Higher Siegel-Weil formulae over function fields

In their seminal work, Feng-Yun-Zhang introduced function field analogues of Kudla-Rapoport cycles for moduli spaces of unitary shtukas, and initiated the study of their intersection theory. They proved a higher Siegel-Weil formula in the case of non-degenerate Fourier coefficients, relating the degrees of these cycles to higher derivatives of Siegel-Eisenstein series. In this thesis, we generalize their result in two directions: we 1) prove a higher Siegel-Weil formula for unitary groups for corank-1 degenerate coefficients, and 2) introduce analogous cycles on moduli spaces of symplectic shtukas, and prove a higher Siegel-Weil formula for such cycles in the non-degenerate case, relating their degrees to derivatives of Siegel-Eisenstein series on split orthogonal groups.