New non-kinetic actions on four-manifolds discovered.
problem Finding non-kinetic smooth homotopy coherent actions on four-manifolds.
method Restricting a nontrivial action on a K3 surface and constructing extensions. result First example of a non-kinetic smooth homotopy coherent action of order two on a four-manifold.
Study shows exotic Dehn twists on certain 3-sphere fillings.
problem Extending group actions from boundaries to interiors of 4-manifolds.
method Analyzing Dehn twists on Seifert homology spheres and their fillings.
result Dehn twists on certain fillings are infinite order exotic.
Study shows equivariant Khovanov homotopy types are equivalent.
problem Understanding equivariant structures in Khovanov homotopy types.
method Investigates group actions on homotopy coherent diagrams to prove equivalence.
result Equivariant Khovanov homotopy types are equivariantly stably homotopy equivalent.
Smooth actions of infinite groups linked to homotopy theory.
problem Connecting infinite-dimensional smooth groups to homotopy theory.
method Two computations: diffeological homotopy groups and localization of a strict category.
result Natural constructions yield homotopically coherent group actions of G.
Morse theory extended to noncompact manifolds with complex geometric data.
problem Extending Morse theory to noncompact manifolds with intricate geometric and homotopy data.
method Defining Morse homology for pairs of manifolds and related geometric/homotopy data, constructing a homotopy coherent diagram of linear maps, and showing it computes Morse homology.
result Morse homology can be computed using a chain complex derived from a homotopy coherent diagram.
Let p be a fibration over a finite simplicial complex, whose fibers have the homotopy type of finite simplicial complexes. Then p is equivalent to an approximate fibration whose total space is a compact ENR. The proof uses homotopy coherent diagrams and their homotopy colimits. We also comment on the simple homotopy ty…
Dehn twists on K3-type 4-manifolds are not homotopy coherently Nielsen realizable.
problem Homotopy coherent Nielsen realization problem for Dehn twists on 4-manifolds
method Using family Seiberg-Witten theory
result Failure of the classical Nielsen realization problem in K3-type 4-manifolds
Study homotopy sheaves on categories and their presheaves, proving descent properties.
problem Homotopy sheaves on categories and their presheaves.
method Homotopy right Kan extension, pretopologies, Yoneda embedding.
result Preserves homotopy sheaves and induces equivalence between sheaves and colimit-preserving sheaves.
We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point p of the cylinder is called {\em coherent} if all three branches intersect at p pairwise with the same index. A {\em triple unknotting} of a classical knot K is a homotopy which connects K with the trivial knot and which has as singu…
New mapping class group actions on Hochschild complexes for modular categories.
problem Understanding actions of mapping class groups on Hochschild complexes of modular categories.
method Construction of a symmetric monoidal functor with excision property.
result Homotopy coherent projective action of mapping class groups on Hochschild complexes.
Study Drinfeld centralizers and Rouquier complexes in homotopy categories.
problem Interplay between Drinfeld centralizers and Rouquier complexes in homotopy categories.
method Study Drinfeld centralizers and Rouquier complexes in homotopy categories.
result Proved folklore facts about conjugation by Rouquier complexes in the Hecke category.
Develops equivariant Chern characters for coherent sheaves with group actions.
problem Computing Chern characters for coherent sheaves on manifolds with group actions.
method Introduces equivariant Chern characters and proves Riemann-Roch-Grothendieck theorem in Bott-Chern cohomology.
result Establishes a Riemann-Roch-Grothendieck theorem for coherent sheaves with finite group actions.
This note is mostly an exposition of an unpublished result of Deligne, which introduces an analogue of perverse t-structure on the derived category of coherent sheaves on a Noetherian scheme with a dualizing complex. Construction extends to the category of coherent sheaves equivariant under an action of an algebraic …
We explore homotopies in quantum field theory formalism.
problem Constructing homotopies in Batalin-Vilkovisky formalism.
method Review and construction of homotopies from renormalization group flow and gauge fixing changes.
result Constructing spans of quantum master actions with isomorphic effective actions using homotopies.
