Chance to land near Cantor set?
Log n over n.
Michigan State University
My research is on the probability that “Buffon’s needle” (or noodle, as the case may be) will land near a self-similar (“Cantor-like”) set of Hausdorff dimension 1 in the plane. (The abstract and first chapter are relatively non-technical.)
Nothing is clear, what will he do?
I would satisfice.
Technion – Israel Institute of Technology
Title: “Strategic interactions under severe uncertainty” (2008)
My thesis in the discipline of Decision Theory explores the application of info-gap decision theory to cases where the uncertainty is not about the properties of the world, but rather concerns the reaction of some (not necessarily adverse) agent. Info-gap decision theory claims that one should not aim for the best performance possible (optimizing), but to be able to withstand great error in estimation and still yield a reasonable result (satisficing).
It should be here but…
When thou art too large
New York University
Title: “Effective stochastic dynamics in deterministic systems” (2006).
My thesis concerns a few examples of deterministic systems that seem random when they are large enough. This is related to the “paradox” that even though the underling dynamics of a system my be deterministic (e.g., described by Newton’s laws), large ensembles can be treated in a probabilistic way using the tools of statistical physics.
C-symmetries are many,
But not infinite.
R. Travis Kowalski
University of California at San Diego
Dissertation Title: “Formal equivalences between real-analytic hypersurfaces”
An investigation (in part) of the complex-analytic equivalences between three-dimensional geometric shapes (called hypersurfaces) inside of a complex, two-dimensional universe (called C^2). My primary result is that the number (or dimension) of the self-symmetries of such shapes with a specific type of degeneracy (called infinite 1-type) can be arbitrarily huge, but nevertheless finite.
Triangles in a
triangular tube; some are
further to the left.
Title: “Tube Representations of Ordered Sets” (2001).
My thesis investigated orderings of abstract geometric shapes in 2, 3, and higher dimensions.
You are elusive,
second homotopy group.
Cough up your secrets!
University of Illinois
Dissertation Title: “A Pi-2 Invariant For Split Complexes”
Split 2-complexes in 4-manifolds sit nicely enough in their hosts to retain features of their cousins resting in 3-manifolds. I have a collection of moves and a measure of the second homotopy group of a split 2-complex which remains invariant under the moves. This is useful in determining if the group is trivial. How strange that groups of order 1 can be so hard to recognize!
All Coxeter groups
have cohomology rings:
I found them (mod 2).
University of Minnesota
My dissertation title, “The mod-two cohomology of Coxeter groups,” seems self-explanatory — a Coxeter group is (for me) a finite bunch of reflections and rotations in an n-dimensional space, and the mod-two cohomology ring of its classifying space is an infinite bunch of polynomial-type gadgets that carry information about the group. I listed them, except in the case of the Coxeter group called E8: that one was too hard.
Prove new conjecture
in Iwasawa theory
of elliptic curves
Dissertation Title: “Iwasawa theory at multiplicative primes” (1987).
My work proved the Mazur-Tate-Teitelbaum conjecture for Iwasawa L-functions of elliptic curves.
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