Abstract:

Skin diseases typically appear as visible information-skin eruptions distributed across the body. However, the biological mechanisms underlying these manifestations are often inferred from fragmented, time-point-specific data such as skin biopsies. The challenge is further compounded for human-specific conditions like urticaria, where animal models are ineffective, leaving researchers to rely heavily on in vitro experiments and sparse clinical observations.

To overcome the current limitations in understanding the pathophysiology of skin diseases, we propose a novel framework that connects the visible morphology of skin eruptions with the underlying pathophysiological dynamics in vivo, using a multidisciplinary approach that integrates mathematical modeling, in vitro experiments, clinical data, and data science. Furthermore, we will introduce an innovative methodology that combines mathematical modeling with topological data analysis and machine learning, allowing for the estimation of patient-specific parameters directly from morphological patterns of skin eruptions. This framework offers a new pathway for personalized analysis and mechanistic insight into complex skin disorders.

Note: this is joint CMCF seminar

Poincaré–Einstein metrics play an important role in geometric analysis and mathematical physics, yet constructing new examples beyond the perturbative regime is difficult. In this talk, I will describe a class of four-dimensional Poincaré–Einstein manifolds that are conformal to Kähler metrics. These metrics admit a natural symmetry generated by a Killing field, which reduces the Einstein equations to a Toda-type system. This approach leads to existence and uniqueness results in the case of complex line bundles over surfaces of genus at least one. The construction produces large-scale, infinite-dimensional families of new Poincaré–Einstein metrics with conformal infinities of non-positive Yamabe type. This is joint work with Mingyang Li.

There is a unique line through 2 points in the plane, and a unique conic through 5. These counts generalize to a count of degree d rational curves in the plane passing through 3d-1 points. Surprisingly, the problem of determining these numbers is connected to mathematical physics, and it was not until the 1990's that it was completely solved. For example, Kontsevich determined them with a celebrated recursive formula. Such formulas are valid when you allow your curves to be defined with complex coefficients.  Some of the solutions may be real, or integral, or defined over Q[i], but the fixed count does not see the difference. Homotopy theory on the other hand, studies continuous deformations of maps. In its modern form, it provides a framework to study shape in many contexts, including the motivic homotopy of algebraic varieties. This talk will introduce some interactions of homotopy theory with the arithmetic of solutions to enumerative problems in geometry. We will use this to completely determine an enriched "count" of degree d rational curves passing through 3d-1 points over an arbitrary field of characteristic not 2 or 3. This enumeration is joint work with Erwan Brugallé and Johannes Rau.