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Abstract. We prove that the higher Frobenius-Schur indicators, introduced by Ng and Schauenburg, give a strong enough invariant to distinguish between any two TambaraYamagami fusion categories. Our proofs are based on computation of the higher indicators in terms of Gauss sums for certain quadratic forms on finite abelian groups and rely on the classification of quadratic forms on finite abelian groups, due to Wall.As a corollary to our work, we show that the state-sum invariants of a Tambara-Yamagami category determine the category as long as we restrict to Tambara-Yamagami categories coming from groups G whose order is not a power of 2. Turaev and Vainerman proved this result under the assumption that G has odd order and they conjectured that a similar result should hold for groups of even order. We also give an example to show that the assumption that |G| is not a power of 2, cannot be completely relaxed.

I would like to thank everyone who has helped me write this thesis or in any other aspect of graduate school. In particular, I want to thank my advisor, Tathagata Basak, for all his help and guidance in the process of earning of my degree. I would also like to especially thank Richard Ng for suggesting the problem and for many encouraging and helpful discussions, as well as several excellent literature suggestions. I also want to thank the other members of my committee, Dr. Clifford Bergman, Dr. Leslie Hogben, and Dr. Amanda Weinstein, for their support. Special thanks goes to Melanie for all the assistance and advice that she has given me and all the other graduate students in the Mathematics department. Lastly, I want to thank my friends at Iowa State for all the fun these past five years. vi