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result(s) for
"Oper"
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q$-opers, $QQ$-systems, and Bethe Ansatz
We introduce the notions of (G,q) -opers and Miura (G,q) -opers, where G is a simply connected simple complex Lie group, and prove some general results about their structure. We then establish a one-to-one correspondence between the set of (G,q) -opers of a certain kind and the set of nondegenerate solutions of a system of Bethe Ansatz equations. This may be viewed as a q DE/IM correspondence between the spectra of a quantum integrable model (IM) and classical geometric objects ( q -differential equations). If g is simply laced, the Bethe Ansatz equations we obtain coincide with the equations that appear in the quantum integrable model of XXZ-type associated to the quantum affine algebra U_q g . However, if g is non-simply-laced, then these equations correspond to a different integrable model, associated to U_q ^Lg where ^L g is the Langlands dual (twisted) affine algebra. A key element in this q DE/IM correspondence is the QQ -system that has appeared previously in the study of the ODE/IM correspondence and the Grothendieck ring of the category O of the relevant quantum affine algebra.
Journal Article
The politics of opera : a history from Monteverdi to Mozart
The Politics of Opera\" takes readers on a fascinating journey into the entwined development of opera and politics, from the Renaissance through the turn of the nineteenth century. What political backdrops have shaped opera? How has opera conveyed the political ideas of its times? Delving into European history and thought and an array of music by such greats as Lully, Rameau, and Mozart, Mitchell Cohen reveals how politics--through story lines, symbols, harmonies, and musical motifs--has played an operatic role both robust and sotto voce. Cohen begins with opera's emergence under Medici absolutism in Florence during the late Renaissance--where debates by humanists, including Galileo's father, led to the first operas in the late sixteenth century. Taking readers to Mantua and Venice, where composer Claudio Monteverdi flourished, Cohen examines how early operatic works like Orfeo used mythology to reflect on governance and policy issues of the day, such as state jurisdictions and immigration. Cohen explores France in the ages of Louis XIV and the Enlightenment and Vienna before and during the French Revolution, where the deceptive lightness of Mozart's masterpieces touched on the havoc of misrule and hidden abuses of power. Cohen also looks at smaller works, including a one-act opera written and composed by philosopher Jean-Jacques Rousseau.
Book
An inertial forward–backward splitting method for solving inclusion problems in Hilbert spaces
In this work, our interest is in investigating the monotone inclusion problems in the framework of real Hilbert spaces. For solving this problem, we propose an inertial forward–backward splitting algorithm involving an extrapolation factor. We then prove its strong convergence under some mild conditions. Finally, we provide some applications including the numerical experiments for supporting our main theorem.
Journal Article
Opera and the city : the politics of culture in Beijing, 1770-1900
In late imperial China, opera was an integral part of life and culture, shared across the social hierarchy. It is in this context that historian Andrea S. Goldman harnesses opera as a lens through which to examine urban cultural history.
Book
Vector bundles and connections on Riemann surfaces with projective structure
Let
B
g
(
r
)
be the moduli space of triples of the form
(
X
,
K
X
1
/
2
,
F
)
, where
X
is a compact connected Riemann surface of genus
g
, with
g
≥
2
,
K
X
1
/
2
is a theta characteristic on
X
, and
F
is a stable vector bundle on
X
of rank
r
and degree zero. We construct a
T
∗
B
g
(
r
)
-torsor
H
g
(
r
)
over
B
g
(
r
)
. This generalizes on the one hand the torsor over the moduli space of stable vector bundles of rank
r
, on a fixed Riemann surface
Y
, given by the moduli space of algebraic connections on the stable vector bundles of rank
r
on
Y
, and on the other hand the torsor over the moduli space of Riemann surfaces given by the moduli space of Riemann surfaces with a projective structure. It is shown that
H
g
(
r
)
has a holomorphic symplectic structure compatible with the
T
∗
B
g
(
r
)
-torsor structure. We also describe
H
g
(
r
)
in terms of the second order matrix valued differential operators. It is shown that
H
g
(
r
)
is identified with the
T
∗
B
g
(
r
)
-torsor given by the sheaf of holomorphic connections on the theta line bundle over
B
g
(
r
)
.
Journal Article
Branched projective structures, branched SO(3,C)-opers and logarithmic connections on jet bundle
We study the branched holomorphic projective structures on a compact Riemann surface X with a fixed branching divisor S=∑i=1dxi , where xi∈X are distinct points. After defining branched SO(3,C) -opers, we show that the branched holomorphic projective structures on X are in a natural bijection with the branched SO(3,C) -opers singular at S. It is deduced that the branched holomorphic projective structures on X are also identified with a subset of the space of all logarithmic connections on J2((TX)⊗OX(S)) singular over S, satisfying certain natural geometric conditions.
Journal Article
Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves?
The geometric Langlands correspondence for complex algebraic curves differs from the original Langlands correspondence for number fields in that it is formulated in terms of sheaves rather than functions (in the intermediate case of curves over finite fields, both formulations are possible). In a recent preprint, Robert Langlands made a proposal for developing an analytic theory of automorphic forms on the moduli space of \\(G\\)-bundles on a complex algebraic curve. Langlands envisioned these forms as eigenfunctions of some analogues of Hecke operators. In these notes I show that if \\(G\\) is an abelian group then there are well-defined Hecke operators, and I give a complete description of their eigenfunctions and eigenvalues. For non-abelian \\(G\\), Hecke operators involve integration, which presents some difficulties. However, there is an alternative approach to developing an analytic theory of automorphic forms, based on the existence of a large commutative algebra of global differential operators acting on half-densities on the moduli stack of \\(G\\)-bundles. This approach (which implements some ideas of Joerg Teschner) is outlined here, as a preview of a joint work with Pavel Etingof and David Kazhdan.
Journal Article
Case Study of an Automated Mower to Support Airport Sustainability
This paper documents a case study of an automated mower to support sustainability at an airport. Mowing is an essential component of an airport’s Wildlife Hazard Management Plan (WHMP), which reduces the risk of birds and other wildlife to aircraft operations. Many airports have large areas of land (hundreds or even thousands of acres), which requires significant resources to manage and mow; experience at the Purdue Airport (KLAF) suggests that automated mowing may support economic and environmental aspects of sustainability. Automated mowing supports economic efficiency by reducing personnel requirements, although personnel are still needed for inspections, maintenance, and “mower rescue” if there is a malfunction (technical or field issue). Automated mowing supports environmental impacts by reducing local emissions since the mower is powered by electricity rather than gasoline; this benefit would be increased with the use of solar-powered mowers. Automated mowing may not be viable everywhere, and factors such as terrain, access to available power, acreage, and location on the airfield (including proximity to protected areas) must be carefully considered. Although automated mowing will not completely replace traditional mowing in the near future, autonomous mowers in remote areas may be an appropriate practice to support airport sustainability.
Journal Article
On the generalized SO(2n,C)-opers
Since their introduction by Beilinson–Drinfeld (Opers, 1993. arXiv math/0501398; Quantization of Hitchin’s integrable system and Hecke eigensheaves, 1991), opers have seen several generalizations. In Biswas et al. (SIGMA Symmetry Integr Geom Methods Appl 16:041, 2020), a higher rank analog was studied, named generalized B-opers, where the successive quotients of the oper filtration are allowed to have higher rank and the underlying holomorphic vector bundle is endowed with a bilinear form which is compatible with both the filtration and the oper connection. Since the definition did not encompass the even orthogonal groups, we dedicate this paper to study generalized B-opers whose structure group is SO(2n,C) and show their close relationship with geometric structures on a Riemann surface.
Journal Article