# Paul Stephen Aspinwall

## Professor of Mathematics

### Overview

String theory is hoped to provide a theory of all fundamental physics encompassing both

quantum mechanics and general relativity. String theories naturally live in a large number of

dimensions and so to make contact with the real world it is necessary to ``compactify'' the

extra dimensions on some small compact space. Understanding the physics of the real

world then becomes a problem very closely tied to understanding the geometry of the space

on which one has compactified. In particular, when one restricts one's attention to

``supersymmetric'' physics the subject of algebraic geometry becomes particularly important.

Of current interest is the notion of ``duality''. Here one obtains the same physics by

compactifying two different string theories in two different ways. Now one may use our limited understanding of one

picture to fill in the gaps in our limited knowledge of the second picture. This appears to be an extremely powerful

method of understanding a great deal of string theory.

Both mathematics and physics appear to benefit greatly from duality. In mathematics one finds hitherto unexpected

connections between the geometry of different spaces. ``Mirror symmetry'' was an example of this but many more

remain to be explored. On the physics side one hopes to obtain a better understanding of nonperturbative aspects

of the way string theory describes the real world.

### Selected Grants

Moduli Spaces & String Theory awarded by National Science Foundation (Principal Investigator). 2012 to 2017

Geometry and Mathematical Physics of D-Branes awarded by National Science Foundation (Principal Investigator). 2009 to 2014

Algebraic Geometry and Quantum Field Theory of D-Branes awarded by National Science Foundation (Principal Investigator). 2006 to 2011

D-Brane Physics and Calabi-Yau Geometry awarded by National Science Foundation (Co-Principal Investigator). 2003 to 2007

Focused Research awarded by National Science Foundation (Co-Principal Investigator). 2000 to 2004

Aspinwall, PS. "Some applications of commutative algebra to string theory." *Commutative Algebra: Expository Papers Dedicated to David Eisenbud on the Occasion of His 65th Birthday.* November 1, 2013. 25-56.
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Aspinwall, PS, and Plesser, MR. "General mirror pairs for gauged linear sigma models." *Journal of High Energy Physics* 2015.11 (November 1, 2015): 1-33.
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Aspinwall, PS. "Exoflops in two dimensions." *Journal of High Energy Physics* 2015.7 (July 31, 2015).
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Aspinwall, PS, and Gaines, B. "Rational curves and (0, 2)-deformations." *Journal of Geometry and Physics* 88 (February 1, 2015): 1-15.
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Aspinwall, PS. "A McKay-like correspondence for (0, 2)-deformations." *Advances in Theoretical and Mathematical Physics* 18.4 (January 1, 2014): 761-797.
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Addington, N, and Aspinwall, PS. "Categories of massless D-branes and del Pezzo surfaces." *Journal of High Energy Physics* 2013.7 (August 19, 2013).
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Aspinwall, PS, and Morrison, DR. "Quivers from Matrix Factorizations." *Communications in Mathematical Physics* 313.3 (August 1, 2012): 607-633.
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Aspinwall, PS, Melnikov, IV, and Ronen Plesser, M. "(0,2) elephants." *Journal of High Energy Physics* 2012.1 (February 27, 2012).
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Aspinwall, PS, and Plesser, MR. "Decompactifications and massless D-branes in hybrid models." *Journal of High Energy Physics* 2010.7 (August 2, 2010).
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Aspinwall, PS, and Ronen Plesser, M. "Decompactifications and massless D-branes in hybrid models." *Journal of High Energy Physics* 2010.7 (July 2010).
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Aspinwall, PS. "Topological D-branes and commutative algebra." *Communications in Number Theory and Physics* 3.3 (January 1, 2009): 445-474.
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