Using the concept of a cohesive module defined by Block, we use the theory of superconnections in the sense of Quillen to construct natural superconnections on Hermitian cohesive modules. By the Chern-Weil construction, we obtain characteristic classes with values in Bott-Chern cohomology which refines the usual deRham…
Let C be an algebraic curve of genus g. A coherent system on C consists of a pair (E,V), where E is an algebraic vector bundle over C of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study t…
Study of 3D partially hyperbolic diffeomorphisms homotopic to identity, proving leaf conjugacy to Anosov flow.
problem Classifying 3D partially hyperbolic diffeomorphisms homotopic to identity.
method Analysis of foliations and dynamics within leaves, proving leaf conjugacy to Anosov flow.
result Every such diffeomorphism on hyperbolic or Seifert fibered 3-manifolds is leaf conjugate to a (topological) Anosov flow.
In the paper of Montgomery, D. and Yang, C.T. [5], they discuss the de-suspension of smooth free actions of S1 on (2n+1)-dimensional homotopy spheres. In this paper we discuss the de-suspension of smooth free actions of S3 on (4n + 3)-dimensional homotopy spheres.
We emphasize some properties of coherent state groups, i.e. groups whose quotient with the stationary groups, are manifolds which admit a holomorphic embedding in a projective Hilbert space. We determine the differential action of the generators of the representation of coherent state groups on the symmetric Fock space…
This thesis extends Hamiltonian actions to multisymplectic geometry, classifying actions on spheres and constructing homotopy comomentum maps.
problem Extending Hamiltonian actions to multisymplectic geometry.
method Explicit constructions and concrete examples of homotopy comomentum maps.
result Complete classification of compact group actions on multisymplectic spheres and explicit construction of homotopy comomentum maps.
Survey on finite group actions on CW-complexes homotopy to spheres.
problem Understanding finite group actions on CW-complexes homotopy equivalent to spheres.
method Survey of extensive literature on finite G-CW-complexes homotopy equivalent to spheres. result Finite G-CW-complexes homotopy equivalent to spheres have finite group actions. Study of symplectomorphisms on ruled surfaces under circle actions.
problem Homotopy type of equivariant symplectomorphisms on rational ruled surfaces.
method Analysis of action on compatible and invariant almost complex structures, use of Delzant's and Karshon's classifications.
result Equivariant symplectomorphisms are homotopy equivalent to tori or their pushout.
The paper identifies conditions for free circle actions on specific 7-manifolds.
problem Determining conditions for free circle actions on certain 7-manifolds.
method Analyzing specific 7-manifolds and their components.
result Identifies conditions for free circle actions on kS2imesS5#lS3imesS4#Σ. New tools prove smooth actions on exotic spheres.
problem Existence of smooth actions on exotic spheres.
method Homotopy-theoretic tools, complex and quaternionic Mahowald invariants.
result Existence of smooth U(1)- and Sp(1)-actions on exotic spheres. For each link L in S^3 and every quantum grading j, we construct a stable homotopy type X^j_o(L) whose cohomology recovers Ozsvath-Rasmussen-Szabo's odd Khovanov homology, H_i(X^j_o(L)) = Kh^{i,j}_o(L), following a construction of Lawson-Lipshitz-Sarkar of the even Khovanov stable homotopy type. Furthermore, the odd Kh…
Study on group actions on spheres using multisymplectic geometry.
problem Existence of homotopy comoment maps for compact Lie group actions on spheres.
method Investigation of multisymplectic actions and comoments on spheres.
result Explicit constructions of comoments for interesting cases.
New actions found on exotic spheres using group theory.
problem Understanding smooth transformations on exotic spheres.
method Recent progress in stable homotopy groups of spheres and group theory.
result Smooth circle and cyclic group actions on exotic spheres produced.
A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is show…
Study modular class of Lie ∞-algebroids and their adjoint actions.
problem Understanding the modular class and adjoint actions of Lie ∞-algebroids.
method Equivalence of descriptions, homotopy invariance, explicit actions and dualities.
result Homotopy invariance of modular classes and explicit adjoint actions.
We show that any 3-dimensional homotopy lens space M^3 that is simple-homotopy equivalent to a lens space L(p,q) is topologically s-cobordant to the lens space. It follows that M has the same multi-signature as L(p,q) and the action of π_1(M) on the universal cover of M embeds in an orthogonal action on S^7.
Identifies LA-groups via VB-group structure and complementary actions.
problem Understanding the structure and integrability of LA-groups.
method Identifies LA-groups via VB-group structure and complementary actions up to homotopy.
result Establishes an equivalence between LA-groups and LA-matched pairs.
