Study Type C skein modules using Sp(2n) webs and construct transparent elements.
problem Understanding Type C skein modules and constructing transparent elements. method Diagrammatic approach using multivariable Chebyshev polynomials and explicit braiding formulas.
result Construction of transparent elements in the skein module at roots of unity.
Explains connections between monopoles and modules on elliptic curves.
problem Understanding relationships between different mathematical objects.
method Explains equivalences between monopoles and polystable bundles and modules.
result Monopoles and polystable difference modules on elliptic curves are equivalent.
Study periodic monopoles via algebraic difference modules.
problem Classify periodic monopoles of GCK type.
method Relate geometric monopoles to algebraic difference modules.
result Established a Kobayashi-Hitchin correspondence for periodic monopoles.
Study Swan modules and homotopy types, resolving Wall and Dyer questions.
problem Homotopy classification of CW-complexes and Swan modules.
method Analysis of Swan modules and their stably free properties.
result Existence of non-free stably free Swan modules and implications for homotopy equivalence.
Study meromorphic open-string vertex algebras and modules over Riemannian manifolds.
problem Characterize meromorphic open-string vertex algebras and their modules over Riemannian manifolds.
method Explicitly determine bases for meromorphic open-string vertex algebras and their modules, using parallel tensors and eigenfunctions of the Laplace-Beltrami operator.
result Every irreducible module of a specific type is completely reducible if every composition factor is generated by eigenfunctions of eigenvalue p(p−1)K for some p∈Z+. Study geodesic orbit property on pseudo-Riemannian H-type nilmanifolds.
problem Characterize geodesic orbit property for pseudo-Riemannian H-type Lie groups.
method Extend results from Riemannian to pseudo-Riemannian H-type Lie groups, focusing on minimal admissible Clifford modules.
result Complete characterization of geodesic orbit property for pseudo-Riemannian H-type Lie groups.
Formula for Dehn twists on HOMFLY-PT skein modules with applications.
problem Understanding the action of Dehn twists on HOMFLY-PT skein modules.
method Introduced a formula for the action of Dehn twists on the HOMFLY-PT type skein module of a surface.
result Constructed an invariant \( z(M) \) for integral homology 3-spheres, finite type invariant of order \( n \).
New invariant for 4-manifolds with framed links, stronger than existing invariants.
problem Distinguishing 4-manifolds with framed links.
method Introducing a new invariant called KLS lasagna homotopy type.
result The new invariant is stronger than existing invariants.
We explain an elementary topological construction of the Springer representation on the homology of (topological) Springer fibers of types C and D in the case of nilpotent endomorphisms with two Jordan blocks. The Weyl group and component group actions admit a diagrammatic description in terms of cup diagrams which app…
We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type D) is a module over the algebra an…
In the bordered Floer theory, gluing thickened torus of positive meridional Dehn twist to the boundary of a knot complement result in the knot complement of increased framing. For a fixed knot K, we construct a direct system of positively framed knot complements and study the direct limit. We also study the morphism sp…
Study on Lie algebras from Clifford modules, focusing on specific types.
problem Characterizing Lie algebras from Clifford modules and their graded structures.
method Analysis of pseudo H-type Lie algebras and their representations through Clifford algebras. result Different types of Lie algebras have varying possibilities of containing pseudo H-type Lie algebras in their negative part. In the conditional setting we provide a complete duality between quasiconvex risk measures defined on L0 modules of the Lp type and the appropriate class of dual functions. This is based on a general result which extends the usual Penot-Volle representation for quasiconvex real valued maps.
Combinatorial approach to compute satellite knot invariants using graph theory.
problem Computing knot invariants for satellite knots using bordered Heegaard Floer homology.
method Construct weighted A∞-modules using decorated planar graphs and prove their isomorphism. result Combinatorial proof of A∞ structure relations for the constructed modules. 3D HQFTs constructed using graded monoidal categories.
problem Constructing 3D HQFTs with specific targets.
method Using spherical χ-fusion categories and the state sum method.
result 3D HQFTs constructed with target Bχ.
The paper uses non-abelian Hodge theory to generalize Kodaira vanishing theorems.
problem Generalizing Kodaira vanishing theorems for non-abelian settings.
method Non-abelian Hodge theory and Mixed Twistor D-modules.
result Generalized Kodaira vanishing theorems for various settings.
