4,042 research outputs found
New perspectives on self-linking
We initiate the study of classical knots through the homotopy class of the
n-th evaluation map of the knot, which is the induced map on the compactified
n-point configuration space. Sending a knot to its n-th evaluation map realizes
the space of knots as a subspace of what we call the n-th mapping space model
for knots. We compute the homotopy types of the first three mapping space
models, showing that the third model gives rise to an integer-valued invariant.
We realize this invariant in two ways, in terms of collinearities of three or
four points on the knot, and give some explicit computations. We show this
invariant coincides with the second coefficient of the Conway polynomial, thus
giving a new geometric definition of the simplest finite-type invariant.
Finally, using this geometric definition, we give some new applications of this
invariant relating to quadrisecants in the knot and to complexity of polygonal
and polynomial realizations of a knot.Comment: 26 pages, 17 figure
Perspectives for proof unwinding by programming languages techniques
In this chapter, we propose some future directions of work, potentially
beneficial to Mathematics and its foundations, based on the recent import of
methodology from the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as well as the
working mathematician, is not a survey of the field, but rather a personal view
of the author who hopes that it may inspire future and fellow researchers
Gauge-theoretic invariants for topological insulators: A bridge between Berry, Wess-Zumino, and Fu-Kane-Mele
We establish a connection between two recently-proposed approaches to the
understanding of the geometric origin of the Fu-Kane-Mele invariant
, arising in the context of 2-dimensional
time-reversal symmetric topological insulators. On the one hand, the
invariant can be formulated in terms of the Berry connection and
the Berry curvature of the Bloch bundle of occupied states over the Brillouin
torus. On the other, using techniques from the theory of bundle gerbes it is
possible to provide an expression for containing the square root
of the Wess-Zumino amplitude for a certain -valued field over the
Brillouin torus.
We link the two formulas by showing directly the equality between the above
mentioned Wess-Zumino amplitude and the Berry phase, as well as between their
square roots. An essential tool of independent interest is an equivariant
version of the adjoint Polyakov-Wiegmann formula for fields , of which we provide a proof employing only basic homotopy theory and
circumventing the language of bundle gerbes.Comment: 23 pages, 1 figure. To appear in Letters in Mathematical Physic
Solving Optimal Control Problems for Delayed Control-Affine Systems with Quadratic Cost by Numerical Continuation
- In this paper we introduce a new method to solve fixed-delay optimal
control problems which exploits numerical homotopy procedures. It is known that
solving this kind of problems via indirect methods is complex and
computationally demanding because their implementation is faced with two
difficulties: the extremal equations are of mixed type, and besides, the
shooting method has to be carefully initialized. Here, starting from the
solution of the non-delayed version of the optimal control problem, the delay
is introduced by numerical homotopy methods. Convergence results, which ensure
the effectiveness of the whole procedure, are provided. The numerical
efficiency is illustrated on an example
The homotopy branching space of a flow
In this talk, I will explain the importance of the homotopy branching space
functor (and of the homotopy merging space functor) in dihomotopy theory. The
paper is a detailed abstract of math.AT/0304112 and math.AT/0305169.Comment: Expository paper ; 11 pages ; to appear in GETCO'03 proceedin
Orientation twisted homotopy field theories and twisted unoriented Dijkgraaf-Witten theory
Given a finite -graded group with ungraded subgroup and a twisted cocycle which restricts to , we construct a lift of -twisted -Dijkgraaf--Witten theory to an unoriented topological quantum field theory. Our construction uses a new class of homotopy field theories, which we call orientation twisted. We also introduce an orientation twisted variant of the orbifold procedure, which produces an unoriented topological field theory from an orientation twisted -equivariant topological field theory
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