New tools found to create hyperbolic links with lower volume bounds.
problem Finding lower bounds on volumes of staked links.
method Defining charm bracelets and constructing hyperbolic links.
result Infinitely many hyperbolic staked links with lower volume bounds.
The paper develops a learning theory for neural network-based CHARME models.
problem Developing a learning theory for CHARME models using neural networks.
method Proves the stationarity and ergodicity of CHARME models under weak conditions, then applies neural networks to derive strong consistency and asymptotic normality of estimators.
result Strong consistency and asymptotic normality of NN-based estimators of CHARME model weights and biases under weak conditions.
Bracelets and theta bases match in various cluster algebras.
problem Matching bracelet and theta bases in cluster algebras.
method Comparing skein and cluster algebras, defining quantum bracelets, and analyzing cluster scattering diagrams.
result Quantum bracelets coincide with theta functions in various cluster algebras.
CHARM creates mock halo catalogs from dark matter density fields using neural networks.
problem Creating accurate mock halo catalogs for cosmological studies is computationally expensive.
method CHARM uses multi-stage neural spline flow networks to learn the mapping from dark matter density fields to halo catalogs.
result Mock halo catalogs have the same statistical properties as those from high-resolution N-body simulations.
An unknotting tunnel in a 3-manifold with boundary is a properly embedded arc, the complement of an open neighborhood of which is a handlebody. A geodesic with endpoints on the cusp boundary of a hyperbolic 3-manifold and perpendicular to the cusp boundary is called a vertical geodesic. Given a vertical geodesic in a h…
We show that the twisted SL(2) skein algebra of a surface has a natural basis (the bracelets basis) that is positive, in the sense that the structure constants for multiplication are positive integers.
Open geometry puzzles keep the author engaged.
problem Open problems in geometry that challenge the author.
method Collection of open problems based on puzzle-charm.
result No spare hands in solving the problems.
Following Goussarov's paper `Interdependent Modifications of Links and Invariants of Finite Degree' [Topology 37 (1998) 595--602] we describe an alternative finite type theory of knots. While (as shown by Goussarov) the alternative theory turns out to be equivalent to the standard one, it nevertheless has its own share…
A common problem in a high energy physics experiment is extracting a signal from a much larger background. Posed as a classification task, there is said to be an imbalance in the number of samples belonging to the signal class versus the number of samples from the background class. In this work we provide a brief overv…
Bistable structures associated with non-linear deformation behavior, exemplified by the Venus flytrap and slap bracelet, can switch between different functional shapes upon actuation. Despite numerous efforts in modeling such large deformation behavior of shells, the roles of mechanical and nonlinear geometric effects …
Quantum duality map extended to general marked surfaces and its compatibility with skein algebras proven.
problem Generalizing quantum duality map to general marked surfaces and proving its compatibility with skein algebras.
method Generalized quantum duality map, reduced stated skein algebras, quantum trace maps, skein lifting.
result Compatibility of quantum duality map with skein algebras proven.
The paper connects two skein algebras and characterizes their representations.
problem Characterizing representations of Roger-Yang skein algebras.
method Calculating Roger-Yang skein algebra of an annulus, establishing a homomorphism to Kauffman bracket skein algebra of a torus, and using these to characterize representations.
result Characterization of irreducible, finite-dimensional representations of Roger-Yang skein algebra of an annulus with two interior punctures.
Spaces of positive and tropical points described as Teichmüller and lamination spaces with pinnings.
problem Describing spaces of positive and tropical points in moduli space with pinnings.
method Topological description of Teichmüller and lamination spaces with pinnings.
result Formulae relating functions on Teichmüller space with pinnings and topological description of tropicalized amalgamation map.
Researchers prove positivity of skein algebra structure constants for specific surfaces.
problem Positivity of structure constants in skein algebras of specific surfaces.
method Mirror symmetry construction based on higher genus Gromov-Witten theory applied to a complex cubic surface.
result Proved positivity of structure constants for skein algebras of the 4-punctured sphere and 1-punctured torus.
Quantum cluster algebra constructed from web skein relations on surfaces.
problem Quantization of cluster structures on moduli spaces of SL3 local systems.
method Constructing a quantum cluster algebra inside the skew-field of a skein algebra of unpunctured surfaces.
result Laurent expressions of webs in clusters have positive coefficients.
The paper constructs bases for cluster varieties using mSL3-webs and laminations.
problem Cluster varieties associated to mSL3-local systems on surfaces. method Introducing mSL3-laminations, developing quantum and classical trace maps, and constructing bases. result Bases of regular functions on mPGL3 cluster varieties constructed from mSL3-laminations. The growing number of low-power smart devices in the Internet of Things is coupled with the concept of "Edge Computing", that is moving some of the intelligence, especially machine learning, towards the edge of the network. Enabling machine learning algorithms to run on resource-constrained hardware, typically on low-p…