Synthetic Petri Dish predicts neural architecture performance faster.
problem Expensive NAS evaluation process with ground-truth data.
method Instantiates motifs in small networks, evaluates with few synthetic samples.
result Significantly higher accuracy in predicting motif performance.
In complex processes, various events can happen in different sequences. The prediction of the next event given an a-priori process state is of importance in such processes. Recent methods have proposed deep learning techniques such as recurrent neural networks, developed on raw event logs, to predict the next event fro…
Neural surrogate predicts SPN rates from token trajectories.
problem Challenging parameter estimation in SPNs with covariates.
method 1D Convolutional Residual Network trained on Gillespie-simulated SPN realizations.
result Surrogate predicts rate-function coefficients with RMSE = 0.043.
The paper studies almost complex torus manifolds using graphs and Hirzebruch genera, proving properties of their fixed points and cohomology.
problem Understanding the fixed points and cohomology of almost complex torus manifolds.
method Using directed labeled multigraphs and Hirzebruch genera to encode and analyze the manifolds.
result Almost complex torus manifolds have positive Todd genus and at least n+1 fixed points.
Let M be a manifold homotopy equivalent to the complex projective space $\C P^m$. Petrie conjectured that M has standard total Pontrjagin class if M admits a non-trivial action by S1. We prove the conjecture for m<12 under the assumption that the action extends to a nice Pin(2)-action with fixed point. The…
Study on length spectrum of random hyperbolic 3-manifolds.
problem Understanding the length spectrum of random hyperbolic 3-manifolds.
method Modeling random hyperbolic 3-manifolds using truncated tetrahedra and analyzing their length spectrum as volume tends to infinity.
result The length spectrum converges in distribution to a Poisson point process with a computable intensity λ as volume increases.
Directly proves logarithmic systolic growth for all hyperbolic surfaces.
problem Proving logarithmic systolic growth for all hyperbolic surfaces.
method Using original Brooks/Buser-Sarnak surfaces through a direct approach.
result Directly proves logarithmic systolic growth for all hyperbolic surfaces.
Study shows systole behavior changes significantly for large genus hyperbolic surfaces.
problem Understanding systole behavior in large genus hyperbolic surfaces.
method Analysis of random surfaces with respect to Weil-Petersson volume.
result Expected value of separating systole behaves like 2logg for large genus. For G an almost-connected Lie group, we study G-equivariant index theory for proper co-compact actions with various applications, including obstructions to and existence of G-invariant Riemannian metrics of positive scalar curvature. We prove a rigidity result for almost-complex manifolds, generalising Hattori's result…
Classifies circle actions on 6D manifolds with 4 fixed points.
problem Classifying circle actions on 6D manifolds with specific fixed points.
method Analyzes fixed point data and proves agreement with known actions.
result Agrees with actions on 6-spheres or CP3. Study on systole of random hyperbolic 3-manifolds, proving limit exists and calculating it.
problem Understanding the systole of random hyperbolic 3-manifolds.
method Modeling random hyperbolic 3-manifolds using truncated tetrahedra, calculating expected systole limit as volume increases.
result Closed formula and numerical approximation for the limit of the expected systole as volume tends to infinity.
Let X be a smooth manifold belonging to one of these three collections: acyclic manifolds (compact or not, possibly with boundary), compact connected manifolds (possibly with boundary) with nonzero Euler characteristic, integral homology spheres. We prove that Diff(X) is Jordan. This means that there exists a const…
Study on geodesics and eigenvalues on random hyperbolic surfaces with cusps.
problem Counting short geodesics and small eigenvalues on random hyperbolic surfaces.
method Rescaling and convergence to a Poisson point process.
result The probability of having at least k=o(n) arbitrarily small eigenvalues tends to 1 as no∞. We propose a network independent, hand-held system to translate and disambiguate foreign restaurant menu items in real-time. The system is based on the use of a portable multimedia device, such as a smartphones or a PDA. An accurate and fast translation is obtained using a Machine Translation engine and a context-speci…
This is a survey of our research on geometric structures of projective embeddings and includes some topics of our talks in several symposia during 1990-99. We clarify our main problem, which is to construct a kind of geometric composition series of projective embeddings. The concept of "geometric composition series" is…
Study on random hyperbolic surfaces with many cusps, focusing on tight geodesics.
problem Understanding length statistics of geodesics on random hyperbolic surfaces with cusps.
method Recursion formula for tight Weil-Petersson volumes and generalization of Mirzakhani's integration formula.
result Recovery of Poisson point process in large genus limit for length statistics of tight geodesics.
Study on hyperbolic surfaces' volumes, proving asymptotic expansion for high genus.
problem Analyzing the volume of moduli spaces of hyperbolic surfaces with varying genus.
method Topological recursion formula by Mirzakhani, asymptotic expansion for high genus.
result Explicit computation of the second term in the asymptotic expansion.
Maps with a single face converge to hyperbolic surfaces in large genus.
problem Understanding geometric properties of high genus maps.
method Analyzing uniformly random maps and their convergence to hyperbolic surfaces.
result Lengths of simple cycles converge to a Poisson process.
If X is a smooth manifold and G is a subgroup of Diff(X) we say that (X,G) has the almost fixed point property if there exists a number C such that for any finite subgroup G≤G there is some x∈X whose stabilizer Gx≤G satisfies [G:Gx]≤C. We say that $X…
The Gaiotto locus for Sp(2n) is shown to lie in the nilpotent cone.
problem Characterizing the Gaiotto locus for Sp(2n) Lie groups.
method Using symplectic representations and moment maps, analyzing Higgs fields and their closures.
result The Gaiotto locus for Sp(2n) is the irreducible component of the nilpotent cone.
Study of straight-line flows on a unique infinite surface.
problem Understanding straight-line flows on a specific infinite surface.
method Geometric description and characterization of periodic and drift orbits; use of rigid symmetries and Veech group.
result Complete characterization of periodic directions and proof of density of periodic and ergodic directions.