MAT477: Seminar in mathematics

Each week, we will have three student presentations of 45 mins each. I expect each student will speak at least twice.

The evaluation will be in four parts:

• (21%) Two days before the presentation, the presenter must hand in approximately 4 pages of (ideally, clearly handwritten) lecture notes via email.
• (30%) The presentation itself will be evaluated based on clarity/pedagogy as well as knowledge/understanding.
• (25%) Participation (this means attendance and engagement, e.g. asking questions)
• (24%) One pedagogical exercise will be assigned by each presenter. These must be submitted at the start of the next class.

The main reference for this seminar will be Enumerative Geometry and String Theory, a book by Sheldon Katz published by the AMS and available in an electronic edition.

The reason I have selected this text is that it provides, with very little required background, an introduction to the key conceptual insights provided by string theory into enumerative geometry. The ideas described in this book are at the heart of the subject of mirror symmetry. You will be forced to learn about many topics along the way, and much of what is in the book is dealt with in an incomplete way, but that is the whole point of this seminar; it is more about showing you what is out there rather than establishing the foundations of enumerative geometry.

Other references for further study:

• An invitation to quantum cohomology: Kontsevich’s formula for rational plane curves, (J. Kock and I. Vainsencher)
• Introduction to Gromov-Witten Theory (S. Rose)
• Notes on stable maps and quantum cohomology (W. Fulton and R. Pandharipande)
• Gromov-Witten classes, quantum cohomology, and enumerative geometry (M. Kontsevich and Y. Manin)
• Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces (Y. Manin)
• Lectures on Complex Manifolds (P. Candelas and X. De la Ossa)
• Lectures on complex geometry, Calabi-Yau manifolds and toric geometry (V. Bouchard)

We may need to adjust depending on the material. Remember that your mission is to summarize the material and highlight the interesting parts, giving detail where it is most crucial. This means that it is supposed to feel like you are squeezing a lot of information into a short presentation. If you feel that the material is too sparse, and there is not enough to cover, then let me know and we may not split your chapter into two lectures.