Proves rational injectivity of K-theory assembly map for group rings.
problem Rational injectivity of Farrell-Jones assembly map for algebraic K-theory.
method Cyclotomic trace map to topological cyclic homology, Bökstedt-Hsiang-Madsen's functor C, and new assembly map results.
result Concrete description of a large direct summand of Kn(Z[G])⊗ZQ in terms of group homology. In this article, we define and study the affine and cyclotomic Yokonuma-Hecke algebras. These algebras generalise at the same time the Ariki-Koike and affine Hecke algebras and the Yokonuma-Hecke algebras. We study the representation theory of these algebras and construct several bases for them. We then show how we can…
The cyclotomic trace of Bökstedt-Hsiang-Madsen, the subject of Bökstedt's lecture at the congress in Kyoto, is a map of pro-abelian groups K_*(A) -> TR_*^.(A;p) from Quillen's algebraic K-theory to a topological refinement of Connes' cyclic homology. Over the last decade, our understanding of the target and its relatio…
In this paper we consider all possible generalizations of the B-type Hecke algebras, namely the cyclotomic and what we call 'generalized', and we construct Markov traces on each of them, so as to obtain all possible different levels of homfly-pt analogues in the solid torus related to the (Hecke) algebras of B-type.
Cyclotomic polynomials help classify mapping classes on surfaces.
problem Characterizing mapping classes on surfaces using cyclotomic polynomials.
method Investigating characteristic polynomials of integral symplectic matrices and using cyclotomic polynomials to classify them.
result For n≥3, the polynomial φn(x) is realized by a mapping class of algebraically finite type if and only if n has at most two distinct prime divisors. We show that for each genus there are only finitely many algebraically primitive Teichmueller curves C, such that i) C lies in the hyperelliptic locus and ii) C is generated by an abelian differential with two zeros of order g-1. We prove moreover that for these Teichmueller curves the trace field of the affine group i…
Proved cyclotomic expansion for double twist knots' HOMFLY-PT invariants.
problem Proving cyclotomic expansion for colored HOMFLY-PT invariants of double twist knots.
method Cyclotomic expansion formula for double twist knots.
result Confirmed cyclotomic expansion conjecture for SU(N)-invariants.
Conjectures closed-form expressions and cyclotomic expansions for knot invariants.
problem Calculating HOMFLY-PT invariants of knots colored by rectangular diagrams.
method Interpolation Macdonald polynomials and cyclotomic expansions.
result Conjectured closed-form expressions and cyclotomic expansions for knot invariants.
Computes Jones polynomial for specific knots.
problem Computing Jones polynomial for double twist knots.
method Using cyclotomic expansion and Kauffman bracket skein theory.
result Answers a question about Jones polynomial for specific knots.
We show that if {L_n} is any infinite sequence of links with twist number tau(L_n) and with cyclotomic Jones polynomials of increasing span, then lim sup tau(L_n)=infty. This implies that any infinite sequence of prime alternating links with cyclotomic Jones polynomials must have unbounded hyperbolic volume. The main t…
Study superpolynomials and volume conjectures for knot homologies.
problem Understanding superpolynomials and their properties for knot homologies.
method Cyclotomic expansion and volume conjecture for superpolynomials of colored HOMFLY-PT and Kauffman homologies.
result Proved volume conjecture for SU(n) specialized superpolynomial associated to reduced colored HOMFLY homology. Quantum trace map defined for 3-manifolds with torus boundaries.
problem Quantifying topological structures of 3-manifolds with torus boundaries.
method Defining a quantum trace map from skein module to a quantum torus module.
result Established a 3D quantum trace map for 3-manifolds with torus boundaries.
New basis for quantum gl_N invariants derived from Macdonald polynomials.
problem Constructing new bases for quantum gl_N invariants.
method Using interpolation Macdonald polynomials and Okounkov's results.
result Cyclotomic expansions for gl_N invariants and knot invariants.
3D quantum trace map connects 3-manifold quantizations.
problem Quantization of 3-manifold character varieties.
method Study of stated skein modules and face suspensions.
result Existence of 3D quantum trace map proved.
