Study links curve singularities to quiver mutations.
problem Understanding the relationship between curve singularities and quiver mutations.
method Investigates the connection between the topology of curve singularities and the mutation equivalence of quivers associated with their morsifications.
result Established a connection between the topology of isolated curve singularities and the mutation equivalence of quivers.
New invariant measures loop iterations in algebraic structures.
problem Measuring the asymptotic behavior of loop iterations in algebraic structures.
method Introduced sign stability and cluster stretch factor to measure loops.
result Cluster algebraic entropies match cluster stretch factor.
EA improves clustering efficiency using crossover and mutation.
problem Efficiently clustering data using model-based methods.
method Developed an evolutionary algorithm with crossover and mutation for model-based clustering.
result EA outperforms EM algorithm and k-means in clustering performance.
Machine learning clusters mutations in cancer exomes, improving diagnostic speed and cost.
problem Extracting stable mutation structures from cancer exome data for early diagnostics.
method Statistically deterministic machine learning algorithm *K-means applied to exome samples.
result Majority of cancer types exhibit stable mutation clustering, while NMF methods are unstable.
The paper computes presentations of cluster modular groups and verifies their generation by Dehn twists.
problem Computing presentations and verifying generation of cluster modular groups.
method A method to compute presentations of saturated cluster modular groups and verification of generation by cluster Dehn twists.
result The cluster modular groups of specified types are virtually generated by cluster Dehn twists.
The paper defines matrices related to cluster transformations and proves certain quivers have no maximal sequences.
problem Proving quivers associated with once-punctured surfaces do not have maximal green or reddening sequences.
method Defining matrices related to cluster transformations and showing their relationships to the Jacobian and C-matrix.
result Quivers associated with once-punctured surfaces do not have maximal green or reddening sequences.
We generalise surface cluster algebras to the case of infinite surfaces where the surface contains finitely many accumulation points of boundary marked points. To connect different triangulations of an infinite surface, we consider infinite mutation sequences. We show transitivity of infinite mutation sequences on tria…
Paper tackles cancer mutation data challenges by creating useful low-dimensional representations.
problem Challenges in analyzing and using cancer mutation data for classification and clustering.
method Flatsomatic: variational autoencoders (VAEs) to create latent representations of somatic profiles.
result VAE embeddings perform better than PCA for clustering and equally well for classification.
Fixed points found in cluster modular groups under specific conditions.
problem Proving fixed points in cluster modular groups.
method Generalizing Kerckhoff's Nielsen realization theorem for cluster modular groups, using convexity of log-cluster variables.
result Finite subgroups of cluster modular groups have fixed points in cluster manifolds under certain conditions.
A cluster variety of Fock and Goncharov is a scheme constructed from the data related to the cluster algebras of Fomin and Zelevinsky. A seed is a combinatorial data which can be encoded as an n×n matrix with integer entries, or as a quiver in special cases, together with n formal variables. A mutation is a c…
Deep generative model for healthcare data identifies coherent substructures and mutational clusters.
problem Analytical challenges in healthcare data, including sparsity, missingness, and small sample sizes.
method Proposes a deep generative Bayesian model with collapsed Gibbs sampling for multinomial count data.
result Identifies coherent substructures and biologically meaningful mutational clusters in cancer data.
We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh and involves mutations of quivers and diagrams defined in the theory of cluster algebras. We generalize our construction by assigning to every …
Geometrically, Legendrian surfaces related by surgery have related skein-valued cluster spaces.
problem Understanding the skein-valued cluster transformation in Legendrian surfaces.
method Geometric considerations of moduli of holomorphic curves.
result Skein-valued cluster transformation of Legendrian surfaces related by surgery.
In this paper, we study the distribution of the genuses of cluster quivers of finite mutation type. First, we prove that in the 11 exceptional cases, the distribution of genuses is 0 or 1. Next, we consider the relationship between the genus of an oriented surface and that of cluster quivers from this surface. It…
New knot invariants derived using quantum cluster algebras.
problem Deriving new knot invariants from quantum cluster algebras.
method Interpreting R-matrix of Uq(sl2) as cluster transformation, introducing auxiliary parameter ε. result Derives perturbed-Alexander invariants with higher-order terms in ε. A cluster variety of Fock and Goncharov is a scheme constructed by gluing split algebraic tori, called seed tori, via birational gluing maps called mutations. In quantum theory, the ring of functions on seed tori are deformed to non-commutative rings, represented as operators on Hilbert spaces. Mutations are quantized …
We try to give a cluster algebraic interpretation of complex volume of knots. We construct the R-operator from the cluster mutations, and we show that it is regarded as a hyperbolic octahedron. The cluster variables are interpreted as edge parameters used by Zickert in computing complex volume.
