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

We give a definition of differentiable cohomology of a Lie group G (possibly infinite-dimensional) with coefficients in any abelian Lie group. This differentiable cohomology maps both to the cohomology of the group made discrete and to Lie algebra cohomology. We show that the secondary characteristic classes of Beilinson lead to differentiable cohomology classes with coefficients in C*. These may be viewed as an enrichment of the Chern-Simons differential forms. 
By transgression, classes in differentiable cohomology of a Lie group G lead to differentiable cohomology classes for gauge groups Map(M,G). These classes generalize the central extensions of loop groups. We also discuss holomorphic cohomology of complex Lie groups as the natural place to construct secondary classes. 
We present several conjectures relating the above cohomology classes to the differential forms of Bott-Shulman-Stasheff.

Differentiable Cohomology of Gauge Groups

We introduce the beta function of a knot in Euclidean three-space. This is a meromorphic function of a complex variable which we prove admits a Bernstein type functional equation. We determine the first residues.

The beta function of a knot

We give general formulae for explicit Cech cocycles representing characteristic classes of real and complex vector bundles, as well as for cocycles representing Chern-Simons classes of bundles with arbitrary connections. Our formulae involve integrating differential forms over moving simplices inside homogeneous spaces. An important feature of our cocycles is that they take integer values (as opposed to real or rational values). We find in particular a formula for the instanton number of a connection over a closed four-manifold with arbitrary structure group. For flat connections, our formulae recover and generalize those of Cheeger and Simons. The methods of this paper apply also to the purely geometric construction of the Quillen line bundle with its metric.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1104286562

Čech cocycles for characteristic classes

The geometry of two-dimensional symbols

We introduce the beta function of a knot in euclidean three-space. This is a meromorphic function of a complex variable which we prove admits a Bernstein type functional equation. We determine the first residues.

Jean-Luc Brylinski

Papers

Differentiable Cohomology of Gauge Groups

The beta function of a knot

Čech cocycles for characteristic classes

The geometry of two-dimensional symbols

The beta function of a knot