This paper generalize [7](math.GT/0601291): We construct new links invariants from g, a type I basic classical Lie superalgebra. The construction uses the existence of an unexpected replacement of the vanishing quantum dimension of typical module. Using this, we get a multivariable link invariant associated to any one …
Defines a new link invariant for type D webs.
problem Invariants for framed unoriented links in type D.
method Positive state sum for type D webs, derived from quantum so(2N) modules.
result Relates to Reshetikhin-Turaev's invariant.
The paper constructs tilting modules for knots using algebraic structures.
problem Understanding the algebraic structure of knot invariants.
method Constructing modules over Jacobian algebras associated with knots.
result The constructed modules M are rigid and τ-rigid, and their endomorphism algebra is isomorphic to the Jacobian algebra. We prove that if M is a CW-complex and M1 is its 1-skeleton then the crossed module Π2(M,M1) depends only on the homotopy type of M as a space, up to free products, in the category of crossed modules, with Π2(D2,S1). From this it follows that, if G is a finite crossed module and M is finite, then th…
Alternative basis for Kauffman bracket skein module of solid torus found using braids.
problem Computing Kauffman bracket skein module of lens spaces.
method Using Temperley--Lieb algebra of type B and braids.
result Alternative basis BmST for ${
m KBSM}\left({
m ST}
ight)$. We give an interpretation of Yetter's Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter's invariant depends only on the homotopy type of M, and the weak homot…
The branching problem for a couple of non-compatible Lie algebras and their parabolic subalgebras applied to generalized Verma modules was recently discussed in \cite{ms}. In the present article, we employ the recently developed F-method, \cite{KOSS1}, \cite{KOSS2} to the couple of non-compatible Lie algebras $({\LieGt…
We identify branched coverings (continuous open surjections p:Y->X of Hausdorff spaces with uniformly bounded number of pre-images) with Hilbert C*-modules C(Y) over C(X) and with faithful unital positive conditional expectations E:C(Y)->C(X) topologically of index-finite type. The case of non-branched coverings corres…
Develops a new model for radar waveform classification and clustering.
problem Classifying and clustering radar waveforms with different modulation types.
method Introduces a generalized multivariate Student-t mixture model with a new prior distribution for hyper-parameters.
result The method is less sensitive to initialization and provides more accurate results.
For a closed Riemannian manifold we extend the definition of analytic and Reidemeister torsion associated to an orthogonal representation of fundamental group on a Hilbert module of finite type over a finite von Neumann algebra. If the representation is of determinant class we prove, generalizing the Cheeger-Müller the…
PLoP optimizes LoRA placement for efficient large model finetuning.
problem Improving efficiency of LoRA adaptation for large models.
method Intuitive theoretical analysis for automatic adapter placement.
result PLoP consistently outperforms existing placement strategies.
Deep neural networks improve modulation recognition accuracy.
problem Improving modulation recognition accuracy in wireless signals.
method Developed and tested deep neural network architectures, including CNN, ResNet, DenseNet, and CLDNN.
result Achieved high accuracy (up to 88.5%) in recognizing wireless signal modulations.
Improves Bayesian optimization efficiency for mixed variable spaces.
problem Boosting sample efficiency in Bayesian optimization for mixed variable spaces.
method Proposes frequency modulated (FM) kernels to model complex dependencies across different types of variables.
result BO-FM outperforms competitors in various optimization problems.
We describe how to formulate Khovanov's functor-valued invariant of tangles in the language of bordered Heegaard Floer homology. We then give an alternate construction of Lawrence Roberts' Type D and Type A structures in Khovanov homology, and his algebra BΓn, in terms of Khovanov's theory of modules over …
MixFT re-partitions data into sub-domains for better TSFM fine-tuning.
problem Improving zero-shot forecasting for new time series domains.
method MixFT re-divides data using Bayesian mixtures into homogeneous sub-domains for separate fine-tuning.
result MixFT outperforms per-dataset fine-tuning methods.
Adversaries can fool deep learning modulator classifiers over wireless channels.
problem Vulnerability of deep learning modulator classifiers to adversarial attacks over wireless channels.
method Presented various adversarial attacks considering channel effects, including targeted and non-targeted attacks, and a universal adversarial perturbation attack.
result Modulation classification is vulnerable to adversarial attacks over wireless channels with realistic channel effects.