Mutation graph of support τ-tilting modules over skew-gentle algebras is connected.
problem Understanding the structure of support τ-tilting modules over skew-gentle algebras.
method Introducing mutation of maximal rigid objects and using exchange triangles to define mutations of support τ-tilting modules.
result The mutation graph of support τ-tilting modules over a skew-gentle algebra is connected.
We construct geometric realization for non-exceptional mutation-finite cluster algebras by extending the theory of Fomin and Thurston to skew-symmetrizable case. Cluster variables for these algebras are renormalized lambda lengths on certain hyperbolic orbifolds. We also compute growth rate of these cluster algebras, p…
Framework uses machine learning to distinguish major COVID-19 variants.
problem Discriminate and visualize associations between major COVID-19 variants based on genome sequences.
method Unsupervised machine learning methods, including k-mer analysis, PCA, t-SNE, UMAP, and agglomerative hierarchical clustering.
result Framework effectively distinguishes between major variants and identifies emerging variants.
In many real life problems, objects are described by large number of binary features. For instance, documents are characterized by presence or absence of certain keywords; cancer patients are characterized by presence or absence of certain mutations etc. In such cases, grouping together similar objects/profiles based o…
Quantum trace maps for surfaces are shown to be compatible under triangulations.
problem Constructing and understanding quantum trace maps for surfaces.
method Developed quantum mutation maps between subalgebras of quantum torus algebras for different triangulations.
result Quantum trace maps are natural and independent of triangulation choices.
We prove the existence of Lagrangian fillings for Dn-type Legendrian links.
problem Exact Lagrangian fillings of Legendrian links of Dn-type. method Legendrian weave calculus and construction of 1-cycles.
result Existence of a Lagrangian filling represented by a weave.
We study iteration maps of recurrence relations arising from mutation periodic quivers of arbitrary period. Combining tools from cluster algebra theory and (pre)symplectic geometry, we show that these cluster iteration maps can be reduced to symplectic maps on a lower dimensional submanifold, provided the matrix repres…
Constructs quivers related to Weyl groups and higher Teichmüller spaces.
problem Understanding the structure of higher Teichmüller spaces.
method Constructs weighted quivers and computes cluster transformations.
result Establishes a correspondence between quivers and higher Teichmüller spaces.
The paper proves there are many Lagrangian fillings for Legendrian links of affine type.
problem Proving the existence of many Lagrangian fillings for Legendrian links of affine type.
method Using cluster structures and Coxeter mutation to prove the existence of fillings.
result There are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of affine type.
We present a global optimization algorithm for clustering data given the ratio of likelihoods that each pair of data points is in the same cluster or in different clusters. To define a clustering solution in terms of pairwise relationships, a necessary and sufficient condition is that belonging to the same cluster sati…
We propose a new description of 3d N=2 theories which do not admit conventional Lagrangians. Given a quiver Q and a mutation sequence m on it, we define a 3d N=2 theory T[(Q,m)] in such a way that the Sb3 partition function of the theory coincides with the cluster partition f…
Study character varieties of surfaces using cluster algebras and Poisson structures.
problem Character varieties of surfaces and their Poisson structures.
method Use Bonahon-Wong's trace map and cluster algebras associated with ideal triangulations.
result Recover Goldman Poisson algebra from cluster algebra structure and show automorphisms.
Characterizes pseudo-Anosov mapping classes on general marked surfaces.
problem Stability of mapping classes on marked surfaces.
method Cluster algebraic description and reduction procedure of mapping classes.
result Characterizes pseudo-Anosov mapping classes in terms of uniform sign stability.
Paper proves conjecture linking knot braid length to representation existence.
problem Existence of geometric knot representations from braid presentations.
method Analyzes cluster mutations and polynomial equations from braid presentations.
result Hikami-Inoue conjecture holds if and only if braid length is odd.
Genus 2 mutation is the process of cutting a 3-manifold along an embedded closed genus 2 surface, twisting by the hyper-elliptic involution, and gluing back. This paper compares genus 2 mutation with the better-known Conway mutation in the context of knots in the 3-sphere. Despite the fact that any Conway mutation can …
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. New algebra for twice-punctured torus curves.
problem Constructing a new algebra for skein theory.
method Using Heegaard dual of Iwahori--Hecke operator, Dehn twists are represented.
result Automorphisms correspond to Dehn twists on the twice-punctured torus.
