New techniques simplify computing geometric index of links in solid tori.
problem Computing the geometric index of links in solid tori.
method Introducing geometric index for solid chambers and proving its computation under specific conditions.
result Simplified computation of geometric index for new examples.
We introduce a basis of the Orlik-Solomon algebra labeled by chambers, so called chamber basis. We consider structure constants of the Orlik-Solomon algebra with respect to the chamber basis and prove that these structure constants recover D. Cohen's minimal complex from the Aomoto complex.
The paper explores genericity of triples of antipodal chambers in affine buildings.
problem Investigating genericity of triples of antipodal ideal chambers in locally finite affine buildings.
method Defined ideal-genericity and affine-genericity at the ideal boundary and within the affine building, respectively. Established a barycenter map and showed its continuity.
result Affine-genericity implies ideal-genericity, but the latter is more suitable for constructing a barycenter map.
Developed causal chambers for AI validation, providing real-world data.
problem Limited real-world datasets for AI method validation.
method Created computer-controlled physical systems (causal chambers) to generate datasets.
result Demonstrated applications in various AI fields, validated causal models.
The paper studies the geometric structure of modular curves X1(N) using meromorphic differentials.
problem Understanding the topological complexity of modular curves X1(N). method Using the walls-and-chambers structure of strata of meromorphic differentials.
result Formulas for the number of chambers and effective means to draw the incidence graph of X1(N). The paper studies chambered invariants of real Cauchy-Riemann operators on Riemann surfaces.
problem Counting pseudo-holomorphic curves in symplectic Calabi-Yau 3-folds.
method Constructs three chambered invariants: nBl, n1,2, n2,1, defined by counting solutions to ADHM vortex equations and pseudo-holomorphic sections of bundles. result Conjectures a relationship between n1,2 and n2,1 and symplectic invariants. The paper analyzes feedback loops in recommender systems causing echo chambers and filter bubbles.
problem Feedback loops in recommender systems leading to echo chambers and filter bubbles.
method Theoretical analysis of user dynamics and recommender system behavior.
result Solutions to slow down system degeneracy and understanding echo chambers and filter bubbles.
The paper studies proper discontinuity of actions on Weyl chamber flow spaces.
problem Properly discontinuous actions on Weyl chamber flow spaces for transverse subgroups.
method Analyzes limit sets and quotient spaces, introduces growth indicators and conformal measures.
result Establishes ergodic dichotomy for Weyl chamber flow and introduces new measures.
Research examines arrangements of hyperplanes in real projective spaces, focusing on specific cases.
problem Analyzing the structure of hyperplane arrangements in real projective spaces.
method Investigates arrangements of m hyperplanes in the n-dimensional real projective space, with a focus on m=n+3 and n=3 or n=4. result Provides insights into the structure of chambers cut out by these specific hyperplane arrangements.
New compactification for character varieties with good topological properties.
problem Compactification of character varieties with good topological properties.
method Announced a new compactification with interpretations of ideal points.
result Relates to Weyl chamber length compactification and applies to maximal and Hitchin representations.
In this paper we set up the family Seiberg-Witten theory. It can be applied to the counting of nodal pseudo-holomorphic curves in a symplectic 4-manifold (especially a Kahler surface). A new feature in this theory is that the chamber structure plays a more prominent role. We derive some wall crossing formulas measuring…
We investigate the geometry in a real Euclidean building X of type A2 of some simple configurations in the associated projective plane at infinity P, seen as ideal configurations in X, and relate it with the projective invariants (from the cross ratio on P). In particular we establish a geometric classification of gene…
We study the space $\nua{m}{d}$ of clouds in $\bbr^d$ (ordered sets of m points modulo the action of the group of affine isometries). We show that $\nua{m}{d}$ is a smooth space, stratified over a certain hyperplane arrangement in $\bbr^m$. We give an algorithm to list all the chambers and other strata (this is indep…
The radius of the star-shaped set need not have been continuous.
The generalized soap bubble problem seeks the least perimeter way to enclose and separate n given volumes in R^m. We study the possible configurations for perimeter minimizing bubble complexes enclosing more than two regions. We prove that perimeter minimizing planar bubble complexes with equal pressure regions and wit…
New algorithm clusters particle tracks for better trajectory recognition in noisy data.
problem Challenging automatic reconstruction of particle tracks from Active Target Time Projection Chambers data.
method Non-parametric algorithm based on hierarchical clustering of point triplets.
result Algorithm identifies and isolates non-analytical particle tracks with high recall and precision.
