Jeffrey Steven Case
The Pennsylvania State University
Local and global conformal invariants
of submanifolds.
I will describe a systematic method for constructing
local and global conformal invariants of submanifolds,
with three applications. First, I will describe
higher-order analogues of the Willmore energy and
their role in computing global invariants of minimal
submanifolds of conformally compact Einstein manifolds.
In particular, this leads to new ways to characterize
totally geodesic submanifolds of hyperbolic space.
Second, I will explain the role of an extrinsic
analogue of the Yamabe constant in computing the
sharp constant in the L2-Michael-Simon
inequality. Third, I will explain the role of this
extrinsic Yamabe constant in obstructing the existence
of compact minimal submanifolds in certain noncompact
manifolds. In particular, this gives a new perspective
on nonexistence results for the singular
σk-Loewner-Nirenberg problem.
This talk is based on joint works with Ayush Khaitan,
Yueh-Ju Lin, Dawit Mengesha, Samuel Perez-Ayala,
Aaron Tyrrell, and Wei Yuan.
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Wei Kang
University of California, Santa Cruz
Data-Driven Approximation of
Hamilton–Jacobi–Bellman Equations for
High-Dimensional Nonlinear Systems
In optimal control theory, solutions of the
Hamilton–Jacobi–Bellman (HJB) equation, which
underpin the design of optimal feedback control
laws, are closely related to invariant manifolds
of an associated Hamiltonian dynamical system.
However, owing to the curse of dimensionality,
the computational cost of approximating these
manifolds typically grows exponentially with
the dimension of the problem.
In the first part of this talk, we introduce a
data-driven approach for approximating semiglobal
solutions of HJB equations for general
high-dimensional nonlinear systems. The value
function is represented by a neural network
trained on data generated without discretizing
the state space. To improve training effectiveness
and data efficiency, we exploit the Pontryagin
maximum principle and use partially trained
neural networks to guide adaptive data generation.
In the second part, time permitting, we turn to
the theoretical complexity of neural networks
and its relationship to the compositional structure
of target functions. We present a mathematically
rigorous upper bound on network complexity,
highlighting how function composition can influence
the complexity of neural network representations.
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Magdalena Toda
Missouri State University
Geometry, Energy, and Life:
From Curvature to Biological Form
Having recently turned 60, I will use this
occasion to reflect on some of the mathematical
ideas that have shaped my work and, more
importantly, on the questions that continue to
motivate it. A recurring theme has been the
geometry of curves and surfaces and the role
of curvature in determining their shape.
I will discuss how curvature energies
(particularly Willmore-type, p-Willmore, and
Helfrich-Canham energies) provide a natural
bridge between geometric analysis and the
modeling of physical and biological structures.
Variational problems associated with these
energies lead to rich geometric questions
concerning critical shapes, stability,
regularity, and the behavior of curvature,
while also providing mathematical models
for elastic membranes and biological forms.
I will illustrate these connections through
selected examples from my work, including
generalized curvature functionals, elastic
curves and surfaces, and models of red blood
cell geometry. Particular attention will be
given to the interplay between geometric
regularity and finite curvature energy,
and to what biologically meaningful shapes
can teach us about the mathematical assumptions
imposed on variational surface models.
This is not intended as a retrospective,
but rather as a look at a continuing journey
between geometry and applications. For the
next decades, I remain very interested in
the problems that are still open, and I hope
to continue exploring how curvature, energy,
and geometry can help us understand the forms
found in mathematics, physics, and life.
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Maximilien Hallgren
University of Arizona
Talk title to be announced
Abstract forthcoming.
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Francesco Maggi
The University of Texas at Austin
Talk title to be announced
Abstract forthcoming.
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