In recent work, Dr. Lynne Walling developed an algebraic proof of Siegel's representation formula, yielding formulas more explicit than those of Siegel to describe average representation numbers of positive definite quadratic forms of even rank and odd level. She is currently working with J.L. Hafner, using the theory of automorphic forms to extend these results to indefinite quadratic forms, obtaining explicit formulas for the measures of the representations of the indefinite forms over number fields and over function fields. In work with O. Imamoglu, Dr. Walling has defined a symplectic theta function over a function field and computed the transformation formula nd will develop some theory of Siegel modular forms and Jacobi forms in the function field setting. Information will be gained regarding representations of quadratic forms by higher dimensional quadratic forms. She also intends to extend the techniques developed in work with J. Hoffstein and K.D. Merrill, explicitly computing the Fourier coefficients of cusp forms of weights 0 and 1/2 for congruence subgroups that admit only one-dimensional spaces of cusp forms. Interactive activities include developing curricula for college/university mathematics courses based on contemplation, precise reasoning and clear communication; developing a summer research institute for women mathematicians; and running a one-week conference for women in harmonic analysis and number theory.
