Lattice points in polytopes

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We will study the topic of counting lattice points in polytopes using various methods from combinatorics and analysis. The connection with toric varieties will also be mentioned and explored.

Basic knowledge of real and complex analysis is recommended. Some knowledge of algebraic geometry will be helpful for part of the course.

Office hour: Sunday, 1--2pm, at Manchester 314

Main reference: Matthias Beck and Sinai Robins -- Computing the continuous discretely (book)

Additional references:

• Alexander Barvinok -- Integer points in polyhedra (book)
• Matthias Beck and Raman Sanyal -- Combinatorial reciprocity theorems (book)
• David Cox, John Little and Hal Schenck -- Toric varieties (book)
• Peter McMullen -- Valuations and Euler-type relations on certain classes of convex polytopes (article)

Course evaluation: The main part of your grade comes from a final presentation/paper (80% total grade). There are biweekly assignments (20% total grade) that are meant to supplement and consolidate your understanding of course materials. You are highly encouraged to collaborate together throughout the course, but you should think carefully about each problem on your own first. You are required to write up your solutions separately.

Summary: We will start with the basic concepts: lattice polytopes and Ehrhart polynomials; and results: Ehrhart’s theorem and Ehrhart–Macdonald reciprocity law. We will discuss a few methods of counting lattice points such as Euler's generating functions and integer-point transforms. Afterwards, we will study Brion's theorem, an important result with interesting applications into computing volumes and moments of polytopes. In the latter part of the course, we will explore some of the following (advanced/extra) topics:

1. McMullen's theory of translation-invariant valuations, which allows us to derive polynomiality and reciprocity from the valuation property. There is also a result involving Minkowski sum of polytopes, in the spirit of the mixed volumes.
2. Combinatorial reciprocity theorems, which give combinatorial meanings to values of some combinatorially interesting polynomials, such as graph polynomials and matroid polynomials, at negative points.
3. Connection with toric varieties, which was a major development in the history of polyhedral geometry and continues to yield new results.

We will spend as much time as we need on the above-mentioned topics. The following topics will be touched on only if time permits.

1. Barvinok's algorithm, which gives us an efficient way to compute the Ehrhart polynomial of a given lattice polytope.
2. Euler--Maclaurin formula for unimodular polytopes, which generalizes the classical Euler--Maclaurin formula which has been extremely useful in analytic number theory.
3. h-polynomials and h*-polynomials of polytopes, which are related to triangulations of the given polytopes.