This seminar will focus on the physics and mathematics underlying gauged linear sigma models, with the goal of understanding the monomial-divisor mirror map for Calabi-Yau hypersurfaces. Broadly speaking, we will cover: - Toric Geometry - Crash-course on toric varieties - Sheaves & Sheaf cohomology on toric varieties - Toric Residues (alg-geom/9410017) and mirror symmetry (math/0203216) - GLSMs - Crash-course in QFT - Phases of N=2 Theories In Two Dimensions (hep-th/9301042) - Summing the Instantons... (hep-th/9412236) Students from both the mathematics and physics departments are encouraged to participate. The student-given talks will aim to develop a healthy mix of the abstract (sheaf cohomology & residues) and the concrete (computing correlators in twisted theories) in a way that is accessible to beginning graduate students and useful for older students as well.