The paper connects volume conjecture with SU(n) invariants and their limits.
problem Understanding the volume conjecture for SU(n) invariants. method Using symmetry properties and cyclotomic expansions, the authors propose and prove conjectural formulas for SU(n) invariants. result Proof of conjectural formulas for SU(n) invariants, including volume conjecture. Propose a new 3d quantum trace map that agrees with Garoufalidis and Yu's construction and extends to certain manifolds with ideal triangulated boundaries.
problem Relationship between two constructions of 3d quantum trace maps.
method Propose a new 3d quantum trace map.
result Proposed 3d quantum trace map agrees with Garoufalidis and Yu's construction and extends to certain manifolds with ideal triangulated boundaries.
Non-injectivity proven for trace map on character varieties.
problem Proving non-injectivity of trace map on character varieties.
method Using Amitsur-Levitzki identity and free homotopy classes.
result Explicit construction of nonzero elements in kernel of trace map.
The paper shows compatibility between two quantum maps for surfaces and 3-manifolds.
problem Connecting quantum trace and UV-IR maps for surfaces and 3-manifolds.
method Analyzing compatibility under triangulation changes and using skein modules.
result Compatibility of quantum trace and UV-IR maps for surfaces and 3-manifolds.
Introduces a new model for mapping matrices to matrices, subsuming linear regression.
problem Learning matrix-to-matrix mappings from data.
method Partial trace regression model, leveraging quantum information theory.
result Relevance demonstrated in matrix-to-matrix regression and positive semidefinite matrix completion.
New method calculates Reidemeister trace for coinciding maps between manifolds.
problem Calculating Reidemeister trace for coinciding maps between manifolds of different dimensions.
method Homotopy invariant construction using shriek maps and homology classes.
result Computes coincidence Reidemeister trace for specific cases.
Study on periodic knots, proving limitations on their Alexander polynomials.
problem Understanding Alexander polynomials of periodic knots.
method Polynomial factorization, number theory interpretation, computational methods.
result Alexander polynomials of freely periodic knots are restricted to products of cyclotomic polynomials.
Quantum Teichmüller spaces and trace map constructed using skein algebra.
problem Constructing quantum trace map and Teichmüller space.
method Using skein algebra to define quantum trace map and Teichmüller space.
result Quantum Teichmüller space realized as a subalgebra of skein algebra.
The author connects Poincaré embeddings to Reidemeister traces and diagonal maps.
problem Existence of Poincaré embeddings for specific spaces.
method Relates total obstruction to Reidemeister trace and uses Poincaré duality.
result Diagonal maps admit Poincaré embeddings under certain conditions.
The study finds all trace field degrees for Torelli group mappings.
problem Identifying all possible trace field degrees for Torelli group mappings.
method Using Thurston-Veech construction of pseudo-Anosov maps, and providing examples of stretch factors with specific algebraic degrees.
result All integers 1≤d≤3g−3 are trace field degrees for g≥2. Study 2-loop part of Johnson cokernel using trace map.
problem Identify components of Johnson cokernel in degree 6.
method Use 2-loop trace map to capture Johnson cokernels.
result Capture all components of Johnson cokernels in degree 6.
Study on heat trace expansion for thermoelastic Dirichlet-to-Neumann map.
problem Asymptotic expansion of heat trace for thermoelastic Dirichlet-to-Neumann map.
method Provided a method to obtain all coefficients of the asymptotic expansion.
result Explicitly gave the first two coefficients involving volume and total mean curvature of the boundary.
Quantum trace map defines invariants for knots and links, confirming a length conjecture.
problem Defining invariants for knots and links in hyperbolic 3-manifolds.
method Introducing a quantum trace map for ideally triangulated knot complements, combining with state-integral models.
result Perturbative invariants determine an asymptotic expansion of the Jones polynomial, confirming the length conjecture.
We construct families of TQFT's over the finite field Z/pZ starting from an integral TQFT obtained by Frohman and Nicas. These TQFT's are likely to describe the constant order contributions of the cyclotomic integer expansions of the Reshetikhin Turaev Ohtsuki theories. Their modular structure is intimately related to …
Quantum traces embed into quantum tori for surface skein algebras.
problem Embedding stated skein algebras into quantum tori.
method Two different embeddings using quantum trace maps and lambda length coordinates.
result Quantum cluster algebra of Muller equals reduced stated skein algebra.
