270 research outputs found
Polyfolds: A First and Second Look
Polyfold theory was developed by Hofer-Wysocki-Zehnder by finding
commonalities in the analytic framework for a variety of geometric elliptic
PDEs, in particular moduli spaces of pseudoholomorphic curves. It aims to
systematically address the common difficulties of compactification and
transversality with a new notion of smoothness on Banach spaces, new local
models for differential geometry, and a nonlinear Fredholm theory in the new
context. We shine meta-mathematical light on the bigger picture and core ideas
of this theory. In addition, we compiled and condensed the core definitions and
theorems of polyfold theory into a streamlined exposition, and outline their
application at the example of Morse theory.Comment: 62 pages, 2 figures. Example 2.1.3 has been modified. Final version,
to appear in the EMS Surv. Math. Sc
Equivariant geometric learning for digital rock physics: estimating formation factor and effective permeability tensors from Morse graph
We present a SE(3)-equivariant graph neural network (GNN) approach that
directly predicting the formation factor and effective permeability from
micro-CT images. FFT solvers are established to compute both the formation
factor and effective permeability, while the topology and geometry of the pore
space are represented by a persistence-based Morse graph. Together, they
constitute the database for training, validating, and testing the neural
networks. While the graph and Euclidean convolutional approaches both employ
neural networks to generate low-dimensional latent space to represent the
features of the micro-structures for forward predictions, the SE(3) equivariant
neural network is found to generate more accurate predictions, especially when
the training data is limited. Numerical experiments have also shown that the
new SE(3) approach leads to predictions that fulfill the material frame
indifference whereas the predictions from classical convolutional neural
networks (CNN) may suffer from spurious dependence on the coordinate system of
the training data. Comparisons among predictions inferred from training the CNN
and those from graph convolutional neural networks (GNN) with and without the
equivariant constraint indicate that the equivariant graph neural network seems
to perform better than the CNN and GNN without enforcing equivariant
constraints
Unfolded Seiberg-Witten Floer spectra, II: Relative invariants and the gluing theorem
We use the construction of unfolded Seiberg-Witten Floer spectra of general
3-manifolds defined in our previous paper to extend the notion of relative
Bauer-Furuta invariants to general 4-manifolds with boundary. One of the main
purposes of this paper is to give a detailed proof of the gluing theorem for
the relative invariants.Comment: 75 pages. Comments are welcomed. v3. Typos fixed. To appear in
Journal of Differential Geometr
Moduli of map germs, Thom polynomials and the Green-Griffiths conjecture
This survey paper is based on my IMPANGA lectures given in the Banach Center,
Warsaw in January 2011. We study the moduli of holomorphic map germs from the
complex line into complex compact manifolds with applications in global
singularity theory and the theory of hyperbolic algebraic varieties
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