Researchers create non-normal affine structures for 3-manifolds, calculating Poincaré series.
problem Calculating Poincaré series for plumbed 3-manifolds.
method Constructing non-normal affine monoids and modules associated with negative definite plumbed 3-manifolds.
result Combinatorial formulas for Seiberg-Witten invariants and polynomial generalizations.
Affine structures on Lie groupoids are studied, showing rich algebraic properties.
problem Understanding affine structures on Lie groupoids.
method Analyzing affine k-vector fields, k-forms, and (p,q)-tensors, and showing their algebraic properties. result The space of affine structures forms a 2-vector space over multiplicative structures, and affine multivector fields have a Lie 2-algebra structure.
New monoids tied to symmetric group and Jones/Brauer monoids discovered.
problem Understanding monoids attached to symmetric group and related monoids.
method Introduced ramified monoids and tied-like monoids, provided presentations.
result Found new monoids that cannot be described as ramified monoids.
Categorifies Hecke algebra subalgebra using sheaves and bimodules.
problem Categorify maximal commutative subalgebra of type A Hecke algebra.
method Construct monoidal functor from coherent sheaves on flag Hilbert scheme to Soergel bimodules, match Hochschild homology with sheaf Euler characteristics.
result Categorified projectors correspond to dg algebras of functions on affine charts of flag Hilbert schemes, conjecturally matching homology of glN projectors. Study on algebraic structures of virtual singular braid monoid and its splittable extension.
problem Algebraic structures of virtual singular braid monoid and its splittable extension.
method Exploration of algebraic structures, construction of representation.
result Monoid VSBn is the splittable extension of VSPn by the symmetric group Sn. We study monoids generated by Zariski-van Kampen generators in the 17 fundamental groups of the complement of logarithmic free divisors in C^3 listed by Sekiguchi (Theorem 1). Five of them are Artin monoids and eight of them are free abelian monoids. The remaining four monoids are not Gaussian and, hence, are neither G…
In the present paper we define dual monoids for all Artin-Tits groups and we prove that for the type A~n we get a (quasi)-Garside structure. Such a structure provides normal forms for the Artin-Tits group elements and allows to solve some questions such as to determine the centralizer of a power of the Coxeter…
In this paper we define a monoid of pseudo braids and prove that this monoid is isomorphic to a singular braid monoid. We also prove an analogue of Markov's theorem for pseudo braids.
The paper explores actions of various monoids on geometric structures.
problem Exploring new definitions and structures in geometric actions.
method Study of smooth and holomorphic actions of specific monoids.
result Recovery of known structures like holomorphic and complex vector bundles.
The paper introduces a Deligne complex for Artin monoids and studies its properties.
problem Understanding geometric structures associated with Artin monoids.
method Constructing a Deligne complex for Artin monoids and analyzing its properties.
result The Deligne complex for Artin monoids is contractible and can be embedded into the Deligne complex for the corresponding Artin group.
Research categorizes 2-cobordisms and finds non-finitely axiomatizable equational theories.
problem Equational theories of local monoids in cobordism categories are not finitely axiomatizable.
method Introduced combinatorial data encoding 2-cobordisms, constructed von-Neumann-regular extensions, and studied various quotients. result Local monoids of 2Cob do not have finitely axiomatizable equational theories. This paper gives a self-contained and complete proof of the isomorphism of freely generated monoids extracted from Temperley-Lieb algebras with monoids made of Kauffman's diagrams.
The paper studies properties of Artin monoid Cayley graphs and their quasi-isometry to Deligne complexes.
problem Investigate properties of Artin monoid Cayley graphs.
method Show quasi-isometry to modified Deligne complex, address infinite diameter conjecture.
result Prove conjecture about infinite diameter for Artin groups containing specific subgroups.
The paper discusses strictification and non-strictification of monoidal categories.
problem Transforming monoidal categories into strict or non-strict ones.
method Explicit models and 2-adjunctions for strictification and non-strictification.
result Strictification and non-strictification are part of a pair of free-forgetful 2-adjunctions.
