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### Page 4673 of Mathematical Reviews Vol. , Issue 92h [page]

1992 Mathematical Reviews
The solutions to the deformed problems are related, and can be tracked as the deformation proceeds. The function describing the deformation is called a homotopy map.  ...  This paper surveys some recent major advances in globally con- vergeni homotopy algorithms for unconstrained and constrained optimization, with applications to the optimal design of compos- ite laminated  ...

### Collision-free motion planning on manifolds with boundary [article]

Cesar A. Ipanaque Zapata
2017 arXiv   pre-print
This paper concerns the study of the homotopy type of the ordered configuration space for manifolds with boundary and as an application we will study the collision free motion planning problem on manifolds  ...  advance; (d) collision between two objects occurs if they are situated at the same point in space; (e) an object touches an obstacle if the point representing the object is situated at the obstacle.  ...  Assume, for instance, that we have k objects (robots) moving in D n := {x ∈ R n | x ≤ 1} with no collisions and avoiding the obstacles whose geometry is prescribed in advance.  ...

### Moduli spaces of (bi)algebra structures in topology and geometry [article]

Sinan Yalin
2016 arXiv   pre-print
To understand the classification and deformation theory of these structures on a given object, a relevant idea inspired by geometry is to gather them in a moduli space with nice homotopical and geometric  ...  Derived geometry provides the appropriate framework to describe moduli spaces classifying objects up to weak equivalences and encoding in a geometrically meaningful way their deformation and obstruction  ...  However, this is generally a highly singular object, and one would like to apply the principle shown in the previous example: treat this singular object as a smooth object in a derived framework.  ...

### Moduli Spaces of (Bi)algebra Structures in Topology and Geometry [chapter]

Sinan Yalin
2018 MATRIX Book Series
To understand the classification and deformation theory of these structures on a given object, a relevant idea inspired by geometry is to gather them in a moduli space with nice homotopical and geometric  ...  Derived geometry provides the appropriate framework to describe moduli spaces classifying objects up to weak equivalences and encoding in a geometrically meaningful way their deformation and obstruction  ...  However, this is generally a highly singular object, and one would like to apply the principle shown in the previous example: treat this singular object as a smooth object in a derived framework.  ...

### Voevodsky's Univalence Axiom in Homotopy Type Theory

Steve Awodey, Álvaro Pelayo, Michael A. Warren
2013 Notices of the American Mathematical Society
Acknowledgments We thank the Institute for Advanced Study for the excellent resources which have been made available to the authors during the preparation of this article.  ...  Awodey is partly supported by NSF Grant DMS-1001191 and AFOSR Grant 11NL035, and was supported by the Friends of the Institute for Advanced Study and the Charles Simonyi Endowment.  ...  The homotopy H may be thought of as a "continuous deformation" of f into g.  ...

### Derived Galois deformation rings

S. Galatius, A. Venkatesh
It is a pro-simplicial ring R classifying deformations of a fixed Galois representation to simplicial coefficient rings; its zeroth homotopy group π_0 R recovers Mazur's deformation ring.  ...  We explain how the Taylor--Wiles method can be used to upgrade such an action to a graded action of π_* R on the homology.  ...  Let R be an object of pro-Art k , and let π 0 R be the object of pro-Art k obtained by applying π 0 level-wise to R.  ...

### OUP accepted manuscript

2020 International mathematics research notices
We give an elementary construction of representation homology parallel to the Loday-Pirashvili construction of higher Hochschild homology; in fact, we establish a direct geometric relation between the  ...  In the case of link complements, we identify the representation homology in terms of ordinary Hochschild homology, which gives a new algebraic invariant of links in R 3 .  ...  The 2nd author expresses his gratitude to the Department of Mathematics, Indian Institute of Science, Bengaluru for conducive working conditions during his visit in the summer of 2016.  ...

### Mechanics and Geometry of Solids and Surfaces

J. D. Clayton, M. A. Grinfeld, T. Hasebe, J. R. Mayeur
As overlooked in the initial review by the author, Finsler geometry was applied towards deforming ferromagnetic crystals by Amari in 1962 [3] and has somewhat recently been applied to fracture mechanics  ...  of defined space-moving objects.  ...

### Equivariant homotopy and deformations of diffeomorphisms [article]

C. E. Durán, A. Rigas
2007 arXiv   pre-print
This is applied to the study of exotic diffeomorphisms and involutions of spheres and to the equivariant homotopy of Lie groups.  ...  We present a way of constructing and deforming diffeomorphisms of manifolds endowed with a Lie group action.  ...  The authors would like to thank T. Püttmann for helpful discussions.  ...

### Deformation theory of objects in homotopy and derived categories I: General theory

Alexander I. Efimov, Valery A. Lunts, Dmitri O. Orlov
This is the first paper in a series. We develop a general deformation theory of objects in homotopy and derived categories of DG categories.  ...  The first two functors describe the deformations (and co-deformations) of E in the homotopy category, and the last two - in the derived category. We study their properties and relations.  ...  Acknowledgments It is our pleasure to thank A. Bondal, P. Deligne, M. Mandell, M. Larsen and P. Bressler for useful discussions. We especially appreciate the generous help of B. Keller.  ...

### Proper L–S category, fundamental pro-groups and 2-dimensional proper co-H-spaces

M. Cárdenas, F.F. Lasheras, F. Muro, A. Quintero
2005 Topology and its Applications
Properly based L-S category In the deformation diagram (1) in Section 1 the proper homotopy class of the map β is unique up to homotopy by Remark 2.2.  ...  Hence each N i ⊂ W i is properly deformable in X to the ray α.  ...  In order to set up the homotopy theory of P one observes that this category is not closed under push-outs but it contains sufficiently many to allow the basic homotopical constructions.  ...

### Page 6413 of Mathematical Reviews Vol. , Issue 87k [page]

1987 Mathematical Reviews
The hull of a deformation functor of an algebraic geometric object is determined in some way by the appropriate cohomology at the object and its Massey products.  ...  When the structural group of the bundle is abelian, this problem has been reduced to the evaluation of forgetful maps in equivariant homotopy theory.  ...

### A Decoupling Algorithm Based on Homotopy Theory for 3-D Tactile Sensor Arrays

Junxiang Ding, Yunjian Ge, Yuan Wang, Zhaohui Wang
2012 Sensors & Transducers
A decoupling algorithm for 3-D tactile sensor arrays based on homotopy theory is described in this paper.  ...  , multi-parameter tactile sensor arrays in real time.  ...  The authors would like to thank financial support from National Natural Science Foundation of China (Grant No. 60910005, 61072032) and National 863 project grants (No. 2007AA04Z200).  ...

J.P. Pridham