The classical Serre-Swan's theorem defines a bijective correspondence between vector bundles and finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an et…
Homflypt skein theory and string topology linked via 2-groupoids.
problem Understanding relations in Homflypt skein theory.
method Defined a 2-groupoid from the fundamental 2-groupoid of singular links, relating it to string topology.
result Relations in Homflypt skein theory are induced from a 2-groupoid.
This paper establishes a correspondence between Einstein-Hermitian metrics and stability for generalized holomorphic vector bundles over symplectic type generalized Kahler manifolds.
problem Establishing a relationship between Einstein-Hermitian metrics and stability for generalized holomorphic vector bundles.
method Using the moment map framework and Poisson modules, the authors prove the Kobayashi-Hitchin correspondence.
result The equivalence of the existence of an Einstein-Hermitian metric and ψ-polystability of a generalized holomorphic vector bundle. This paper is devoted to an elementary new construction of 1-singular Gelfand-Tsetlin modules using complex geometry. We introduce a universal ring Do together with the vector space S=S(Do) with basis Bo=B(Do) formed from some local distributi…
New mathematical proposal for TQFTs using TMF-modules.
problem Constructing new types of TQFTs at the intersection of topology, algebra, physics, and homotopy theory.
method Defines TMF-modules associated with symmetric bilinear forms and assigns them to closed 3-manifolds and maps of TMF-modules to 4-dimensional cobordisms.
result Invariants of 4-manifolds arising from 6-dimensional superconformal field theories, conjecturally generalizing the theta function of a lattice.
Invariants for 3D manifolds with boundaries using crossed modules.
problem Develop invariants for compact 3D manifolds with boundaries.
method Employ crossed modules to count correct colors over triangulations.
result Validated invariants for compact 3D manifolds with boundaries.
Several formulas for computing coarse indices of twisted Dirac type operators are introduced. One type of such formulas is by composition product in E-theory. The other type is by module multiplications in K-theory, which also yields an index theoretic interpretation of the duality between Roe algebra and stable Hi…
HyperST-Net uses hypernetworks to improve spatio-temporal forecasting.
problem Forecasting spatio-temporal data is challenging due to complex spatial and temporal factors.
method Proposes a framework based on hypernetworks with three modules: spatial, temporal, and deduction.
result Models achieve significant improvements over state-of-the-art baselines.
Paper defends deep learning classifiers against channel-aware adversarial attacks.
problem Deep learning classifiers are vulnerable to adversarial attacks.
method Channel-aware adversarial attacks are presented and defended against.
result Certified defense based on randomized smoothing makes classifiers robust.
We give a geometric interpretation of the building associated to the real Lie group E_6(-14) in terms of its 54-dimensional module.
An invariant for knots with colored bonds respects HOMFLYPT relation.
problem Modeling and distinguishing knots with colored bonds.
method Introducing a HOMFLYPT skein module for colored bonded knots.
result The non-rigid version of the module provides information about knottedness of bonds.
Explains Khovanov homology and its applications.
problem Understanding Khovanov homology and its applications.
method Expository lecture notes covering Jones polynomial, Khovanov homology, cobordism category, spectral sequences, and skein lasagna modules.
result Explains the Jones polynomial, Khovanov homology, and their applications.
Study long-term behavior of semi-Markov modulated processes using integral functions.
problem Analyzing long-term behavior of semi-Markov modulated processes involving integral functions.
method Using ergodic semi-Markovian environment and affine stochastic recurrence equation.
result Mixture type laws emerge in long-term limit for processes.
GNN-FiLM uses feature-wise linear modulation to improve graph neural networks.
problem Improving graph neural networks for better performance.
method Feature-wise linear modulation applied to target node representations in GNNs.
result GNN-FiLM outperforms baseline methods on a regression task for molecular graphs.
We introduce Dynamic Deep Neural Networks (D2NN), a new type of feed-forward deep neural network that allows selective execution. Given an input, only a subset of D2NN neurons are executed, and the particular subset is determined by the D2NN itself. By pruning unnecessary computation depending on input, D2NNs provide a…
For a closed manifold equipped with a Riemannian metric, a triangulation, a representation of its fundamental group on an Hilbert module of finite type (over of finite von Neumann algebra), and a Hermitian structure on the flat bundle associated to the representation, one defines a numerical invariant, the relative tor…