The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in the simplicial category, are submonoids of monoids one finds in Temperley-Lieb algebras, and as the monoids of Temperley-Lieb algebras are linked to situations where an endofunctor is adjoint to itself, so the monoids of simplicial endomorphi…
Defines Hopf monoid of directed graphs and its invariant.
problem Defining a Hopf monoid for directed graphs.
method Defines Hopf monoid of directed graphs and shows embedding in GP.
result Invariant of directed graphs coincides with strict chromatic polynomial.
The abstract discusses model structures and modules in simplicial spaces and rings.
problem Model structures and modules in simplicial spaces and rings.
method Proves model structures and properties of modules in simplicial spaces and rings.
result Weak equivalences between monoids induce Quillen equivalences between categories of modules.
Researchers found faithful representations for surface singular braid monoids.
problem Linearity problem for surface singular braid monoids.
method Derived new presentations with reduced relations and generators.
result Representations are not faithful for at least two or three strands.
The paper constructs semistrict monoidal 2-categories from foam evaluations.
problem Creating examples of semistrict monoidal 2-categories.
method Using a closed foam evaluation formula as input, the paper rigorously constructs semistrict monoidal 2-categories.
result The constructed monoidal 2-categories are semistrict, have duals and adjoints, and carry a spatial duality structure.
New method calculates growth rates of Artin-Tits monoids, linking to partial theta function.
problem Growth rates of Artin-Tits monoids and their connection to partial theta function.
method Determining growth functions using simple matrix determinants and atomic generators.
result Exponential growth rates of Artin-Tits monoids of type An tend to 3.233636... as n increases. Study groups formed by words in braid monoid.
problem Invariance properties of braid group presentations.
method Generalized presentations of braid groups and studied their invariance under operations.
result Groups depend only on conjugacy class in braid group under mild hypotheses.
The main new notions are the notions of tangent-like spaces and local monoids. The main result is the pasage from a local monoid to its tangent-like space which is a local Leibniz algebra.
In this paper we show that the singular braid monoid of an orientable surface can be embedded in a group. The proof is purely topological, making no use of the monoid presentation.
Extends Jones' construction to map Thompson group to pointed links.
problem Constructing a map from Thompson group to pointed links.
method Extended Jones' construction, introduced operations, defined new monoid.
result Surjective map from central monoid to linked monoid.
We show that a certain linear representation of the singular braid monoid on three strands is faithful. Furthermore we will give a second - group theoretically motivated - solution to the word problem in this monoid.
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 studies submonoids of singular twisted virtual braids and their properties.
problem Examining submonoids within singular twisted virtual braid monoid.
method Established epimorphisms, identified generators and defining relations, constructed other epimorphisms.
result Identified generators and defining relations for STVPn and other submonoids. Monoidal categorifies genus zero skein algebra using K-theory.
problem Relating skein algebra to K-theory.
method Using quantized K-theoretic Coulomb branch and Grothendieck ring of equivariant coherent sheaves.
result Monoidal categorification of the skein algebra.
The paper presents new algebraic structures on the 2-sphere using topological field theories.
problem Understanding algebraic structures on the 2-sphere through topological field theories.
method Defined P-monoids and L-monoids, analyzed their properties, and related them to 3-dimensional topological field theories.
result New algebraic structures (P-monoids and L-monoids) on the 2-sphere are equivalent and have strong constraints.
We initiate the study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary. The monoid strictly contains the monoid of products of positive Dehn twists. We explain the relationship to tight contact structures and open book decompositions.
Abstract: Generalizes Milnor-Schwarz lemma to inverse monoids.
problem Applying Milnor-Schwarz lemma to inverse monoids.
method Two proofs provided: elementary and using Vietoris-Rips complex.
result Generalization of Milnor-Schwarz lemma to inverse monoids.
Garside's results and the existense of the greedy normal form for braids are shown to be true for the singular braid monoid. An analogue of the presentation of J. S. Birman, K. H. Ko and S. J. Lee for the braid group is also obtained for this monoid.
