# 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.

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.
Full Text

Aspinwall, PS, and Plesser, MR. "General mirror pairs for gauged linear sigma models." *Journal of High Energy Physics* 2015.11 (November 2015).
Full Text

Aspinwall, PS. "Exoflops in two dimensions." *Journal of High Energy Physics* 2015.7 (July 2015).
Full Text

Aspinwall, PS, and Gaines, B. "Rational curves and (0, 2)-deformations." *Journal of Geometry and Physics* 88 (February 2015): 1-15.
Full Text

Aspinwall, PS, and Morrison, DR. "Quivers from Matrix Factorizations." *Communications in Mathematical Physics* 313.3 (2012): 607-633.
Full Text

Aspinwall, PS, Melnikov, IV, and Plesser, MR. "(0,2) elephants." *Journal of High Energy Physics* 2012.1 (2012).
Full Text

Aspinwall, PS, and Plesser, MR. "Decompactifications and massless D-branes in hybrid models." *Journal of High Energy Physics* 2010.7 (2010).
Full Text

Aspinwall, PS. "Topological D-branes and commutative algebra." *Communications in Number Theory and Physics* 3.3 (2009): 445-474.

Aspinwall, PS, Maloney, A, and Simons, A. "Black hole entropy, marginal stability and mirror symmetry." *Journal of High Energy Physics* 2007.7 (2007).
Full Text

Aspinwall, PS. "Landau-Ginzburg to Calabi-Yau dictionary for D-branes." *Journal of Mathematical Physics* 48.8 (2007).
Full Text

Aspinwall, PS, and Katz, S. "Computation of superpotentials for D-branes." *Communications in Mathematical Physics* 264.1 (2006): 227-253.
Full Text