6 research outputs found
Integral Arnol'd Conjecture
We explain how to adapt the methods of Abouzaid-McLean-Smith to the setting
of Hamiltonian Floer theory. We develop a language around equivariant
``-manifolds'', which are a type of manifold-with-corners
that suffices to capture the combinatorics of Floer-theoretic constructions. We
describe some geometry which allows us to straightforwardly adapt Lashofs's
stable equivariant smoothing theory and Bau-Xu's theory of FOP-perturbations to
-manifolds. This allows us to compatibly smooth global
Kuranishi charts on all Hamiltonian Floer trajectories at once, in order to
extract a Floer complex and prove the Arnol'd conjecture over the integers. We
also make first steps towards a further development of the theory, outlining
the analog of bifurcation analysis in this setting, which can give short
independence proofs of the independence of Floer-theoretic invariants of all
choices involved in their construction.Comment: 56 page
Holomorphic Floer Theory and the Fueter Equation
We outline a proposal for a -category associated to a
hyperk\"ahler manifold , which categorifies the subcategory of the Fukaya
category of generated by complex Lagrangians. Morphisms in this
-category are formally the Fukaya--Seidel categories of holomorphic
symplectic action functionals. As such, is based on counting
maps to satisfying the Fueter equation with boundary values on holomorphic
Lagrangians. We make the first step towards constructing this category by
establishing some basic analytic results about Fueter maps, such as the energy
bound and maximum principle. When is the cotangent bundle of a
K\"ahler manifold and are the zero section and the graph of
the differential of a holomorphic function , we prove that
all Fueter maps correspond to the complex gradient trajectories of in ,
which relates our proposal to the Fukaya--Seidel category of . This is a
complexification of Floer's theorem on pseudo-holomorphic strips in cotangent
bundles. Throughout the paper, we suggest problems and research directions for
analysts and geometers that may be interested in the subject.Comment: 81 pages, 14 figures. Submitted version. Parts of Section 2 moved to
appendix and several small updates mad
Renormalizing Diffusion Models
We explain how to use diffusion models to learn inverse renormalization group
flows of statistical and quantum field theories. Diffusion models are a class
of machine learning models which have been used to generate samples from
complex distributions, such as the distribution of natural images. These models
achieve sample generation by learning the inverse process to a diffusion
process which adds noise to the data until the distribution of the data is pure
noise. Nonperturbative renormalization group schemes in physics can naturally
be written as diffusion processes in the space of fields. We combine these
observations in a concrete framework for building ML-based models for studying
field theories, in which the models learn the inverse process to an
explicitly-specified renormalization group scheme. We detail how these models
define a class of adaptive bridge (or parallel tempering) samplers for lattice
field theory. Because renormalization group schemes have a physical meaning, we
provide explicit prescriptions for how to compare results derived from models
associated to several different renormalization group schemes of interest. We
also explain how to use diffusion models in a variational method to find ground
states of quantum systems. We apply some of our methods to numerically find RG
flows of interacting statistical field theories. From the perspective of
machine learning, our work provides an interpretation of multiscale diffusion
models, and gives physically-inspired suggestions for diffusion models which
should have novel properties.Comment: 69+15 pages, 8 figures; v2: figure and references added, typos
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Diffusion with Forward Models: Solving Stochastic Inverse Problems Without Direct Supervision
Denoising diffusion models are a powerful type of generative models used to
capture complex distributions of real-world signals. However, their
applicability is limited to scenarios where training samples are readily
available, which is not always the case in real-world applications. For
example, in inverse graphics, the goal is to generate samples from a
distribution of 3D scenes that align with a given image, but ground-truth 3D
scenes are unavailable and only 2D images are accessible. To address this
limitation, we propose a novel class of denoising diffusion probabilistic
models that learn to sample from distributions of signals that are never
directly observed. Instead, these signals are measured indirectly through a
known differentiable forward model, which produces partial observations of the
unknown signal. Our approach involves integrating the forward model directly
into the denoising process. This integration effectively connects the
generative modeling of observations with the generative modeling of the
underlying signals, allowing for end-to-end training of a conditional
generative model over signals. During inference, our approach enables sampling
from the distribution of underlying signals that are consistent with a given
partial observation. We demonstrate the effectiveness of our method on three
challenging computer vision tasks. For instance, in the context of inverse
graphics, our model enables direct sampling from the distribution of 3D scenes
that align with a single 2D input image.Comment: Project page: https://diffusion-with-forward-models.github.io
Renormalization Group Flow as Optimal Transport
We establish that Polchinski's equation for exact renormalization group flow
is equivalent to the optimal transport gradient flow of a field-theoretic
relative entropy. This provides a compelling information-theoretic formulation
of the exact renormalization group, expressed in the language of optimal
transport. A striking consequence is that a regularization of the relative
entropy is in fact an RG monotone. We compute this monotone in several
examples. Our results apply more broadly to other exact renormalization group
flow equations, including widely used specializations of Wegner-Morris flow.
Moreover, our optimal transport framework for RG allows us to reformulate RG
flow as a variational problem. This enables new numerical techniques and
establishes a systematic connection between neural network methods and RG flows
of conventional field theories.Comment: 34+11 pages, 4 figures; v2: typos fixed, references and comments
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