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Affine Hecke Algebras and Quantum Symmetric Pairs
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Affine Flag Varieties and Quantum Symmetric Pairs
The quantum groups of finite and affine type $A$ admit geometric realizations in terms of partial flag varieties of finite and affine type $A$. Recently, the quantum group associated to partial flag varieties of finite type $B/C$ is shown to be a coideal subalgebra of the quantum group of finite type $A$.
Twenty-Four Hours of Local Cohomology
This book is aimed to provide an introduction to local cohomology which takes cognizance of the breadth of its interactions with other areas of mathematics. It covers topics such as the number of defining equations of algebraic sets, connectedness properties of algebraic sets, connections to sheaf cohomology and to de Rham cohomology, Gröbner bases in the commutative setting as well as for $D$-modules, the Frobenius morphism and characteristic $p$ methods, finiteness properties of local cohomology modules, semigroup rings and polyhedral geometry, and hypergeometric systems arising from semigroups. The book begins with basic notions in geometry, sheaf theory, and homological algebra leading to the definition and basic properties of local cohomology. Then it develops the theory in a number of different directions, and draws connections with topology, geometry, combinatorics, and algorithmic aspects of the subject.
Local Well-Posedness and Break-Down Criterion of the Incompressible Euler Equations with Free Boundary
In this paper, we prove the local well-posedness of the free boundary problem for the incompressible Euler equations in low regularity Sobolev spaces, in which the velocity is a Lipschitz function and the free surface belongs to C 3 2 +ε. Moreover, we also present a Beale-Kato-Majda type break-down criterion of smooth solution in terms of the mean curvature of the free surface, the gradient of the velocity and Taylor sign condition.
Linear Dynamical Systems on Hilbert Spaces: Typical Properties and Explicit Examples
We solve a number of questions pertaining to the dynamics of linear operators on Hilbert spaces, sometimes by using Baire category arguments and sometimes by constructing explicit examples. In particular, we prove the following results. (i) A typical hypercyclic operator is not topologically mixing, has no eigen-values and admits no non-trivial invariant measure, but is densely distri-butionally chaotic. (ii) A typical upper-triangular operator with coefficients of modulus 1 on the diagonal is ergodic in the Gaussian sense, whereas a typical operator of the form “diagonal with coefficients of modulus 1 on the diagonal plus backward unilateral weighted shift” is ergodic but has only count...
Bounded Littlewood Identities
We describe a method, based on the theory of Macdonald–Koornwinder polynomials, for proving bounded Littlewood identities. Our approach provides an alternative to Macdonald’s partial fraction technique and results in the first examples of bounded Littlewood identities for Macdonald polynomials. These identities, which take the form of decomposition formulas for Macdonald polynomials of type (R, S) in terms of ordinary Macdonald polynomials, are q, t-analogues of known branching formulas for characters of the symplectic, orthogonal and special orthogonal groups. In the classical limit, our method implies that MacMahon’s famous ex-conjecture for the generating function of symmetric plane partitions in a box follows from the identification of GL(n, R), O(n) as a Gelfand pair. As further applications, we obtain combinatorial formulas for characters of affine Lie algebras; Rogers–Ramanujan identities for affine Lie algebras, complementing recent results of Griffin et al.; and quadratic transformation formulas for Kaneko–Macdonald-type basic hypergeometric series.
Twistors, Quartics,and del Pezzo Fibrations
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Representation Theory of Geigle-Lenzing Complete Intersections
View the abstract. https://www.ams.org/bookstore/pspdf/memo-285-1412-abstract.pdf?
The 2D Compressible Euler Equations in Bounded Impermeable Domains with Corners
We study 2D compressible Euler flows in bounded impermeable domains whose boundary is smooth except for corners. We assume that the angles of the corners are small enough. Then we obtain local (in time) existence of solutions which keep the L2 Sobolev regularity of their Cauchy data, provided the external forces are sufficiently regular and suitable compatibility conditions are satisfied. Such a result is well known when there is no corner. Our proof relies on the study of associated linear problems. We also show that our results are rather sharp: we construct counterexamples in which the smallness condition on the angles is not fulfilled and which display a loss of L2 Sobolev regularity with respect to the Cauchy data and the external forces.
