7,995 research outputs found
Microlocal sheaves and quiver varieties
We relate Nakajima Quiver Varieties (or, rather, their multiplicative
version) with moduli spaces of perverse sheaves. More precisely, we consider a
generalization of the concept of perverse sheaves: microlocal sheaves on a
nodal curve X. They are defined as perverse sheaves on normalization of X with
a Fourier transform condition near each node and form an abelian category M(X).
One has a similar triangulated category DM(X) of microlocal complexes. For a
compact X we show that DM(X) is Calabi-Yau of dimension 2. In the case when all
components of X are rational, M(X) is equivalent to the category of
representations of the multiplicative pre-projective algebra associated to the
intersection graph of X. Quiver varieties in the proper sense are obtained as
moduli spaces of microlocal sheaves with a framing of vanishing cycles at
singular points. The case when components of X have higher genus, leads to
interesting generalizations of preprojective algebras and quiver varieties. We
analyze them from the point of view of pseudo-Hamiltonian reduction and
group-valued moment maps.Comment: 49 page
Characteristic foliation on a hypersurface of general type in a projective symplectic manifold
Given a projective symplectic manifold and a non-singular hypersurface , the symplectic form of induces a foliation of rank 1 on ,
called the characteristic foliation. We study the question when the
characteristic foliation is algebraic, namely, all the leaves are algebraic
curves. Our main result is that the characteristic foliation of is not
algebraic if is of general type. For the proof, we first establish an
\'etale version of Reeb stability theorem in foliation theory and then combine
it with the positivity of the direct image sheaves associated to families of
curves
Manifold interpolation and model reduction
One approach to parametric and adaptive model reduction is via the
interpolation of orthogonal bases, subspaces or positive definite system
matrices. In all these cases, the sampled inputs stem from matrix sets that
feature a geometric structure and thus form so-called matrix manifolds. This
work will be featured as a chapter in the upcoming Handbook on Model Order
Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A.
Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the
numerical treatment of the most important matrix manifolds that arise in the
context of model reduction. Moreover, the principal approaches to data
interpolation and Taylor-like extrapolation on matrix manifolds are outlined
and complemented by algorithms in pseudo-code.Comment: 37 pages, 4 figures, featured chapter of upcoming "Handbook on Model
Order Reduction
Discontinuity induced bifurcations of non-hyperbolic cycles in nonsmooth systems
We analyse three codimension-two bifurcations occurring in nonsmooth systems,
when a non-hyperbolic cycle (fold, flip, and Neimark-Sacker cases, both in
continuous- and discrete-time) interacts with one of the discontinuity
boundaries characterising the system's dynamics. Rather than aiming at a
complete unfolding of the three cases, which would require specific assumptions
on both the class of nonsmooth system and the geometry of the involved
boundary, we concentrate on the geometric features that are common to all
scenarios. We show that, at a generic intersection between the smooth and
discontinuity induced bifurcation curves, a third curve generically emanates
tangentially to the former. This is the discontinuity induced bifurcation curve
of the secondary invariant set (the other cycle, the double-period cycle, or
the torus, respectively) involved in the smooth bifurcation. The result can be
explained intuitively, but its validity is proven here rigorously under very
general conditions. Three examples from different fields of science and
engineering are also reported
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