The geometry and combinatorics of Springer fibers

This survey paper describes Springer fibers, which are used in one of the earliest examples of a geometric representation. We will compare and contrast them with Schubert varieties, another family of subvarieties of the flag variety that play an important role in representation theory and combinatorics, but whose geometry is in many respects simpler. The end of the paper describes a way that Springer fibers and Schubert varieties are related, as well as open questions.Comment: 18 page

Permutation actions on equivariant cohomology

This survey paper describes two geometric representations of the permutation group using the tools of toric topology. These actions are extremely useful for computational problems in Schubert calculus. The (torus) equivariant cohomology of the flag variety is constructed using the combinatorial description of Goresky-Kottwitz-MacPherson, discussed in detail. Two permutation representations on equivariant and ordinary cohomology are identified in terms of irreducible representations of the permutation group. We show how to use the permutation actions to construct divided difference operators and to give formulas for some localizations of certain equivariant classes. This paper includes several new results, in particular a new proof of the Chevalley-Monk formula and a proof that one of the natural permutation representations on the equivariant cohomology of the flag variety is the regular representation. Many examples, exercises, and open questions are provided.Comment: 24 page

Billey's formula in combinatorics, geometry, and topology

In this expository paper we describe a powerful combinatorial formula and its implications in geometry, topology, and algebra. This formula first appeared in the appendix of a book by Andersen, Jantzen, and Soergel. Sara Billey discovered it independently five years later, and it played a prominent role in her work to evaluate certain polynomials closely related to Schubert polynomials. Billey's formula relates many pieces of Schubert calculus: the geometry of Schubert varieties, the action of the torus on the flag variety, combinatorial data about permutations, the cohomology of the flag variety and of the Schubert varieties, and the combinatorics of root systems (generalizing inversions of a permutation). Combinatorially, Billey's formula describes an invariant of pairs of elements of a Weyl group. On its face, this formula is a combination of roots built from subwords of a fixed word. As we will see, it has deeper geometric and topological meaning as well: (1) It tells us about the tangent spaces at each permutation flag in each Schubert variety. (2) It tells us about singular points in Schubert varieties. (3) It tells us about the values of Kostant polynomials. Billey's formula also reflects an aspect of GKM theory, which is a way of describing the torus-equivariant cohomology of a variety just from information about the torus-fixed points in the variety. This paper will also describe some applications of Billey's formula, including concrete combinatorial descriptions of Billey's formula in special cases, and ways to bootstrap Billey's formula to describe the equivariant cohomology of subvarieties of the flag variety to which GKM theory does not apply.Comment: 14 pages, presented at the International Summer School and Workshop on Schubert Calculus in Osaka, Japan, 201

The Future of Central Banking in the Changing Financial Environment

During the last decades there were deep technology-driven changes in financial systems of many countries. The result was the decreasing demand for cash and commercial banks’ liquid reserves. The decreasing demand for central bank money has changed the operational side of the monetary policy. The maturities of open market operations have been shortened substantially. The reserve requirements were lowered or eliminated. The changes lead to a decrease in the volume of the monetary base, but the demand for central bank money will not vanish. Even with zero reserve ratio commercial banks will keep cash balances with central banks to settle their transactions. Thus, there will be a demand for banks’ liquid reserves and cash. Despite the decreasing demand for the monetary base, the central banks ability to influence the level of short-term interest rates will not be impaired, because central banks will continue to play the role of clearing houses for the banking systems. The central bank ability to conduct monetary policy does not depend on the volume of the monetary base, but on the demand for interbank settlements made by central bank. This is not the size of the monetary base either, that decides about the effectiveness of the monetary policy. Monetary authorities can influence interest rates via the payment system, which is typically based in central bank. The consequence of the decrease in the monetary base will be a fall in seigniorage income, but this will not impair central banks ability to conduct monetary policy. Interbank settlements must be located in central banks due to their reliability and the ability to play the role of lenders of last resort. As the central banks will stay as the institutions refinancing the payment system, they will decide about the level of short-term interest rates. Electronic money will not change the situation, because if this kind of money is to be widely accepted, it has to be exchanged into central bank money.monetary policy, monetary base, interbank settlements