- Associate Professor Emeritus of Mathematics

**External address:**233 Physics Bldg, Durham, NC 27708

**Internal office address:**Box 90320, Durham, NC 27708-0320

**Phone:**(919) 660-2849

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# David P. Kraines

**External address: **233 Physics Bldg, Durham, NC 27708**Internal office address: **Box 90320, Durham, NC 27708-0320**Phone: **(919) 660-2849
### Research Areas and Keywords

##### Biological Modeling

##### Topology

Dr. Kraines contributed to the theory of homology and cohomology operations, particularly to Massey products and loop operations. Among the applications of his work has been his construction of the counterexample to the transfer conjecture of Quillen. He has also studied the variational bicomplex of Vinogradov and introduced the cohomology of quantum electrodynamics. Dr. Kraines has applied game theoretical techniques to study the evolution of cooperation. With Dr. Vivian Kraines, he introduced a stochastic learning approach, dubbed the Pavlov strategy, for the iterated Prisoner's Dilemma. They show that, in a noisy environment, agents using the Pavlov strategy may achieve a higher level of cooperation than those using Tit for Tat type strategies. Using computer simulations, dynamic systems and Markov chains, they extend their analysis to the evolution of the rate of learning in a society of Pavlov type agents. Recently, they have explored the natural selection of stochastic strategies in the simultaneous and the alternating Prisoner's Dilemma and identified several evolutionarily stable strategies.

##### Education & Training

## Pages

## Pages

## Mathematics

## Academics

## Current Students

## Future Students

- Associate Professor Emeritus of Mathematics

Prisoner's dilemma, sequestering CO2

cohomology operations

Ph.D., University of California at Berkeley 1965

M.A., University of California at Berkeley 1963

B.A., Oberlin College 1961

Drachman, B, and Kraines, D. "A duality between transpotence elements and Massey products." *Pacific Journal of Mathematics* 39.1 (October 1, 1971): 119-123.
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Kraines, D. "Rational cohomology operations and Massey products." *Proceedings of the American Mathematical Society* 22.1 (January 1, 1969): 238-238.
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Kraines, D. "Primitive chains and H∗(ΩX)." *Topology* 8.1 (January 1969): 31-38.
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Kraines, D. "Primitive chains and H*(ΩX)." *Topology* 8.1 (1969): 31-38.

Kraines, D. "Massey higher products." *Transactions of the American Mathematical Society* 124.3 (March 1, 1966): 431-431.
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Kraines, D. "Higher products." *Bulletin of the American Mathematical Society* 72.1 (January 1, 1966): 128-132.
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After 45 years teaching at Duke, David Kraines will become emeritus at Duke this summer. Kraines attended Oberlin College and the University of California at Berkeley where he received his doctorate in 1965 under the supervision of Ed Spanier. After... read more »

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Six students have been named **2016 PRUV Fellows** and will work on a variety of research projects from algebraic geometry and number theory to modeling biological systems. They are **Trung Can, Matthew Gherman, Feng Gui, Mendel...** read more »

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For the 22th time since 1990, the Duke Putnam team placed among the top ten on the WL Putnam Mathematical Competition. Duke achieved 3 first place finishes, 2 seconds and 6 thirds. A total of 4275 students from 554 colleges and universities in... read more »

120 Science Drive

117 Physics Building

Campus Box 90320

Durham, NC 27708-0320

phone: 919.660.2800

fax: 919.660.2821

dept@math.duke.edu