# Adding Funds to the Equation

Aravind Asok, assistant professor of mathematics in USC College, is awarded a federal stimulus grant. His research draws together several branches of mathematics — arithmetic, topology and algebraic geometry — illustrating the fundamental unity of the subject.

Arriving in USC College in August, Aravind Asok, assistant professor of mathematics in USC College, has been awarded a $135,552 federal stimulus grant by the National Science Foundation.

The grant will enable Asok to continue his cutting-edge research on the developing area of homotopy theory of algebraic varieties.

In mathematics, two objects are considered homotopic if one can be continuously deformed into the other. The concept of homotopy was first formulated around the end of the 19th century by Henri Poincaré, considered a founder of the field of topology.

But the relatively new subject of A^1-homotopy theory provides a framework in which one may apply techniques of algebraic topology to algebraic varieties. Spaces having deep arithmetic structure are, from the standpoint of invariants, put on equal footing with those that are easy to visualize.

“My research is focused on applying newly developed mathematical tools to some very old problems in geometry,” Asok said. “Thus, my research contributes to the development of mathematics as a subject, which historically has had many, often unexpected, applications.”

The aim of his research is to integrate ideas of arithmetic, topology, and algebraic geometry by transplanting the successful method of classifying topological spaces via surgery — a cutting-and-pasting procedure carefully controlled by appropriate invariants — into an algebraic geometry by means of A^1-homotopy theory.

“I think the societal contributions of my research are closely tied to the goal of producing more mathematically educated people,” he said.

For example, Asok organizes seminars and conferences focused on his research, which help to train graduate students who then use the mathematical expertise in various fields.

The grant will also provide access to travel funds and money for materials and supplies. The award does not cover the cost of graduate researchers, but Asok expects it will allow more flexibility for graduate students to attend important scientific meetings.

“Much of the mathematics I do is the product of a collaborative effort,” Asok said. “The travel funds allow me, for example, to invite visitors and visit collaborators. Even equipped with modern tools, Web conferencing for instance, there is no substitute for face-to-face interaction.”

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