Classifies limits of groups of involutions in SL(2,F) over local fields.
problem Classifying limits of groups of involutions in SL(2,F) over local fields.
method Classifying involutions, proving polar decompositions, classifying limits.
result Classification of Chabauty limits of various groups of involutions.
Define web algebras for annular SL(2) and SL(3) using foam TQFTs.
problem Define invariant algebras for tangles in the annulus.
method Use annular foam TQFTs to assign complex of bimodules to tangles.
result Establish properties of algebras and bimodules, and provide a bijection between webs and paths.
Classifies actions of SL(n,R) and SL(n,Z) on closed n-manifolds.
problem Classifying actions of SL(n,R) and SL(n,Z) on closed n-manifolds.
method Classification and construction of actions, using geometric structures.
result New exotic actions of SL(n,Z) on n-tori and their connected sums.
Study of Lagrangian submanifolds in pseudo-nearly Kähler SL(2,R)×SL(2,R).
problem Characterize Lagrangian submanifolds in pseudo-nearly Kähler space.
method Analyze isometry group and classify extrinsically homogeneous Lagrangian submanifolds.
result Complete classification of extrinsically homogeneous Lagrangian submanifolds.
Classifies SL(2;C) representations of a Brieskorn homology 3-sphere.
problem Classifying SL(2;C) representations of a Brieskorn homology 3-sphere.
method Shows irreducible representations are conjugate to SU(2) or SL(2;R) representations. Constructs SL(2;R) representations from PSL(2;R) representations of the orbifold.
result Classifies SL(2;C) representations and provides a construction method.
We show the correspondence between left invariant flat projective structures on Lie groups and certain prehomogeneous vector spaces. Moreover by using the classification theory of prehomogeneous vector spaces, we classify complex Lie groups admitting irreducible left invariant flat complex projective structures. As a r…
Explicitly found generators of cohomology for SL_n(Z) using sharbly cycles and cosharbly cocycles.
problem Finding explicit generators for the cohomology of SL_n(Z).
method Using sharbly cycles and cosharbly cocycles, and applying Borel-Serre duality.
result Explicitly found generators of H_t(SL_n(Z),St) in terms of sharbly cycles and cosharbly cocycles.
Extends s l ( 3 ) sl(3) s l ( 3 ) invariant to tangles via webs and functors.
problem Extending s l ( 3 ) sl(3) s l ( 3 ) invariant to tangles. method Defined a category of s l ( 3 ) sl(3) s l ( 3 ) webs and a functor from tangles to webs. result Recover the s l ( 3 ) sl(3) s l ( 3 ) -polynomial invariant for links from tangles. We construct an equivariant colored sl(N)-homology for links, which generalizes both the colored sl(N)-homology defined by the author and the equivariant sl(N)-homology defined by Krasner. The construction is a straightforward generalization of that of the colored sl(N)-homology. The proof of invariance is based on a s…
The paper classifies topological properties of specific geometric forms on manifolds.
problem Classifying topological properties of closed G ~ 2 \widetilde{\mathrm{G}}_2 G 2 , S L ( 3 ; C ) \mathrm{SL}(3;\mathbb{C}) SL ( 3 ; C ) and S L ( 3 ; R ) 2 \mathrm{SL}(3;\mathbb{R})^2 SL ( 3 ; R ) 2 forms. method Algebraic and topological techniques, including characteristic classes and obstruction theory, with recent h h h -principles. result Complete classification of closed S L ( 3 ; C ) \mathrm{SL}(3;\mathbb{C}) SL ( 3 ; C ) forms up to homotopy. New sl(2) action defined on a mathematical module.
problem No specific problem stated; focuses on mathematical construction.
method Construction of sl(2)-action on equivariant skein lasagna module.
result Infinitesimal sl(2)-symmetries constructed.
Study Chabauty limits of subgroups of S L ( n , Q p ) SL(n, \mathbb{Q}_p) S L ( n , Q p ) , focusing on parahoric and conjugate subgroups.
problem Classify and understand Chabauty limits of subgroups of S L ( n , Q p ) SL(n, \mathbb{Q}_p) S L ( n , Q p ) . method Use various Levi decompositions and Bruhat–Tits buildings to classify limits.
result Found infinitely many non-conjugate Chabauty limits for n ≥ 7 n \geq 7 n ≥ 7 . Characterizes groups for specific types of representations into SL(d,R).
problem Understanding groups admitting certain types of representations into SL(d,R).
method Characterization and bounds on cohomological dimension.
result Bounds on cohomological dimension and characterizations of representations.
It is showed that many examples of AMD submanifolds of higher dimensions come from SL normal bundles. A symmetry property of SL submanifolds and Björling type problem for SL normal bundles are also mentioned.
Classifies real-analytic SL(n,R) actions on closed manifolds.
problem Classifying real-analytic SL(n,R) actions on closed manifolds.
method Analytic and smooth classification methods.
result Extensions and classifications of Fisher--Melnick's work.
