Proves finiteness of Lagrangian fibrations with specific invariants.
problem Finiteness of hyperkaehler Lagrangian fibrations with fixed invariants.
method Proves finiteness of fibrations with fixed Fujiki constant, discriminant, and line bundle degree.
result Finiteness of hyperkaehler Lagrangian fibrations in any fixed dimension.
The paper shows that certain geometric structures remain unchanged under specific twists.
problem The rational Beauville-Bogomolov-Fujiki lattices of related fibrations are similar.
method Analytic and étale topologies, Hodge structures, and degenerate twistor deformations.
result Isomorphisms of graded vector spaces and Hodge-similar lattices.
The paper examines symplectic structures and their relation to the partial derivative lemma.
problem Understanding the relationship between complex symplectic structures and the partial derivative lemma.
method Analyzes complex symplectic manifolds and their associated Beauville-Bogomolov-Fujiki quadric.
result The quadric is smooth if and only if h 2 , 0 ( X ) = 1 h^{2,0}(X) = 1 h 2 , 0 ( X ) = 1 and irreducible if and only if h 1 , 1 ( X ) > 0 h^{1,1}(X) > 0 h 1 , 1 ( X ) > 0 when the partial derivative lemma holds. Study non-Kahler symplectic manifolds, proving deformation and Torelli theorems.
problem Topology and deformation theory of non-Kahler holomorphically symplectic manifolds.
method Investigation of topology and deformation theory, proving local Torelli theorem and Fujiki formula.
result Holomorphically symplectic deformations of BG-manifolds are unobstructed, and the period map is locally a diffeomorphism.
Hybrid subgroups found in non-arithmetic PU(2,1) lattices.
problem Exploring hybrid subgroups in non-arithmetic PU(2,1) lattices.
method Exploring hybrid subgroups of certain non-arithmetic lattices in PU(2,1). Showing that Mostow's lattices are virtually hybrids and some are hybrids of two non-commensurable arithmetic lattices in PU(1,1).
result Mostow's lattices are virtually hybrids and some are hybrids of two non-commensurable arithmetic lattices in PU(1,1).
New non-arithmetic lattice found in PU(3,1)
problem Arithmeticity of Couwenberg-Heckman-Looijenga lattices
method Study of arithmeticity and non-arithmetic lattices in PU(n,1)
result Found a non-arithmetic lattice in PU(3,1) not commensurable to Deligne-Mostow lattice
New property identifies arithmetic lattices from nonuniform lattices.
problem Characterizing arithmetic lattices among nonuniform lattices.
method Introduced Bounded Clustering (B-C) property.
result B-C property uniquely identifies arithmetic lattices.
New lattice stick knot condition identified.
problem Determining if 2D lattice knots project to 3D lattice sticks.
method Provided a necessary and sufficient condition.
result Identified a condition for 2D lattice knots to project to 3D lattice sticks.
Simplified proof for lattice link projections.
problem Necessary and sufficient condition for lattice link projections.
method Shorter and simpler proof of existing result.
result Simplified proof for lattice link projections.
This work compares lattice-free and lattice-based training criteria for LVCSR.
problem Improving acoustic model performance in speech recognition.
method Direct comparison of lattice-free and lattice-based sequence discriminative training criteria using GPU.
result Lattice-free MMI performance is comparable to lattice-based criteria, while lattice-based sMBR remains superior.
Research finds bounds for knots in hexagonal lattice and classifies 11-stick knots.
problem Determining the stick number and edge length of knots in a hexagonal lattice.
method Introducing a linear transformation between lattices to prove strict inequalities and classifying knots.
result Only trefoil and figure-eight knots are 11-stick knots in the hexagonal lattice.
We outline the theory of sets with distributive operations: multishelves and multispindles, with examples provided by semi-lattices, lattices and skew lattices. For every such a structure we define multi-term distributive homology and show some of its properties. The main result is a complete formula for the homology o…
New lattices from 4-manifolds show manifold properties.
problem Identifying distinct 4-manifolds from their lattices.
method Examined smooth, simply-connected 4-manifolds bounded by fixed homology 3-spheres.
result Found two distinct sets of Niemeier lattices.
Course on arithmetic lattices at EPFL.
problem Understanding arithmetic lattices.
method Introductory course on arithmetic lattices.
result Introduction to arithmetic lattices.
Upper bound for lattice stick number of spatial graphs.
problem Finding the minimum number of sticks in a cubic lattice to represent spatial graphs.
method Defined lattice stick number for spatial graphs and presented an upper bound in terms of crossing number.
result An upper bound for the lattice stick number of spatial graphs.
New thin subgroups found in special linear groups via bending techniques.
problem Finding thin subgroups of lattices in special linear groups.
method Techniques from convex projective geometry.
result Infinitely many non-commensurable lattices with thin subgroups.