Identifies Heegaard Floer homology solid tori via Dehn fillings.
problem Characterizing Heegaard Floer homology solid tori.
method Using Dehn fillings to identify solid tori.
result Characterized Seifert fibered Heegaard Floer solid tori.
The paper proves an infinite double bubble theorem in higher dimensions.
problem Characterizing minimizing partitions of infinite and finite volumes in Rn. method Proves a variant of the double bubble theorem for configurations with infinite and finite chambers.
result Locally minimizing (1,2)-clusters are unique in Rn for n≤7 and n≥8 under certain conditions. The paper characterizes when numerical criteria for PDE solvability fail and provides effective criteria for existence.
problem Characterizing when numerical criteria for PDE solvability fail.
method Finite number of subvarieties violating Nakai type criterion, and their rigidity.
result Finite number of subvarieties violating the Nakai type criterion, and these subvarieties are rigid.
We investigate discrete groups G of isometries of a complete connected Riemannian manifold M which are generated by reflections, in particular those generated by disecting reflections. We show that these are Coxeter groups, and that the the orbit space M/G is isometric to a Weyl chamber C which is a Riemannian …
The paper studies HYM connections on stable vector bundles over Kähler manifolds.
problem Analyzing stability and convergence of Hermitian Yang-Mills connections.
method Semialgebraic decomposition of the Kähler cone into stability chambers.
result HYM connections converge to a stable HYM connection as polarisation converges.
In this paper we present a topological way of building a compactification of a symmetric space from a compactification of a Weyl Chamber.
New framework models echo chamber learning, proving tight bounds on algorithm performance.
problem Echo chambers in machine learning where systems learn from self-annotated data.
method Online Learning in the Replay Setting, Extended Threshold dimension, closure-based learner.
result Proves tight bounds on algorithm performance against replay adversaries.
Explicitly constructs moduli spaces of stable parabolic bundles.
problem Understanding moduli spaces of stable parabolic bundles over the Riemann sphere.
method Explicit construction and quotient of stable parabolic structures by bundle automorphisms.
result Explicit models of moduli spaces as smooth, compact complex manifolds.
New self-shrinkers of Platonic solids found.
problem Finding new embedded self-shrinkers of specific genus.
method Variational methods, numerical discovery by D. Chopp.
result Constructed self-shrinkers resembling doublings of Platonic solids.
Proves NP and co-NP status for knot core recognition in solid torus.
problem Determining if a knot is the core of a solid torus.
method Alternate proof and corollary of Hopf link recognition problem.
result Proves NP and co-NP status for solid torus core recognition problem.
Rational knots and links in solid torus characterized by continued fractions.
problem Characterizing rational knots and links in solid torus.
method Using rational tangles and continued fractions, and generalizing to skein module invariants.
result Rational links in solid torus fully characterized by rational tangles and continued fractions.
New theorem for 4D links simplifies characterisation problem.
problem Long-standing open problem in link characterisation.
method Reidemeister Theorem for solid ribbon torus links.
result Complete characterisation of a related class of links.
We consider the solid angle that a planar compact subset subtends at a point in a level set of height h and study two extremal problems for the solid angle. One of the variables is a point in such a plane, that is, we study the properties of the solid angle maximizer. The other is the pair of a planar compact subset an…
Study solid angles to find Seifert hypersurfaces.
problem Understanding solid angles and their relation to Seifert hypersurfaces.
method Analyze the solid angle function and its critical levels.
result Non-critical level sets of the solid angle function are Seifert hypersurfaces.
BCAE-2D compresses 3D data from a time projection chamber at high speed.
problem Compressing high-speed, sparse 3D data from a time projection chamber.
method 2D Bicephalous Convolutional Autoencoder (BCAE-2D) approach.
result 3x speedup in compression throughput with improved reconstruction accuracy.
Classifies small links in an unmarked solid torus.
problem Classifying knots and links in an unmarked solid torus.
method Invariants and Dehn twists to detect and transform links.
result Classification of all non-split links up to 6 crossings.
New method constructs Seifert solids from bridge trisections.
problem Constructing Seifert solids from bridge trisections.
method Adapting Seifert's algorithm to tri-plane diagrams.
result Classification results on surface decomposability and unknottedness.