New representations of Lie algebras via monoidal category actions.
problem Constructing representations of Lie algebras using monoidal categories.
method Using crossed homomorphisms and monoidal categories to generate representations.
result Established new bifunctor for weak and admissible representations of Lie-Rinehart algebras.
Study on endomorphism and automorphism groups of specific quandles.
problem Characterizing endomorphism and automorphism groups of residually finite and profinite quandles.
method Proved properties of endomorphism monoids and automorphism groups for residually finite and profinite quandles.
result Endomorphism and automorphism groups of residually finite quandles are residually finite.
We give presentations, in terms of generators and relations, for the monoids of singular braids on closed surfaces. The proof of the validity of these presentations can also be applied to verify, in a new way, the presentations given by Birman for the monoids of Singular Artin braids.
In this paper we study the growth rates of Artin monoids and we show that 4 is a universal upper bound. We also show that the generating functions of the associated right-angled Artin monoids are given by families of Chebyshev polynomials. Applications to Artin groups and positive braids are given.
A homology cylinder over a surface consists of a homology cobordism between two copies of the surface and markings of its boundary. The set of isomorphism classes of homology cylinders over a fixed surface has a natural monoid structure and it is known that this monoid can be seen as an enlargement of the mapping class…
In this paper we study regular irreducible algebraic monoids over $\fldc$ equipped with the euclidean topology. It is shown that, in such monoids, the Green classes and the spaces of idempotents in the Green classes all have natural manifold structures. The interactions of these manifold structures and the semigroup st…
Involutive Hopf monoids yield surface invariants.
problem Involutive Hopf monoids in symmetric monoidal categories.
method Construction of invariants via (co)equalizers and images.
result Categorical generalization of quantum double models.
Inverse braid monoid describes a structure on braids where the number of strings is not fixed. So, some strings of initial n may be deleted. In the paper we show that many properties and objects based on braid groups may be extended to the inverse braid monoids. Namely we prove an inclusion into a monoid of partial m…
The article studies a monoid of smooth maps on Lie groupoids and their properties.
problem Investigating a monoid of smooth mappings on Lie groupoids and their properties.
method Generalization of A. Stacey's result (Stacey-Roberts Lemma) to infinite-dimensional manifolds.
result The group of units of a Lie groupoid is an infinite-dimensional Lie group and a Lie subgroup of the monoid.
We give a solution to the word problem for the singular braid monoid SB_n. The complexity of the algorithm is quadratic in the product of the word length and the number of the singular generators in the word. Furthermore we algebraically reprove a result of Fenn, Keyman and Rourke that the monoid embeds into a group an…
Newly found infinite linearly independent set in 2-spheres.
problem Linear independence of knotted 2-spheres in S4. method Relating 0-concordance monoid to linear independence of Seifert solids. result Contains an infinite linearly independent set of knotted 2-spheres.
In this article we survey, and make a few new observations about, the surprising connection between sub-monoids of mapping class groups and interesting geometry and topology in low-dimensions.
Researchers found a regular language for maximal lexicographic representatives in braid monoids.
problem Understanding the language of maximal lexicographic representatives in braid monoids.
method Detailed description of the smallest Finite State Automaton and analysis of the proportion of elements.
result The proportion of elements of length k whose maximal lexicographic representative finishes with the first generator tends to a number P_{n,1} ≥ 1/8 as k tends to infinity.
The paper defines biquandles for groups and constructs a homomorphism.
problem Describing dynamical systems in configuration systems.
method Defined biquandles on groups Gnk and constructed a homomorphism. result Homomorphism from surface singular braid monoid to Gn2. We show that the map obtained by viewing a geometric (ie. representative) braid as a string link induces an isomorphism of the n-strand braid group onto the group of units of the n-strand string link monoid.
The set of homology cobordisms from a surface to itself with markings of their boundaries has a natural monoid structure. To investigate the structure of this monoid, we define and study its Magnus representation and Reidemeister torsion invariants by generalizing Kirk-Livingston-Wang's argument over the Gassner repres…