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Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
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The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
Nonnegative pinching, moduli spaces and bundles with infinitely many souls
 J. of Diff. Geometry
"... Abstract. We show that in each dimension n ≥ 10 there exist infinite sequences of homotopy equivalent but mutually nonhomeomorphic closed simply connected Riemannian nmanifolds with 0 ≤ sec ≤ 1, positive Ricci curvature and uniformly bounded diameter. We also construct open manifolds of fixed diff ..."
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Abstract. We show that in each dimension n ≥ 10 there exist infinite sequences of homotopy equivalent but mutually nonhomeomorphic closed simply connected Riemannian nmanifolds with 0 ≤ sec ≤ 1, positive Ricci curvature and uniformly bounded diameter. We also construct open manifolds of fixed diffeomorphism type which admit infinitely many complete nonnegatively pinched metrics with souls of bounded diameter such that the souls are mutually nonhomeomorphic. Finally, we construct examples of noncompact manifolds whose moduli spaces of complete metrics with sec ≥ 0 have infinitely many connected components. 1.
Superbranching degrees
 Proceedings Oberwolfach 1989, Springer Verlag Lecture Notes in Mathematics
, 1990
"... Solovay ..."
METAPHORS IN SYSTOLIC GEOMETRY
"... This essay is about Gromov’s systolic inequality. We will discuss why the inequality is difficult, and we will discuss several approaches to proving the inequality based on analogies with other parts of geometry. The essay does not contain proofs. It is supposed to be accessible to a broad audience. ..."
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This essay is about Gromov’s systolic inequality. We will discuss why the inequality is difficult, and we will discuss several approaches to proving the inequality based on analogies with other parts of geometry. The essay does not contain proofs. It is supposed to be accessible to a broad audience. The story of the systolic inequality begins in the 1940’s with Loewner’s theorem. Loewner’s systolic inequality. (1949) If (T 2, g) is a 2dimensional torus with a Riemannian metric, then there is a noncontractible curve γ ⊂ (T 2, g) whose length obeys the inequality where C = 2 1/2 3 −1/4. length(γ) ≤ CArea(T 2, g) 1/2, To get a sense of Loewner’s theorem, let’s look at some pictures of 2dimensional tori in R 3.
On minimal wttdegrees and computably enumerable Turing degrees
, 2006
"... Computability theorists have studied many different reducibilities between sets of natural numbers including one reducibility (≤1), manyone reducibility (≤m), truth table reducibility (≤tt), weak truth table reducibility (≤wtt) and Turing reducibility (≤T). The motivation for studying reducibilitie ..."
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Computability theorists have studied many different reducibilities between sets of natural numbers including one reducibility (≤1), manyone reducibility (≤m), truth table reducibility (≤tt), weak truth table reducibility (≤wtt) and Turing reducibility (≤T). The motivation for studying reducibilities stronger that Turing reducibility stems from internally motivated