L4

In this talk, I will present a result proved in my recent paper arXiv:2502.13580. I will discuss biharmonic maps between (and submanifolds of) conformally compact manifolds, a large class of complete manifolds generalizing hyperbolic space. After an introduction to conformally compact geometry, I will discuss one of the main results of the paper: if S is a properly embedded sub-manifold of a conformally compact manifold (N,h), and moreover S is transverse to the boundary and (N,h) has non-positive curvature, then S must be minimal. This result confirms a conjecture known as the Generalized Chen’s Conjecture, in the conformally compact context.

Symplectic capacities are symplectic invariants that measure the “size” of symplectic manifolds and are designed to capture phenomena of symplectic rigidity.

In this talk, I will focus on symplectic capacities of fiberwise convex domains in cotangent bundles. This setting provides a natural link to the systolic geometry of the base manifold. I will survey current results and discuss the variety of techniques used to compute symplectic capacities, ranging from billiard dynamics to pseudoholomorphic curves and symplectic homology. I will illustrate these techniques using disk tangent bundles of ellipsoids as an example.

I will discuss an ongoing joint work with Luigi Appolloni and Andrea Malchiodi concerning the above objects. Minimal surfaces are critical points of the area functional, which is analytic in this case, so we should expect critical points (minimal surfaces) to be either isolated or to belong to smooth nearby minimal foliations. On the other hand, the flat plane of multiplicity two in $\mathbb{R}^3$ can be (in compact regions) approximated by a blown-down catenoid, which will converge back to the plane with multiplicity two in the limit. Hence a plane of multiplicity two cannot be thought of as being isolated, or belonging solely to a smooth family, because there are “nearby” minimal surfaces of distinct topology weakly converging to it. We will nevertheless prove that, when the ambient manifold is closed and analytic, this type of local degeneration is impossible amongst closed and embedded minimal surfaces of bounded topology: such surfaces, even with multiplicity are either isolated or belong to smooth families of nearby minimal surfaces.  

We consider a class of Lagrangian sections L contained in certain Calabi-Yau Lagrangian fibrations (mirrors of toric weak Fano manifolds). We prove that a form of the Thomas-Yau conjecture holds in this case: L is isomorphic to a special Lagrangian section in this class if and only if a stability condition holds, in the sense of a slope inequality on objects in a set of exact triangles in the Fukaya-Seidel category. This agrees with general proposals by Li. On
surfaces and threefolds, under more restrictive assumptions, this result can be used to show a precise relation with Bridgeland stability, as predicted by Joyce. Based on arXiv:2505.07228 and arXiv:2508.17709.

I will discuss the paper "Construction of Gross-Neveu model using Polchinski flow equation" by Pawel Duch (https://arxiv.org/abs/2403.18562).

In this talk, I will introduce the nilpotent cohomological Hall algebra COHA(S, Z) of coherent sheaves on a smooth quasi-projective complex surface S that are set-theoretically supported on a closed subscheme Z. This algebra can be viewed as the "largest" algebra of cohomological Hecke operators associated with modifications along a subscheme Z of S. When S is the minimal resolution of an ADE singularity and Z is the exceptional divisor, I will describe how to characterize COHA(S, Z) in terms of the Yangian of the corresponding affine ADE quiver Q (based on joint work with Emanuel Diaconescu, Mauro Porta, Oliver Schiffmann, and Eric Vasserot, arXiv:2502.19445). More generally, I will discuss nilpotent COHAs arising from Bridgeland stability conditions on the bounded derived category of nilpotent representations of the preprojective algebra of Q, following joint work with Olivier Schiffmann and Parth Shimpi (arXiv:2511.08576).

Everything you have been taught about Turing patterns is wrong! (Well, not everything, but qualifying statements tend to weaken a punchy first sentence). Turing patterns are universally used to generate and understand patterns across a wide range of biological phenomena. They are wonderful to work with from a theoretical, simulation and application point of view. However, they have a paradoxical problem of being too easy to produce generally, whilst simultaneously being heavily dependent on the details. In this talk I demonstrate how to fix known problems such as small parameter regions and sensitivity, but then highlight a new set of issues that arise from usually overlooked issues, such as boundary conditions, initial conditions, and domain shape. Although we’ve been exploring Turing’s theory for longer than I’ve been alive, there’s still life in the old (spotty) dog yet.

We survey the life of a Turing pattern, from initial diffusive instability through the emergence of dominant spatial modes and to an eventual spatially heterogeneous pattern. While many mathematically ideal Turing patterns are regular, repeating in structure and remaining of a fixed length scale throughout space, in the real world there is often a degree of irregularity to patterns. Viewing the life of a Turing pattern through the lens of spatial modes generated by the geometry of the bounded space domain housing the Turing system, we discuss how irregularity in a Turing pattern may arise over time due to specific features of this space domain or specific spatial dependencies of the reaction-diffusion system generating the pattern.