A group theoretical approach to hydrodynamics considers hydrodynamics to be the differential geometry of diffeomorphism groups. The principle of least action implies that the motion of a fluid is described by the geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy. Investigation of the geometry and structure of such groups turns out to be useful for describing the global behavior of fluids for large time intervals.

Topological methods in hydrodynamics

Monomial Ideals.- Squarefree monomial ideals.- Borel-fixed monomial ideals.- Three-dimensional staircases.- Cellular resolutions.- Alexander duality.- Generic monomial ideals.- Toric Algebra.- Semigroup rings.- Multigraded polynomial rings.- Syzygies of lattice ideals.- Toric varieties.- Irreducible and injective resolutions.- Ehrhart polynomials.- Local cohomology.- Determinants.- Plucker coordinates.- Matrix Schubert varieties.- Antidiagonal initial ideals.- Minors in matrix products.- Hilbert schemes of points.

https://www.springer.com/productFlyer_978-0-387-22356-8.pdf?SGWID=0-0-1297-34952457-0

Combinatorial Commutative Algebra

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmuller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. It is related to the motivic dilogarithm.

/pdf/moduli-spaces-of-local-systems-and-higher-teichmuller-theory-36h32nsutb.pdf

Moduli spaces of local systems and higher Teichmüller theory

We analyze global anomalies for elementary Type II strings in the presence of D-branes. Global anomaly cancellation gives a restriction on the D-brane topology. This restriction makes possible the interpretation of D-brane charge as an element of K-theory.

https://www.intlpress.com/site/pub/files/_fulltext/journals/ajm/1999/0003/0004/AJM-1999-0003-0004-a006.pdf

Anomalies in string theory with D-branes

We announce a reformulation of the conjecture in [8,9,10]. The advantage of the new version is that it is simpler and applies more generally than the earlier statement. A key point is to use the universal example for proper actions introduced in [10]. There, the universal example seemed somewhat peripheral to the main issue. Here, however, it will play a central role.

http://www.personal.psu.edu/ndh2/math/Papers_files/Baum,%20Connes,%20Higson%20-%201994%20-%20Classifying%20space%20for%20proper%20actions%20and%20K-theory%20for%20group%20C*-algebras.pdf

Classifying Space for Proper Actions and K-Theory of Group C*-algebras

This book deals with the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Recent developments in mathematical physics (e.g., in knot theory, gauge theory and topological quantum field theory) have led mathematicians and physicists to look for new geometric structures on manifolds and to seek a synthesis of ideas from geometry, topology and category theory. In this spirit this book develops the differential geometry associated to the topology and obstruction theory of certain fibre bundles (more precisely, associated to gerbes). The new theory is a 3-dimensional analogue of the familiar Kostant-Weil theory of line bundles. In particular the curvature now becomes a 3-form. Applications presented in the book involve anomaly line bundles on loop spaces and anomaly functionals, central extensions of loop groups, Kaehler geometry of the space of knots, Cheeger-Chern-Simons secondary characteristic classes, and group cohomology. Finally, the last chapter deals with the Dirac monopole and Dirac's quantization of the electrical charge. The book will be of interest to topologists, geometers, Lie theorists and mathematical physicists, as well as to operator algebraists. It is written for graduate students and researchers, and will be an excellent textbook. It has a self-contained introduction to the theory of sheaves and their cohomology, line bundles and geometric prequantization a la Kostant-Souriau.

Loop Spaces, Characteristic Classes and Geometric Quantization

In [7], D. Kazhdan and G. Lusztig gave a conjecture on the multiplicity of simple modules which appear in a Jordan-H61der series of the Verma modules. This multiplicity is described in the terms of Coxeter groups and also by the geometry of Schubert cells in the flag manifold (see [8]). The purpose of this paper is to give the proof of their conjecture. The method employed here is to associate holonomic systems of linear differential equations with R.S. on the flag manifold with Verma modules and to use the correspondance of holonomic systems and constructible sheaves. Let G be a semi-simple Lie group defined over • and g its Lie algebra. We take a pair (B,B-) of opposed Borel subgroups of G and let T=B~Bbe a maximal torus and W the Weyl group. Let b, b and f the corresponding Lie algebras and 9l the nilpotent radical of b. Let us denote by Jg the category of holonomic systems with R.S. on X=G/B whose characteristic varieties are contained in the union of the conormal bundles of Xw=BWB/B (we W). On the other hand, let (9 denote the category of finitely-generated U(g)-modules which are Tl-finite. By (gtrlv we denote the category of the modules in (9 with the trivial central character. We shall prove that J / / a n d (~trlv are equivalent by the correspondances S0l ~--*F(X;gJI) and M~--~,~| Here ~ is the sheaf of differential operators on X. Let us denote by M w the Verma module with highest weight -w(p)-p and let ~Jl w be the dual g -module of ~codimXwt/~ ~ Then, ~ w and Mw ~ [X~] ~,~X]\" correspond by the above correspondence. For any 9 J l e ~ , we can calculate the character of F(X; 93l) by the formula

Kazhdan-Lusztig Conjecture and Holonomic Systems.

SummaryIn this paper, we extend recent work of one of us [Br] to investigate an old problem of the other one [B2]. Given a connected semisimple complex Lie-groupG with Lie-algebrag, we study the representation\n$$\\psi _X :U(\\mathfrak{g}) \\to D(X)$$\n of the enveloping algebra of\n$$\\mathfrak{g}$$\n by global differential operators on a complete homogeneous spaceX=G/P. It turns out that the kernelIx of ψX is the annihilator of a generalizedVerma-module. On the other hand, we study the associated graded ideal grIx, and relate it to the geometry of a generalizedSpringer-resolution, that is a map\n$$\\pi _X :T^* (X) \\to \\mathfrak{g}$$\n of the cotangent-bundle ofX onto a nilpotent variety in\n$$\\mathfrak{g}$$\n, as studied e.g. in [BM1]. We prove, for instance, that grIx is prime if and only if πX is birational with normal image. In general, we show that\n$$\\sqrt {grI_X }$$\n is prime. Equivalently, the associated variety ofIx in\n$$\\mathfrak{g}$$\n is irreducible: In fact, it is the closure of theRichardson-orbit determined byP. For the homogeneous spaceY=G/(P, P), we prove that the analogous idealIy has for associated variety the closure of theDixmier-sheet determined byP. From this main result, we derive as a corollary, that for any module induced from a finitedimensional LieP-module the associated variety of the annihilator is irreducible, proving an old conjecture [B2], 2.5. Finally, we give some applications to the study of associated varieties of primitive ideals.

Differential Operators on Homogeneous Spaces. I. Irreducibility of the Associated Variety for Annihilators of Induced Modules.

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Jean-Luc Brylinski

Papers

Loop Spaces, Characteristic Classes and Geometric Quantization

Kazhdan-Lusztig Conjecture and Holonomic Systems.

Differential Operators on Homogeneous Spaces. I. Irreducibility of the Associated Variety for Annihilators of Induced Modules.

Central extensions of reductive groups by K2

The geometry of degree-four characteristic classes and of line bundles on loop spaces I