1,720,985 research outputs found
Special Lagrangian conifolds, I: Moduli spaces
No full textWe discuss the deformation theory of special Lagrangian (SL) conifolds in C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. This category
allows for the simultaneous presence of conical singularities and of non-compact, asymptotically conical, ends.
Our main theorem is the natural next step in the chain of results
initiated by McLean and continued by the author
and Joyce. We emphasize a unifying framework for studying the
various
cases and discuss analogies and differences between them. This paper also lays down the geometric foundations for
our paper "Special Lagrangian conifolds, II" concerning gluing constructions for SL conifolds in C^m
Desingularizing isolated conical singularities: Uniform estimates via weighted Sobolev spaces
Get PDFWe define a very general ``parametric connect sum'' construction which can be
used to eliminate isolated conical singularities of Riemannian manifolds.
We then show that various important
analytic and elliptic estimates, formulated in terms of weighted Sobolev spaces,
can be obtained independently of the parameters used in the construction.
Specifically, we prove uniform estimates related to (i) Sobolev Embedding
Theorems, (ii) the invertibility of the Laplace operator and (iii)
Poincare' and Gagliardo-Nirenberg-Sobolev type inequalities.
Our main tools are the well-known theories of weighted Sobolev spaces and elliptic operators on ``conifolds''. We provide an overview of both, together with an extension of the former to general Riemannian manifolds.
For a geometric application of our results we refer the reader to our paper "Special Lagrangian conifolds, II" concerning desingularizations of special Lagrangian
conifolds in C^m
Special Lagrangian conifolds, II: Gluing constructions in C^m
Get PDFWe prove two gluing theorems for special Lagrangian (SL) conifolds in C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds.
In particular, our theorems yield the first examples of smooth SL conifolds with 3 or more planar ends and the first (non-trivial) examples of SL conifolds which have a conical singularity but are not, globally, cones. We also obtain: (i) a desingularization procedure for transverse intersection and self-intersection points, using ``Lawlor necks''; (ii) a construction which completely desingularizes any SL conifold by replacing isolated conical singularities with non-compact asymptotically conical (AC) ends; (iii) a proof that there is no upper bound on the number of AC ends of a SL conifold; (iv) the possibility of replacing a given collection of conical singularities with a completely different collection of conical singularities and of AC ends.
As a corollary of (i) we improve a result by Arezzo and Pacard concerning minimal desingularizations of certain configurations of SL planes in C^m, intersecting transversally
Complex Structures on SO(M,g)
Get PDFData una varieta’ Riemanniana orientata (M, g), il fibrato princi-
pale SOg (M ) di basi ortonormali positive su (M, g) ha una paralleliz-
zazione canonica dipendente dalla connessione di Levi-Civita. Questo
fatto suggerisce la definizione di una classe molto naturale di strutture
quasi-complesse su (M, g). Dopo le necessarie definizioni, discutiamo
qui l’integrabilita’ di queste strutture, esprimendola in termini della
struttura Riemanniana g
Mean Curvature Flow, Orbits, Moment Maps
Get PDFGiven a compact Riemannian manifold together with a group of isometries, we discuss MCF of the orbits and some applications: eg, finding minimal orbits. We then specialize to Lagrangian orbits in Kaehler manifolds. In particular, in the Kaehler-Einstein case we find a relation between MCF and moment maps which, for example, proves that the minimal Lagrangian orbits are isolated
Deformations of Asymptotically Conical Special Lagrangian Submanifolds
Get PDFMcLean studied the deformations of compact special Lagrangian submanifolds, showing in particular that they come in moduli spaces whose dimension depends only on the topology of the submanifold. In this article we study the analogous problem for non-compact, "asymptotically conical" SL submanifolds, with respect to various "boundary conditions at infinity"
Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems
No full textLet M denote the space of probability measures on R^D endowed with the
Wasserstein metric. A differential calculus for a certain class of absolutely continuous curves in M was introduced by Ambrosio, Gigli, and Savare'. In this paper
we develop a calculus for the corresponding class of differential forms on M. In