We construct a categorification of the quantum sl_3 projectors, the sl_3 analog of the Jones-Wenzl projectors, as the stable limit of the complexes assigned to k-twist torus braids (as k goes to infinity) in a suitably shifted version of Morrison and Nieh's geometric formulation of sl_3 link homology (math.GT/0612754).…
Decomposes S L 3 SL_3 S L 3 skein algebras for surfaces.
problem Decomposing S L 3 SL_3 S L 3 skein algebras for surfaces. method Splitting surfaces into triangles and analyzing the resulting algebras.
result Explicit basis and injective splitting morphisms for S L 3 SL_3 S L 3 stated skein algebras. The study proves non-existence of hypercomplex structures on SL(3,R) and finds one on SL(2n+1,C).
problem Proving the non-existence of hypercomplex structures on specific Lie groups.
method Revising the classification of complex structures and using a complex product structure to find hypercomplex structures.
result No left-invariant hypercomplex structures on SL(3,R), and a new hypercomplex structure on SL(2n+1,C).
We define two functors from Elias and Khovanov's diagrammatic Soergel category, one targeting Clark-Morrison-Walker's category of disoriented sl(2) cobordisms and the other the category of (universal) sl(3) foams.
Vanishing of cohomology for SL_2 groups over special fields.
problem Vanishing of cohomology for SL_2 groups over non-Archimedean local fields and S-integers.
method Analyzing continuous bounded cohomology of SL_2(k) and generalizing to tree automorphisms.
result Continuous bounded cohomology of SL_2(k) vanishes in all positive degrees for non-Archimedean local fields k.
Two specific Einstein metrics found on a product of SL(2,R) groups.
problem Classifying left-invariant Einstein metrics on a specific group product.
method Analyzing bi-invariant metrics under a one-parameter subgroup.
result Found two specific Einstein metrics: the Killing form and a nearly pseudo-Kähler metric.
Odd-dimensional SL(n,Q) contains dense surface subgroups.
problem Finding dense subgroups in SL(n,Q) for odd n.
method Constructing a continuous path of representations.
result Existence of dense surface subgroups in SL(n,Q) for odd n.
The paper calculates colored Jones polynomials for specific link configurations.
problem Computing colored Jones polynomials in general is difficult, but the paper provides explicit formulas.
method Uses Kuperberg's A 2 A_2 A 2 skein relation and one-row Young diagrams. result Derives the s l 3 \mathfrak{sl}_3 sl 3 tail of ( 2 , 2 m ) (2,2m) ( 2 , 2 m ) -torus links and false theta series. Given a finite volume hyperbolic 3-manifold, we compose a lift of the holonomy in SL(2,C) with the n-dimensional irreducible representation of SL(2,C) in SL(n,C). In this paper we give local coordinates of the SL(n,C)-character variety around this representation. As a corollary, this representation is isolated among al…
Study local limits of subgroups in S L 3 ( R ) SL_3(\mathbb{R}) S L 3 ( R ) .
problem Understanding subgroups of S L 3 ( R ) SL_3(\mathbb{R}) S L 3 ( R ) in a specific topology. method Conjugation and Chabauty topology analysis.
result Described local limits of all connected subgroups.
Formula for evaluating foams related to s l N \mathfrak{sl}_N sl N .
problem Evaluating foams decorated with s l N \mathfrak{sl}_N sl N . method Purely combinatorial formula connecting to link homology.
result Integral polynomial evaluation directly related to link homology.
The paper constructs bases for cluster varieties using m S L 3 {
m SL}_3 m S L 3 -webs and laminations.
problem Cluster varieties associated to m S L 3 {
m SL}_3 m S L 3 -local systems on surfaces. method Introducing m S L 3 {
m SL}_3 m S L 3 -laminations, developing quantum and classical trace maps, and constructing bases. result Bases of regular functions on m P G L 3 {
m PGL}_3 m P G L 3 cluster varieties constructed from m S L 3 {
m SL}_3 m S L 3 -laminations. Study on cohomology of SL_n(Z) for n>=3, proving vanishing of certain cohomology groups.
problem Determine the cohomology of SL_n(Z) for n>=3.
method Construct a partial resolution of the Steinberg module to show vanishing of specific cohomology groups.
result Vanishing of codimension-2 rational cohomology group H^{{n \choose 2} -2} for n >= 3.
SL2MF predicts synthetic lethality using logistic matrix factorization.
problem Predicting synthetic lethality in human cancers from limited experimental data.
method Logistic matrix factorization incorporating biological knowledge.
result SL2MF effectively predicts known and unknown SL interactions.
Lectures introduce evaluation of SL(3) foams and link homology.
problem Categorification of quantum s l 3 \mathfrak{sl}_3 sl 3 web and link invariant. method Introduction and review of S L ( 3 ) \mathsf{SL}(3) SL ( 3 ) foams and their evaluation. result Categorification of Kuperberg quantum s l 3 \mathfrak{sl}_3 sl 3 web and link invariant. The paper classifies totally geodesic Lagrangian submanifolds in a specific pseudo-nearly Kähler space.
problem Understanding Lagrangian submanifolds in pseudo-nearly Kähler spaces.
method Examining four classes of submanifolds based on their behavior with respect to an almost product structure, then classifying totally geodesic ones.
result A complete classification of totally geodesic Lagrangian submanifolds in the pseudo-nearly Kähler S L ( 2 , R ) i m e s S L ( 2 , R ) \mathrm{SL}(2,\mathbb{R}) imes\mathrm{SL}(2,\mathbb{R}) SL ( 2 , R ) im es SL ( 2 , R ) . Study SL(2,C) character schemes for finitely generated groups.
problem Characterize SL(2,C) representations of finitely generated groups.
method Define coordinate rings and equations for SL(2,C) character schemes.
result Explicit equations for character schemes of finitely presented groups.