The study finds many Lagrangian fillings for Legendrian links of specific types.
problem Understanding the number and types of Lagrangian fillings for Legendrian links.
method Proved the existence of at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of finite or affine Dynkin type.
result Found many Lagrangian fillings with rotational and conjugation symmetries for specific types of Legendrian links.
Study of machine learning in quiver gauge theories and Seiberg duality.
problem Determining dualities in quiver gauge theories using machine learning.
method Defined and explored various questions related to binary and multi-class duality determination, evaluated performance of different classifiers, and analyzed effects of additional data.
result High accuracy and confidence achieved in determining dualities using machine learning.
Modeling correlated mutations in cancer for personalized treatment.
problem Identifying mutations for personalized cancer therapy in heterogeneous profiles.
method Proposed correlated zero-inflated negative binomial process with mixed beta-Bernoulli and variational inference.
result Identified biologically relevant correlations between somatic mutations.
Flatsomatic compresses cancer mutation data with VAEs, maintaining predictive power.
problem Compressing somatic mutation profiles in cancer while preserving predictive power.
method Flatsomatic uses a Variational Auto Encoder (VAE) with MLP architecture, optimizing evidence lower bound and beta-VAE for latent space regularization.
result Flatsomatic embeddings maintain predictive power of original data, reducing dimensionality from 8,298 to 64.
The paper connects Legendrian links to cluster algebras via microlocal methods.
problem Understanding the relationship between Legendrian links and cluster algebras.
method Microlocal parallel transport of sheaf quantizations of Lagrangian fillings.
result Existence of quasi-cluster A-structures and cluster Poisson structures. Improved genetic programming by optimizing mutation operators for continuous program search.
problem Small syntactic mutations in genetic programming can lead to unpredictable behavioral shifts.
method Learned a compact trading-strategy DSL, created a block-factorized embedding, and designed geometry-compiled mutation operators.
result Geometry-compiled mutation operators discover strong strategies using fewer evaluations and achieve higher Sharpe ratios.
Develops quantum cluster algebra approach to solve tetrahedron equation.
problem Investigates a three-dimensional generalization of the Yang-Baxter equation.
method Quantum cluster algebra approach with realization of quantum Y-variables in terms of q-Weyl algebras.
result Obtains a solution with three spectral parameters and reproduces Sergeev's R matrix.
Consensus Monte Carlo clusters big data with shared anchors.
problem Clustering and feature allocation in large datasets.
method Bayesian nonparametric models, Dirichlet process, Indian buffet process, consensus Monte Carlo.
result Valid for various sampling models and priors.
Automatically extracts phenotypes from cancer clinical notes for genetic studies.
problem Lack of structured patient representations in EHRs.
method Clustering of medical terms and sentences in clinical notes.
result 341 significant associations between clinical features and somatic mutations.
We define a link homology theory that is readily seen to be both isomorphic to reduced odd Khovanov homology and fully determined by data impervious to Conway mutation. This gives an elementary proof that odd Khovanov homology is mutation invariant, and therefore that mod 2 Khovanov homology is mutation invariant. We a…
Mutation Validation method assesses model fit using mutated training labels without validation sets.
problem Model selection and hyperparameter tuning accuracy in supervised learning.
method Mutation Validation (MV) method using mutated training labels to assess model fit.
result MV provides more accurate model selection and stable hyperparameter tuning results.
We give a new, elementary proof that Khovanov homology with Z/2Z--coefficients is invariant under Conway mutation. This proof also gives a strategy to prove Baldwin and Levine's conjecture that δ--graded knot Floer homology is mutation--invariant. Using the Clifford module structure on $\widetilde…
Mathematician summarizes protein geometry and mutation effects.
problem Understanding how proteins mutate and their structure-function relationship.
method Mathematical analysis of protein structures and functions, focusing on hydrogen bonds and secondary structure.
result Protein secondary structure regulates mutation by stabilizing or destabilizing regions.
We study the effect of mutation on link concordance and 3-manifolds. We show that the set of links concordant to sublinks of homology boundary links is not closed under positive mutation. We show that mutation does not preserve homology cobordism classes of 3-manifolds. A significant consequence is that there exist 3-m…