arXiv research

TVS-FNNs can approximate any continuous function on expanded input spaces.

Two-dimensional RG acts like Ricci flow to model expanding universe.

Uniform spectral gap for convex cocompact hyperbolic surfaces and expanders.

In this paper, we study self-expanders for mean curvature flows. First we show the discreteness of the spectrum of the drifted Laplacian on them. Next we give a universal lower bound of the bottom of the spectrum of the drifted Laplacian and prove that this lower bound is achieved if and only if the self-expander is th…

Discussing curvature flows and their applications.

These lecture notes review the topological string theory and its applications to mathematics and physics. They expand on material presented at the Takagi Lectures of the Mathematical Society of Japan on 21 June 2008 at Department of Mathematics, Kyoto University.

For all 0<t \leq 1, we define a locally Euclidean metric ρ_t on R^3. These metrics are invariant under Euclidean isometries and, if t increases to 1, converges to the Euclidean metric d_E. This research is motivated by expanding universe.

We perform a rescaling analysis to analyze the future behavior of a class of $T^2$-symmetric vacuum spacetimes. We show that on the universal cover, there is $C^0$-convergence to a spatially homogeneous spacetime that does not satisfy the vacuum Einstein equations.

If g(t) is a three-dimensional Ricci flow solution, with sectional curvatures that decay like the inverse of t and diameter that increases at most like the square root of t, then the pullback Ricci flow solution on the universal cover approaches a homogeneous expanding soliton.

We present sufficient conditions for the cohomology of a closed aspherical manifold to be proper Lipschitz in sense of Connes-Gromov-Moscovici [CGM]. The conditions are stated in terms of the Stone-Čech compactification of the universal cover of a manifold. We show that these conditions are formally weaker than the suf…

Streets and Tian introduced a parabolic flow of pluriclosed metrics. We classify the long time behavior of homogeneous solutions of this flow on closed complex surfaces including minimal Hopf, Inoue, Kodaira, and non-Kahler, properly elliptic surfaces. We also construct expanding soliton solutions to the flow on the un…

We calculate the optimal solutions of the fully heterogeneous Von Neumann expansion problem with $N$ processes and $P$ goods in the limit $N\to\infty$. This model provides an elementary description of the growth of a production economy in the long run. The system turns from a contracting to an expanding phase as $N$ in…

Paper develops efficient methods for estimating Hessian inverses in stochastic optimization.

This is an expanded and updated version of a lecture series I gave at Seoul National University in September 1997. It is in some sense an update of the 1979 Griffiths and Harris paper with a similar title. I discuss: Homogeneous varieties, Topology and consequences Projective differential invariants, Varieties with deg…

Graphs with non-negative Ollivier-Ricci curvature cannot be expanders.

New energy definition for expanding de Sitter spacetime with umbilic boundaries.

Universal AI seeks high-optionality states through empowerment and curiosity.

This lecture notes are an expanded version of the course given at the ERC-School on Geometric Measure Theory and Real Analysis, held in Pisa, September 30th - October 30th 2013. The lectures aim to explain the main steps of a new proof of the partial regularity of area minimizing integer rectifiable currents in higher …

LDTA expands LDA's topic modeling capacity with tree-structured priors.

Modeling videos and image-sets as linear subspaces has proven beneficial for many visual recognition tasks. However, it also incurs challenges arising from the fact that linear subspaces do not obey Euclidean geometry, but lie on a special type of Riemannian manifolds known as Grassmannian. To leverage the techniques d…

These notes represent a much expanded and updated version of the \textquotedblleft mini course\textquotedblright that the author gave at the ETH (Zürich) and the University of Zürich in February of 1995. The purpose of these notes is to first provide some basic background to Riemannian geometry and stochastic calculus …

Measurements of cosmic microwave background (CMB) anisotropy are ideal experiments for discovering the non-trivial global topology of the universe. To evaluate the CMB anisotropy in multiply-connected compact cosmological models, one needs to compute the eigenmodes of the Laplace-Beltrami operator. Using the direct bou…

This paper shows universality in spectrum behavior for random inner-product kernel matrices in polynomial regime.

In his work on singularities, expanders and topology of maps, Gromov showed, using isoperimetric inequalities in graded algebras, that every real valued map on the $n$-torus admits a fibre whose homological size is bounded below by some universal constant depending on $n$. He obtained similar estimates for maps with va…

Constructs flow lines connecting unstable to stable self-expanders.

New expanders found using origami surfaces with spectral gap.

The paper extends rigidity results for $λ$-self-expanders to hyperplanes, spheres, and cylinders.

Unified analytical tool for non-Markovian jump processes.

New self-expander found between two given asymptotic ones.

These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at Zhejiang University in July, 2008. Their goal is to introduce and motivate basic concepts and constructions (such as orbifolds and stacks) important in the study of moduli spaces of curves and abelian varieties thro…

No expanding breathers found in noncompact Ricci flows with certain curvature conditions.

Sharp curvature estimates for expanding Ricci solitons in various dimensions.

Gravitational wave memory increases faster than Brownian motion in early universe and astrophysical sources.

New expanders for mean curvature flow contradict genus-reduction conjecture.

New degree theory proves existence of solitons on 4D manifolds.

New expanding Ricci solitons found starting in dimension four.

Study on the spectrum of drift Laplacian on Ricci expanders.

Expander graphs have been a focus of attention in computer science in the last four decades. In recent years a high dimensional theory of expanders is emerging. There are several possible generalizations of the theory of expansion to simplicial complexes, among them stand out coboundary expansion and topological expand…

The paper classifies expanding gradient Yamabe solitons based on scalar curvature.

Constructs self-expanders of positive genus for cones in R^3.

Study of complete space-like self-expanders in Minkovski space.

Study finds unique self-expanders for mean curvature flow.

Study cohomogeneity one expanding Ricci solitons on specific topologies.

We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. …

We give a cohomological characterisation of expander graphs, and use it to give a direct proof that expander graphs do not have Yu's property A.

The paper examines properties and rigidity of self-expanders in Euclidean space.

We show compactness in the locally smooth topology for certain natural families of asymptotically conical self-expanding solutions of mean curvature flow. Specifically, we show such compactness for the set of all two-dimensional self-expanders of a fixed topological type and, in all dimensions, for the set of self-expa…

Strict convexity proven for certain self-expanders in high dimensions.