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- Manfredo do Carmo, Nolan Wallach
- Proceedings of the National Academy of Sciences…
- 1969

In this paper we announce a qualitative description of an important class of closed n-dimensional submanifolds of the m-dimensional sphere, namely, those which locally minimize the n-area in the same way that geodesics minimize the arc length and are themselves locally n-spheres of constant radius r; those r that may appear are called admissible. It is… (More)

The purpose of this department is to provide early announcement of significant new results, with some indications of proof. Although ordinarily a research announcement should be a brief summary of a paper to be published in full elsewhere, papers giving complete proofs of results of exceptional interest are also solicited. Manuscripts more than eight… (More)

- Nolan Wallach
- 2007

- Adriano M. Garsia, Nolan Wallach
- J. Comb. Theory, Ser. A
- 2003

Astract We study here the ring QSn of Quasi-Symmetric Functions in the variables x1, x2, . . . , xn. F. Bergeron and C. Reutenauer [4] formulated a number of conjectures about this ring, in particular they conjectured that it is free over the ring Λn of symmetric functions in x1, x2, . . . , xn. We present here an algorithm that recursively constructs a… (More)

- Nolan Wallach
- 2006

These notes are an expanded form of lectures to presented at the C.I.M.E. summer school in representation theory in Venice, June 2004. The sections of this article roughly follow the five lectures given. The first three lectures (sections) are meant to give an introduction to an audience of mathematicians (or mathematics graduate students) to quantum… (More)

In this paper, we classify the four-dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite-time singularities of Ricci flow on compact four-manifolds with positive isotropic curvature. As a corollary, we generalize a result of Perelman on three-dimensional gradient shrinking solitons to… (More)

In this paper the analytic continuation of the holomorphic discrete series is defined. The most elementary properties of these representations are developed. The study of when these representations are unitary is begun.

- Lei Ni, Nolan Wallach
- 2008

The main purpose of this article is to provide an alternate proof to a result of Perelman on gradient shrinking solitons. In dimension three we also generalize the result by removing the κ-non-collapsing assumption. In high dimension this new method allows us to prove a classification result on gradient shrinking solitons with vanishing Weyl curvature… (More)

- Lei Ni, Nolan Wallach, Nolan Wallach
- 2008

The main purpose of this article is to provide an alternate proof to a result of Perelman on gradient shrinking solitons. Moreover in dimension three our proof generalizes Perelman’s result by removing the κ-non-collapsing assumption and allowing general curvature growth. The method also allows us to prove a classification result on gradient shrinking… (More)

- Esther Cabezas-Rivas, Peter M. Topping, Nolan Wallach
- 2008

To every Ricci flow on a manifold M over a time interval I ⊂ R, we associate a shrinking Ricci soliton on the space-timeM×I. We relate properties of the original Ricci flow to properties of the new higher-dimensional Ricci flow equipped with its own time-parameter. This geometric construction was discovered by consideration of the theory of optimal… (More)