Study automorphisms of pure braid groups on sphere homotopy groups.
problem Understanding automorphisms' effect on sphere homotopy groups.
method Examined Delta-group structure, proved invariance of cycle and boundary groups, computed action for few strands.
result Induced action of all automorphisms of pure braid groups on sphere homotopy groups.
Study of embedding spaces using homotopy theory and operads.
problem Understanding the stable homotopy type of embedding spaces.
method Analysis of cubes of framed configuration spaces, homotopy theory of presheaves, operadic structures.
result Induced action of the Poisson operad on the homology of configuration spaces is a homotopy invariant.
Let M be a manifold homotopy equivalent to the complex projective space $\C P^m$. Petrie conjectured that M has standard total Pontrjagin class if M admits a non-trivial action by S1. We prove the conjecture for m<12 under the assumption that the action extends to a nice Pin(2)-action with fixed point. The…
Improved sample-efficient learning for non-coherent digital jamming.
problem Learning optimal jamming strategies in non-coherent digital modulation schemes without prior knowledge.
method Introduced a linear bandit algorithm that accounts for action similarities and integrates context features.
result Significantly improved convergence behavior compared to prior art.
Curved 10-manifolds with torus symmetry are spheres or complex projective spaces.
problem Characterizing positively curved manifolds with torus symmetry.
method Analyzing actions of 3-dimensional tori on closed, simply connected 10-manifolds. result Closed, simply connected, positively curved 10-manifolds with T3-symmetry are homotopy spheres or complex projective spaces. The paper calculates actions of string link operations for 4- and 5-component links.
problem Classifying link-homotopy classes of string links.
method Explicit calculation of actions of partial conjugations and conjugations for 4- and 5-component string links.
result Presentations of link-homotopy classes for 4- and 5-component links.
We consider the problem of integration of L_\infty-algebroids (differential graded manifolds) to L_\infty-groupoids. We first construct a "big" Kan simplicial manifold (Fréchet or Banach) whose points are solutions of a (generalized) Maurer-Cartan equation. The main analytic trick in our work is an integral transformat…
Representations of coherent state Lie algebras on coherent state manifolds as first order differential operators are presented. The explicit expressions of the differential action of the generators of semisimple Lie groups determine for linear Hamiltonians in the generators of the groups first order differential equati…
Symmetries of Poisson manifolds are in general quantized just to symmetries up to homotopy of the quantized algebra of functions. It is therefore interesting to study symmetries up to homotopy of Poisson manifolds. We notice that they are equivalent to Poisson principal bundles and describe their quantization to symmet…
If M and N are equivariantly homotopy equivalent G-manifolds, then the fixed sets M^G and N^G are also homotopy equivalent. The replacement problem asks the converse question: If F is homotopy equivalent to the fixed set M^G, is F = N^G for a G-manifold equivariantly homotopy equivalent to M? We prove that for locally …
Homotopy momentum map extends Noether's theorem in general relativity.
problem Extending Noether's theorem to spacetime vector fields.
method Using homotopy momentum map and L∞-algebras. result Extension of conserved currents to spacetime vector fields.
Implemented Habegger-Lin algorithm for 4- and 5-component links.
problem Determining link-homotopy of links.
method Explicit computation of group actions and implementation of algorithm.
result Found new pairs of links not distinguishable by Milnor's invariants.
Introduces homotopy momentum sections on multisymplectic manifolds.
problem No specific problem stated; focuses on introducing a new concept.
method Introduces a new concept of homotopy momentum sections on multisymplectic manifolds.
result Shows that a gauged nonlinear sigma model with Wess-Zumino term has homotopy momentum section structure.
In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an a…
The paper studies Goeritz equivalence in lens spaces, describing actions and obstructions.
problem Understanding Goeritz equivalence in lens spaces L(p,1) for genus two Heegaard splittings. method Describes the Goeritz group action on the homology of the Heegaard surface and provides obstructions.
result Homology and homotopy obstructions for Goeritz equivalence of curves in the Heegaard surface.
Study cyclic group actions on specific high-dimensional manifolds.
problem Classify smooth actions of cyclic groups on certain high-dimensional manifolds.
method Analyzes smooth orientation-preserving actions of Z/m on (n−1)-connected 2n-manifolds. result Classifications up to smooth conjugation for specific cases of n and m. Detects free group automorphisms using homology of covers.
problem Identifying free group automorphisms via homology.
method Analyzes the Endomorphism monoid action on homology of covers.
result Homological characterization of homotopy equivalences.