande, concerning the relationship between Gromov-Witten theory and the cohomology of the moduli space of curves.

CANCELLED DUE TO ILLNESS

The response of the Greenland and Antarctic ice sheets to a changing climate is one of the largest sources of uncertainty in future sea level predictions.  The behaviour of the subglacial environment, where ice meets hard rock or soft sediment, is a key determinant in the flux of ice towards the ocean, and hence the loss of ice over time.  Predicting how ice sheets respond on a range of timescales brings together mathematical models of the elastic and viscous response of the ice, subglacial sediment and water and is a rich playground where the simplified models of the contact between ice, rock and ocean can shed light on very large scale questions.  In this talk we’ll see how these simplified models can make sense of a variety of field and laboratory data in order to understand the dynamical phenomena controlling the transient response of large ice sheets.

The relationship between the topological or homotopy-invariant properties of a symplectic manifold X and the set of possible immersed or embedded Lagrangian submanifolds of X is rich and mostly mysterious.  In 2020, D. Alvarez-Gavela, Y. Eliashberg, and D. Nadler proved that any Weinstein manifold (e.g. an affine variety) admitting a Lagrangian plane field retracts onto a Lagrangian submanifold with arboreal singularities (a certain class of singularities which can be described combinatorially). I will discuss work in progress with D. Alvarez-Gavela and T. Large investigating the other direction, in which we prove a partial converse to the AGEN result and show that most Weinstein manifolds do not admit such skeleta. This suggests that the Floer-theoretic invariants of some well-known open symplectic manifolds may be more complicated than expected.