525 citations · 1k across the 33 of their papers we have counts for
35 papers
A Teacher-Student Perspective on the Dynamics of Learning Near the Optimal Point
Carlos Couto, José Mourão, Mário A. T. Figueiredo +1
Near an optimal learning point of a neural network, the learning performance of gradient descent dynamics is dictated by the Hessian matrix of the loss function with respect to the…
Fibering polarizations and Mabuchi rays on symmetric spaces of compact type
Thomas Baier, Ana Cristina Ferreira, Joachim Hilgert +2
In this paper, we describe holomorphic quantizations of the cotangent bundle of a symmetric space of compact type , along Mabuchi rays of $…
Wide neural networks: From non-gaussian random fields at initialization to the NTK geometry of training
Luís Carvalho, João Lopes Costa, José Mourão +1
Recent developments in applications of artificial neural networks with over parameters make it extremely important to study the large behaviour of such networks. Mo…
Quantization in fibering polarizations, Mabuchi rays and geometric Peter--Weyl theorem
Thomas Baier, Joachim Hilgert, Oğuzhan Kaya +2
In this paper we use techniques of geometric quantization to give a geometric interpretation of the Peter--Weyl theorem. We present a novel approach to half-form corrected geometri…
Laughlin states change under large geometry deformations and imaginary time Hamiltonian dynamics
Gabriel Matos, Bruno Mera, José M. Mourão +2
We study the change of the Laughlin states under large deformations of the geometry of the sphere and the plane, associated with Mabuchi geodesics on the space of metrics with Hami…
Partial coherent state transforms, -invariant Kähler structures and geometric quantization of cotangent bundles of compact Lie groups
José M. Mourão, João P. Nunes, Miguel B. Pereira
In this paper, we study the analytic continuation to complex time of the Hamiltonian flow of certain -invariant functions on the cotangent bundle of a compact connected…
Holomorphic fractional Fourier transforms
William D. Kirwin, José Mourão, João P. Nunes +1
The Fractional Fourier Transform (FrFT) has widespread applications in areas like signal analysis, Fourier optics, diffraction theory, etc. The Holomorphic Fractional Fourier Trans…
Segal-Bargmann transforms from hyperbolic Hamiltonians
William D. Kirwin, José Mourão, João P. Nunes +1
We consider the imaginary time flow of a quadratic hyperbolic Hamiltonian on the symplectic plane, apply it to the Schrödinger polarization and study the corresponding evolution of…
Picard group and quantization of toric orbifolds
Thomas Baier, José M. Mourão, João P. Nunes
In the classical theory of toric manifolds polytopes appear in two guises -- as Newton polytopes of line bundles on the complex, and as moment polytopes on the symplectic side, the…
A new approximation method for geodesics on the space of Kähler metrics using complexified symplectomorphisms and Gröbner Lie series
José Mourão, João P. Nunes, Tomás Reis
It has been shown that the Cauchy problem for geodesics in the space of Kähler metrics with a fixed cohomology class on a compact complex manifold can be effectively reduced to…
Clifford Coherent State Transforms on Spheres
Pei Dang, José Mourão, João P. Nunes +1
We introduce a one-parameter family of transforms, , , from the Hilbert space of Clifford algebra valued square integrable functions on the --dimensional sphere,…
Coherent State Transforms and the Weyl Equation in Clifford Analysis
José Mourão, João P. Nunes, Tao Qian
We study a transform, inspired by coherent state transforms, from the Hilbert space of Clifford algebra valued square integrable functions $L^2({\mathbb R}^m,dx)\otimes {\mathbb C}…
Extending coherent state transforms to Clifford analysis
William D. Kirwin, José Mourão, João P. Nunes +1
We introduce two extensions of the Segal-Bargmann coherent state transform from to Hilbert spaces of slice monogenic and axial monogenic functions and study t…
Quantization in singular real polarizations: Kähler regularization, Maslov correction and pairings
João N. Esteves, José M. Mourão, João P. Nunes
We study the Maslov correction to semiclassical states by using a Kähler regularized BKS pairing map from the energy representation to the Schrödinger representation. For general s…
Complex symplectomorphisms and pseudo-Kähler islands in the quantization of toric manifolds
William D. Kirwin, José M. Mourão, João P. Nunes
Let be a Delzant polytope. We show that the quantization of the corresponding toric manifold in toric Kähler polarizations and in the toric real polarization are relate…
On complexified analytic Hamiltonian flows and geodesics on the space of Kahler metrics
Jose M. Mourao, Joao P. Nunes
In the case of a compact real analytic symplectic manifold M we describe an approach to the complexification of Hamiltonian flows [Se, Do1, Th1] and corresponding geodesics on the…
Coherent state transforms and the Mackey-Stone-Von Neumann theorem
William D. Kirwin, José M. Mourão, João P. Nunes
Mackey showed that for a compact Lie group , the pair has a unique non-trivial irreducible covariant pair of representations. We study the relevance of this resul…
Complex time evolution in geometric quantization and generalized coherent state transforms
William D. Kirwin, José M. Mourão, João P. Nunes
For the cotangent bundle of a compact Lie group , we study the complex-time evolution of the vertical tangent bundle and the associated geometric quantization Hilbert s…
Quantization of some moduli spaces of parabolic vector bundles on CP^1
Indranil Biswas, Carlos Florentino, José Mourão +1
We address quantization of the natural symplectic structure on a moduli space of parabolic vector bundles of parabolic degree zero over with four parabolic points a…
Degeneration of Kaehler structures and half-form quantization of toric varieties
William D. Kirwin, José M. Mourão, João P. Nunes
We study the half-form Kaehler quantization of a smooth symplectic toric manifold , such that and is nonnegative. We define the…
Quantization of Abelian Varieties: distributional sections and the transition from Kähler to real polarizations
Thomas Baier, José M. Mourão, João P. Nunes
We study the dependence of geometric quantization of the standard symplectic torus on the choice of invariant polarization. Real and mixed polarizations are interpreted as degenera…
Toric Kähler metrics seen from infinity, quantization and compact tropical amoebas
Thomas Baier, Carlos Florentino, José M. Mourão +1
We consider the metric space of all toric Kähler metrics on a compact toric manifold; when "looking at it from infinity" (following Gromov), we obtain the tangent cone at infinity,…
On the BKS pairing for Kahler quantizations of the cotangent bundle of a Lie group
Carlos Florentino, Pedro Matias, Jose Mourao +1
A natural one-parameter family of Kähler quantizations of the cotangent bundle of a compact Lie group , taking into account the half-form correction, was studied in \cite…
Geometric quantization, complex structures and the coherent state transform
Carlos Florentino, Pedro Matias, Jose Mourao +1
It is shown that the heat operator in the Hall coherent state transform for a compact Lie group is related with a Hermitian connection associated to a natural one-parameter fam…