A recurring theme in differential geometry is understanding when a set of geometric conditions uniquely characterizes an object. In this talk, I will discuss this question for minimal surfaces in the unit ball, focusing on those meeting the boundary orthogonally. I will explain recent joint work with Jonathan Zhu in which we prove that every properly embedded, free-boundary, minimal annulus is congruent to the critical catenoid. I will also describe closely related results for axially symmetric Alt-Caffarelli cones and free-boundary minimal annuli in spherical caps, and discuss connections with other uniqueness results in differential geometry. 

Professor Raj Shukla will talk about: 'Fixed and Adaptive Topological DeepONets: Functional Measurements on Hausdorff Locally Convex Spaces'

Adaptive Topological DeepONets: Functional Measurements in Locally Convex Spaces
Deep Operator Networks (DeepONets) typically encode an input function through its point values on a fixed discretization. Building on the Topological DeepONet framework of Ismailov, we replace point samples with continuous linear functionals drawn from the continuous dual of a Hausdorff locally convex space. The topology of this space is generated by a point-separating family of seminorms rather than a single norm. Within this setting, we develop both fixed and adaptive functional measurement systems. These measurements are combined with the coefficient-space Two-Step training procedure of Lee and Shin, and a training-only decoder with regularization stabilizes the adaptive coordinates.
On the theoretical side, we derive a discrete error decomposition that separates measurement error, output-basis error, and neural-approximation error, together with a refinement based on Barron-type approximation rates.
We evaluate the framework on the antiderivative operator, a non-normable locally convex input space, heterogeneous Darcy flow, a controlled operator, and both fixed-time and time-evolving Navier–Stokes vorticity operators. For heterogeneous Darcy flow, the functional models keep nearly resolution-independent errors of 5.5–5.6% on unseen grids. For the controlled problem, adaptive measurements reduce the mean error below 1.2%. For fixed-time Navier–Stokes, the Adaptive Topological DeepONet is the most accurate DeepONet-based model, reaching a mean relative L2 error of 1.685% ± 0.017% with only 128 functional coordinates. A Fourier neural operator (FNO) of comparable size achieves a lower error of 0.832% ± 0.172%, but it needs the full 64 × 64 input field, twice the training time, and 10.7 times the peak GPU memory.
Overall, the formulation yields compact, interpretable, and discretization-portable coordinates in the continuous dual space, including for non-normable input spaces.