This article classifies reflective hyperbolic lattices of rank 4.
problem Classifying reflective hyperbolic lattices of a specific rank.
method Using automorphism groups and reflections to classify lattices.
result Complete classification of 1.2-reflective maximal anisotropic lattices of rank 4.
New rigidity theorem for product of lattices.
problem Understanding quasi-isometry of product lattices.
method Demonstrated rigidity for product of non-uniform rank one lattice and nilpotent lattice.
result Any quasi-isometric group is an extension of a non-uniform rank one lattice by a nilpotent lattice.
This paper studies the covolumes of nonuniform arithmetic lattices in PU(n, 1). We determine the smallest covolume nonuniform arithmetic lattices for each n, the number of minimal covolume lattices for each n, and study the growth of the minimal covolume as n varies. In particular, there is a unique lattice (up to conj…
This paper proves the trefoil and figure-8 knots have the smallest lattice stick numbers.
problem Determining the minimum number of straight line segments for knot constructions in a cubic lattice.
method Mathematical proof for specific knot types (trefoil and figure-8) with lattice stick numbers less than 15.
result The trefoil and figure-8 knots are the only knot types with lattice stick numbers less than 15.
The paper finds incommensurable lattices in complex models of Baumslag-Solitar groups.
problem Locally finite 2-complexes and their automorphism groups contain incommensurable lattices.
method Constructing lattices in combinatorial models of Baumslag-Solitar groups and analyzing their properties.
result The constructed lattices are incommensurable and have specific properties like isomorphic Cayley graphs.
Proves a lattice version of the Atiyah-Singer index theorem.
problem Index problems of Wilson-Dirac operators on lattice approximations of manifolds.
method Formulates and proves a K K K -theoretic formula for an index-type invariant. result Main theorem gives a formula for an index-type invariant of operators on lattice approximations of closed integral affine manifolds.
Vertex distortion measures how far lattice knots deviate from straight lines.
problem Measuring how much lattice knots deviate from straight paths.
method Analogous to smooth knots, study vertex distortion in lattice knots.
result Vertex distortion is 1 only for the unknot and can be arbitrarily high.
Regular integer lattices are characterized by k unit vectors that build up their generator matrices. These have rank k for D-lattices, and are rank-deficient for A-lattices, for E_6 and E_7. We count lattice points inside hypercubes centered at the origin for all three types, as if classified by maximum infinity norm i…
The paper proves residual finiteness of certain lattices and constructs surfaces with specific fundamental groups.
problem Residual finiteness of lattices in P U ( 2 , 1 ) ~ \widetilde{\mathrm{PU}(2,1)} PU ( 2 , 1 ) and existence of smooth projective surfaces. method Proved residual finiteness of certain lattices and constructed surfaces using central extensions.
result First examples of residually finite lattices in P U ( 2 , 1 ) ~ \widetilde{\mathrm{PU}(2,1)} PU ( 2 , 1 ) and construction of surfaces with specific fundamental groups. Proves properties of arithmetic lattices and hyperbolic manifolds.
problem Properties of arithmetic lattices and hyperbolic manifolds.
method Study of normalizers of lattices and subgroup growth theory.
result Every arithmetic lattice has the property of being the normalizer of many sublattices.
A celebrated theorem of Hadwiger states that the Euler-Poincaré characteristic is the the unique invariant and continuous valuation on the distributive lattice of compact polyhedra in R^n that assigns value one to each convex non-empty such polyhedron. This paper provides an analogue of Hadwiger's result for finitely p…
This paper proves positivity of Riemann-Roch polynomials for hyperkähler manifolds.
problem Positivity of Riemann-Roch polynomials for hyperkähler manifolds.
method Lefschetz-type decomposition of the root of the Todd genus of hyperkähler manifolds via Rozansky-Witten theory.
result All coefficients of the Riemann-Roch polynomial of a hyperkähler manifold are positive.
The cubic lattice stick index of a knot type is the least number of sticks necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p,p+1)-torus knots, and show how composing and taking satellites can be used to obtain t…
New method finds lattice polygons that can be dissected into triangles with integer areas.
problem Finding lattice polygons that can be dissected into triangles with integer areas.
method A new version of Sperner's Lemma.
result Simple and complete description of lattice polygons that can be dissected into triangles with integer areas.
Study lattices in specific Lie groups for geometric structures.
problem Existence of lattices in Lie groups with certain geometric structures.
method Analyzing left invariant locally conformal Kähler or symplectic structures.
result Existence of lattices only in dimension 4 for Kähler structures, and in any even dimension for symplectic structures.
New discrete system from ellipsoidal billiards on honeycomb lattices.
problem Understanding dynamics in ellipsoidal billiards.
method Introduces a new discrete system related to double reflection nets on a honeycomb lattice.
result The system incorporates dynamics in both original and dual spaces in 2D.