The paper classifies all tight contact structures on a solid torus.
problem Classifying tight contact structures on a solid torus with specified dividing sets.
method Writing down a closed formula for the number of non-isotopic tight contact structures with any given dividing set.
result The complete classification of tight contact structures on a solid torus.
We show that in any triangulation of a solid torus, there is a pre-core curve that lies in the 2-skeleton and that intersects the interior of each face in at most 10 straight arcs. By definition, a pre-core curve is a simple closed curve that becomes a core curve when a collar is attached to the boundary of the solid t…
Study Murasugi sum in 4D for knotted surfaces, defining arborescent surfaces.
problem Defining and understanding Murasugi sum in 4D for knotted surfaces.
method Introduced a 4D Murasugi sum to define arborescent knotted surfaces.
result Defined and studied arborescent knotted surfaces using 4D Murasugi sum.
New invariant defined for tied links in solid torus.
problem Defining an invariant for tied links in solid torus.
method Using skein relations and Jones' method over bt-algebra of type B with Markov trace.
result Recovery of invariant defined for tied links in solid torus.
New resonance theory for Anosov flows connects spectral properties to mixing measures.
problem Defining and analyzing Ruelle-Taylor resonances for Anosov actions.
method Combining microlocal methods and J. Taylor's cohomological theory, defining Ruelle-Taylor resonances and proving Fredholm theory.
result Ruelle-Taylor resonances form a discrete subset of Cκ with λ=0 being a leading resonance. We consider the family of harmonic measures on a lamination L of a compact space X by locally symmetric spaces L of noncompact type, i.e. L≃ΓL\G/K. We establish a natural bijection between these measures and the measures on an associated lamination foliated by G-orbits, $\hat{\mathc…
Geometrically reformulates Cosserat solid mechanics using differential geometry.
problem Formalizing Cosserat solid mechanics in modern differential geometry.
method Formulation as a principal fibre bundle, using Cartan's magic formula, and integrating infinitesimal strains.
result Reveals strain as a Lie algebra-valued one-form and finite strain through integration.
Characterizes solutions to Z-critical equations on surfaces using effective conditions.
problem Characterizing solutions to Z-critical equations on compact Kähler surfaces.
method Uses effective conditions and Picard number bounds to characterize solutions.
result Characterizes optimally destabilizing curves for Donaldson's J-equation and deformed Hermitian Yang-Mills equation.
The paper introduces surfaces with constant solid angle for designing shell structures.
problem Designing shell structures with balanced structural, spatial, aesthetic, and construction requirements.
method Proposes surfaces defined by constant solid angle at all points, using Gauss-Bonnet theorem and Newton's method.
result Constant solid angle surfaces enable control over boundary slope and span-to-height ratio, making them structurally viable.
New topological invariant distinguishes real line arrangements with same combinatorics.
problem Determining topological properties from combinatorial data of real line arrangements.
method Introducing chamber weight invariant based on dual configuration points in real projective plane.
result New Zariski pairs of 13, 15, and 17 lines with different topological embeddings.
We describe the deformation space of a solid torus with boundary modelled on convex ideal hyperbolic polyhedra. This deformation space is given by natural Gauss--Bonnet type inequalities on the dihedral angles. The result extends to solid tori with an arbitrary conical singularity along the core. Our method is to decom…
The paper studies how to convert the motion of rolling solids into a Hamiltonian system.
problem Hamiltonization of the dynamics of rolling solids of revolution.
method Geometric methods and gauge transformations to reduce the system to a Hamiltonian form.
result Genuine Poisson brackets are obtained for the reduced dynamics of rolling solids.
This note generalizes the visual angle to convex sets in 3D space.
problem Analyzing geometric properties of convex sets in 3D space.
method Generalizing the visual angle to convex sets in Euclidean space and expressing geometric quantities in terms of integrals of functions related to the solid angle.
result Invariant quantities of the original convex set can be expressed by integrals of functions related to the solid angle.
We interpret Coxeter's truncated braid groups in terms of Platonic solids.
problem Understanding the structure of truncated braid groups.
method Topological interpretation using orbifolds.
result Connection between truncated braid groups and Platonic solids.
Study convex hulls of orbits for compact groups, defining new invariants related to polynomial degrees.
problem Understanding properties of convex hulls of coadjoint orbits of compact groups.
method Introduce partial convex hulls and use them to define numerical invariants.
result Orbits with new invariants form rational convex polyhedral cones related to Littlewood-Richardson cones.