Research characterizes traces of Sobolev mappings into manifolds, identifying analytical obstructions and proving surjectivity under certain conditions.
problem Characterizing traces of Sobolev mappings into compact manifolds.
method Analytical and topological methods, including homotopy and fundamental groups.
result Identification of an analytical obstruction when p<m is an integer and πp(N) is non-trivial, and proof of surjectivity under specific conditions. We give a self-contained treatment of Le and Habiro's approach to the Jones function of a knot and Habiro's cyclotomic form of the Ohtsuki invariant for manifolds obtained by surgery around a knot. On the way we reproduce a state sum formula of Garoufalidis and Le for the colored Jones function of a knot. As a corollar…
Defines a map connecting 3d-index and skein module.
problem Connecting mathematical physics predictions with topological quantum field theory.
method Defines a map from skein module to Laurent series ring.
result The map fulfills a supersymmetry prediction and is part of a conjectural topological quantum field theory.
Polynomial-time algorithm finds lens spaces in 3D manifolds.
problem Identifying lens spaces in 3D manifolds with unknown parameters.
method Algorithm calculates Reidemeister torsion using numerical analysis over complex numbers.
result Both n and k can be computed in polynomial time.
Study quantized Coulomb branches of Jordan quiver gauge theories and their connections to Cherednik algebras.
problem Understanding the structure of quantized Coulomb branches in Jordan quiver gauge theories.
method Proved isomorphisms between quantized Coulomb branches and spherical graded Cherednik and cyclotomic rational Cherednik algebras.
result Quantized Coulomb branches are deformations of subquotients of Yangians of affine gl(1). 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.
Maps with many singularities found in complex space.
problem Constructing maps with singularities in complex space.
method Created a map with infinitely many Schoen-Wolfson singularities on a disc.
result Found a Ck map with smooth trace in C2. Quantum trace maps abelian character varieties to SL2 character varieties.
problem Classifying irreducible representations of Chekhov-Fock algebras.
method Non-commutative deformation of algebraic morphisms.
result Induces birational morphism between torus and SL2 character variety.
Defines a map from lamination spaces to quantum cluster varieties.
problem Quantum cluster varieties from surfaces.
method Using a quantum trace map, we define a canonical map.
result The map satisfies properties conjectured by Fock and Goncharov.
New link homology theory supercategorifies Jones polynomial.
problem Constructing link homology theories.
method Skew version of KLR algebras for type A quiver, cyclotomic quotients of supercategory.
result Invariant is distinct from Khovanov homology and conjectured to be isomorphic to odd Khovanov homology.
Unified formula for higher traces of linear maps on finite-dimensional normed spaces.
problem Unified trace-average formula for higher traces of linear maps.
method Unified trace-average formula for the k-th higher trace of a linear operator A on a finite-dimensional normed space.
result Unified trace-average formula holds for all A if and only if the operator-valued average equals the identity.
Generalizes Escobar-Riemann mapping problem for smooth metric measure spaces.
problem Finding a function that attains the Escobar weighted constant.
method Introducing Escobar quotient, infimum, and resolving the problem when the weighted constant is negative.
result Obtained an Aubin type inequality connecting weighted Escobar constant and optimal constant for trace inequality.
We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl_2 and sl_3 and by Mazorchuk-Stroppel and Sussan for sl_n. Our technique is to study 2-representations of 2-quantum gr…
In this report I discuss the relations between systoles and volumes of hyperbolic manifolds and a conjecture of Lehmer about the Mahler measure of non-cyclotomic polynomials.
We give an overview over several constructions of TQFT's over finite fields and cyclotomic integers and their applications to characterizing 3-manifolds and their fundamental groups.
New parametrization of 3-spheres using Johnson subgroups.
problem Constructing integral homology 3-spheres.
method Intrinsic description of equivalence relation on fourth Johnson subgroup.
result Intrinsic description of equivalence relation using fourth Johnson handlebody subgroups.
Abstracts Conjecture, tools controlled algebra, trace methods.
problem Farrell-Jones Conjecture in algebraic K-theory.
method controlled algebra, trace methods
result Current status of the conjecture.
The Burau representation fails to be faithful at roots of unity for n ≥ 3.
problem The faithfulness of the Burau representation at roots of unity.
method Analyzing the braid element σiσi+1σi and its powers. result The Burau representation is unfaithful at any primitive root of unity, excepting the first three.
Quantum traces map skein algebras to Fock-Goncharov spaces.
problem Establishing quantum traces between skein algebras and Fock-Goncharov spaces.
method Defining and proving properties of quantum traces for SLn-skein algebras. result Existence and properties of quantum traces for SLn-skein algebras.