L-CNNs approximate gauge actions, revealing fixed points with no lattice artifacts.
problem Approximating gauge actions with lattice artifacts.
method Lattice gauge-equivariant convolutional neural networks (L-CNNs).
result L-CNNs provide fixed point actions with no lattice artifacts.
Hybrid lattices produce subgroups of PU(n,1) from PU(n-1,1) lattices.
problem Understanding hybrid lattices and their properties in PU(n,1) groups.
method Hybridization construction of subgroups from pairs of lattices in PU(n-1,1).
result Hybrid subgroups are lattices for specific d d d and thin subgroups for others. The paper refines transformations of lattice diagrams and introduces dotted diagrams.
problem Investigating transformations and deformations of lattice diagrams and their associated dotted diagrams.
method Introducing dotted diagrams and investigating deformations of these diagrams, relating them to transformations of lattice diagrams.
result Refined results on the relation between deformations of admissible dotted diagrams and transformations of lattice diagrams.
Let G G G be a simply connected, solvable Lie group and Γ Γ Γ a lattice in G G G . The deformation space D ( Γ , G ) \mathcal{D}(Γ,G) D ( Γ , G ) is the orbit space associated to the action of $\Aut(G)$ on the space X ( Γ , G ) \mathcal{X}(Γ,G) X ( Γ , G ) of all lattice embeddings of Γ Γ Γ into G G G . Our main result generalises the classical rigidity theorems of Mal'tsev…
Classifies knots by lattice size, finding unknot ratios and crossing numbers.
problem Understanding the distribution of knots within different lattice sizes.
method Introduced a new knot classification by lattice size, analyzed ratios of unknots and knots with more than 10 crossings, and compared with theoretical estimates.
result Ratio of unknots decreases exponentially with lattice size, and computational results match theoretical estimates.
Proves effective slope gaps for lattice surfaces.
problem Proving effective slope gaps for lattice surfaces.
method Proves effective slope gap distribution for square torus and general lattice surfaces.
result Effective slope gap distribution result for lattice surfaces.
Improved lattice field theory simulations with local-Autoregressive Conditional Normalizing Flow.
problem Efficiently sampling lattice field theories with computational challenges.
method Integrates locality into autoregressive conditional normalizing flows.
result Autocorrelation times improved by orders of magnitude for φ 4 φ^{4} φ 4 theory on a 2D lattice. Proves actions of higher rank lattices on hyperbolic spaces are elementary.
problem Understanding actions of higher rank lattices on hyperbolic spaces.
method Proves actions are either elliptic or parabolic, generalizing tree actions.
result Any morphism from a higher rank lattice to a hierarchically hyperbolic group has finite image.
Study answers arithmeticity question for normal subgroup of lattices.
problem Arithmeticity of discrete subgroups of semisimple Lie groups with dense commensurators.
method Examined normal subgroups of lattices in semisimple Lie groups.
result Positive answer to Greenberg-Shalom's question for lattices.
Study investigates lattices fibring over the circle, focusing on BNSR invariants.
problem Investigating BNSR invariants of irreducible uniform lattices.
method Examines BNSR invariants and Bestvina-Brady groups to understand lattice properties.
result Irreducibility of lattices is linked to the vanishing of BNSR invariants for all finite-index subgroups.
A new surgery formula for knot lattice homology.
problem Developing a new surgery formula for knot lattice homology.
method Provided an iterable version of the surgery formula using doubly-filtered spaces and involutive data.
result Computed knot lattice spaces for specific knots and three-manifolds.
Defines and analyzes the holonomy Lie algebra of geometric lattices.
problem Holonomy Lie algebra of geometric lattices.
method Combinatorial definition and analysis of solvable pairs of lattices.
result Holonomy Lie algebra structure of hypersolvable lattices.
One type of switch simplifies operations on lattice knots.
problem Operations on lattice knots are complex.
method Reduced operations to one type of local switch.
result Simplified set of operations on lattice knots.
New lattices in higher dimensions have dense surface subgroups.
problem Finding dense subgroups in higher-dimensional arithmetic lattices.
method Exhibited nonuniform arithmetic lattices in SO(n,1).
result Contain Zariski-dense surface subgroups.
We construct an infinite commutative lattice of groups whose dual spaces give Kauffman finite-type invariants of long virtual knots. The lattice is based "horizontally" upon the Polyak algebra and extended "vertically" using Manturov's functorial map f f f . For each n n n , the n n n -th vertical line in the lattice contains a…
The (isothermic) compressibility of lattice knots can be examined as a model of the effects of topology and geometry on the compressibility of ring polymers. In this paper, the compressibility of minimal length lattice knots in the simple cubic, face centered cubic and body centered cubic lattices are determined. Our r…