SL(n) covariant valuations on Orlicz spaces are represented and characterized.
problem Representing SL(n) covariant valuations on Orlicz spaces.
method Representation theorem established for continuous, SL(n) covariant vector-valued valuations.
result Unique characterization of SL(n) covariant valuations as moment vectors.
Efficient method computes m S L ( 3 , C ) {
m SL}(3,\mathbb{C}) m S L ( 3 , C ) -character varieties for two-generator groups.
problem Computing m S L ( 3 , C ) {
m SL}(3,\mathbb{C}) m S L ( 3 , C ) -character varieties for two-generator groups. method Efficient method for two-generator groups.
result Efficient computation of m S L ( 3 , C ) {
m SL}(3,\mathbb{C}) m S L ( 3 , C ) -character varieties. Extends knot invariant computation to symmetrically colored sl_N.
problem Computing quantum knot invariants for s l N \mathfrak{sl}_N sl N . method Develops symmetrically colored R matrix for s l N \mathfrak{sl}_N sl N . result Defines F K s l N , s y m F^{\mathfrak{sl}_N, sym}_K F K sl N , sy m for positive braid knots. Paper proves zero stability for one-row colored sl₃-Jones polynomials.
problem Stability of coefficients in colored Jones polynomials.
method Linear skein theory based on Kuperberg's sl₃-webs.
result Zero stability for B-adequate links with anti-parallel twist regions.
New central elements found in a quantum algebra related to knot theory.
problem Understanding the algebraic structure of S L d \mathrm{SL}_d SL d -skein algebras. method Threaded polynomials from symmetric functions.
result Extraction of central elements in S L d \mathrm{SL}_d SL d -skein algebra. 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.
New operator reveals unique features of sl(N) link homology.
problem Understanding unique features of sl(N) link homology.
method Constructing an operator on sl(N) link homology with mod N coefficients.
result Structural features of sl(P) link homology are revealed.
In this paper I define certain interesting 2-functors from the Khovanov-Lauda 2-category which categorifies quantum sl(k), for any k>1, to a 2-category of universal sl(3) foams with corners. For want of a better name I use the term "foamation" to indicate those 2-functors. I conjecture the existence of similar 2-functo…
New representations of Whitehead link group found in SL(3, C).
problem Characterizing SL(3, C) representations of Whitehead link group.
method Using pairs of order three elements in SL(3, C) and exceptional Dehn surgery.
result Found algebraic component in SL(3, C) character variety.
Combining known spectral sequences with a new spectral sequence relating reduced and unreduced sl(N)-homology yields a relationship between the Homflypt-homology of a knot and its sl(N)-concordance invariants. As an application, some of the sl(N)-concordance invariants are shown to be linearly independent.
SL(2,Z) forms lead to new anomaly formulas.
problem Anomaly cancellation in physics.
method SL(2,Z) modular forms.
result New anomaly cancellation formulas derived.
The main purpose of this paper is calculation of differential invariants which arise from prolonged actions of two Lie groups SL(2) and SL(3) on the n n n th jet space of R 2 R^2 R 2 . It is necessary to calculate n n n th prolonged infenitesimal generators of the action.
Researchers calculate colored Jones polynomials for pretzel links using Kuperberg's theory.
problem Calculating colored Jones polynomials for general oriented links is difficult.
method Using Kuperberg's linear skein theory, they focus on one-row polynomials for pretzel links.
result Existence of tails for specific pretzel knots' Jones polynomials is shown.
Quantum Frobenius map for S L 3 SL_3 S L 3 skein modules constructed and described.
problem Constructing a quantum Frobenius map for S L 3 SL_3 S L 3 skein modules. method Using threading polynomials and the Frobenius map of Parshall-Wang for quantum group O q ( S L 3 ) . \mathcal{O}_q(SL_3). O q ( S L 3 ) . result Described the quantum Frobenius map for S L 3 SL_3 S L 3 skein modules. A new invariant for links generalizes Alexander polynomial for sl_3.
problem Defining a non-abelian generalization of the Alexander polynomial.
method Using quantum s l 3 \mathfrak{sl}_3 sl 3 representations and Laurent polynomials. result Established a direct relation between Δ s l 3 Δ_{\mathfrak{sl}_3} Δ sl 3 and the Alexander polynomial. Study character varieties for real forms of SL_n(C).
problem Characterize representations of groups in real forms of SL_n(C).
method Define and analyze G-character varieties, study involutions, and examples.
result Irreducible representations fixed by involutions are conjugate to real forms.