particular we prove an analogue of Green’s theorem for 1-forms and show that
the corresponding first cohomology group, in the sense of de Rham, vanishes. For
D = 2d we then define a symplectic distribution on M in terms of this calcu-
lus, thus obtaining a rigorous framework for the notion of Hamiltonian systems as
introduced by Ambrosio and Gangbo. Throughout the paper we emphasize the
geometric viewpoint and the role played by certain diffeomorphism groups of R^D
Obstructions to special Lagrangian desingularizations and the Lagrangian prescribed boundary problem
Get PDFWe exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/ nonexistence of such smoothings is an important component of the current efforts to understand which singular special Lagrangians arise as limits of smooth special Lagrangians. We also use soft methods from symplectic geometry (the relative version of the h--principle) and tools from algebraic topology to prove (both positive and negative) results about Lagrangian desingularizations of Lagrangian submanifolds with isolated singularities; we view the Lagrangian desingularization problem as the natural soft analogue of the special Lagrangian smoothing problem
A Post-training Quantization Method for the Design of Fixed-Point-Based FPGA/ASIC Hardware Accelerators for LSTM/GRU Algorithms
No full textRecurrent Neural Networks (RNNs) have become important tools for tasks such as speech recognition, text generation, or natural language processing. However, their inference may involve up to billions of operations and their large number of parameters leads to large storage size and runtime memory usage. These reasons impede the adoption of these models in real-time, on-the-edge applications. Field-Programmable Gate Arrays (FPGAs) and Application-Specific Integrated Circuits (ASICs) have emerged as promising solutions for the hardware acceleration of these algorithms, thanks to their degree of customization of compute data paths and memory subsystems, which makes them take the maximum advantage from compression techniques for what concerns area, timing, and power consumption. In contrast to the extensive study in compression and quantization for plain feed forward neural networks in the literature, little attention has been paid to reducing the computational resource requirements of RNNs. This work proposes a new effective methodology for the post-training quantization of RNNs. In particular, we focus on the quantization of Long Short-Term Memory (LSTM) RNNs and Gated Recurrent Unit (GRU) RNNs. The proposed quantization strategy is meant to be a detailed guideline toward the design of custom hardware accelerators for LSTM/GRU-based algorithms to be implemented on FPGA or ASIC devices using fixed-point arithmetic only. We applied our methods to LSTM/GRU models pretrained on the IMDb sentiment classification dataset and Penn TreeBank language modelling dataset, thus comparing each quantized model to its floating-point counterpart. The results show the possibility to achieve up to 90% memory footprint reduction in both cases, obtaining less than 1% loss in accuracy and even a slight improvement in the Perplexity per word metric, respectively. The results are presented showing the various trade-offs between memory footprint reduction and accuracy changes, demonstrating the benefits of the proposed methodology even in comparison with other works from the literature
G(2)-manifold and associative submanifolds via semi-fano 3-folds
Get PDFWe construct many new topological types of compact G2G2-manifolds, that is, Riemannian 77-manifolds with holonomy group G2G2. To achieve this we extend the twisted connected sum construction first developed by Kovalev and apply it to the large class of asymptotically cylindrical Calabi–Yau 33-folds built from semi-Fano 33-folds constructed previously by the authors. In many cases we determine the diffeomorphism type of the underlying smooth 77-manifolds completely; we find that many 22-connected 77-manifolds can be realized as twisted connected sums in a variety of ways, raising questions about the global structure of the moduli space of G2G2-metrics. Many of the G2G2-manifolds we construct contain compact rigid associative 33-folds, which play an important role in the higher-dimensional enumerative geometry (gauge theory/calibrated submanifolds) approach to defining deformation invariants of G2G2-metrics. By varying the semi-Fanos used to build different G2G2-metrics on the same 77-manifold we can change the number of rigid associative 33-folds we produce
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