Math 5810 ACSV course page, Fall 2025
CURRENT TO-DO LIST
READING for April 23: Chapter 11.
This is a condensation of a condensation of large parts of [ABG70].
It is difficult to take in. You might try Baryshnikov-Pemantle (2011)
on my web page.
Resources:
de Loera and Sturmfels (2001):
counting lattice points in polytopes
Atiyah, Bott and Garding (1970) : very long,
very good paper, with foundational constructions for generalized Fourier
transforms and deformations of cycles
READING for April 14-16: Chapter 10. This is the first time we're
having any nontrivial discussion of critical points where the variety
is not smooth. The case in Chapter 10, namely nicely intersecting sheets,
is reasonably well understood, so it's a good place to see how the topology
works in advance of much harder cases. I hope to spend the second half of
April on "cone points", which are the pinnacle of ACSV acheivement at this
time (Chapter 11).
READING for April 2: Chapter 5. You might also want to look at
Section 9.4.2.
UPCOMING READING for the week of April 7-9: Chapter 8 (effective computation).
For computational algebra, I recommend these sources as well:
- Steve Melczer (Springer)
- Invitation to Analytic Combinatorics in Several Variables
- Cox, Little and O'Shea (Springer GTM)
- Using Algebraic Geometry
- Bernd Sturmfels (CBMS)
- Solving Systems of Polynomial Equations
My plan is to start MON 3/31 with an explanation
of the main result in Chapter 5 (Theorem 5.2: multivariate saddle point
integrals in the quadratically nondegenerate case), then go through one
or more two-dimensional quantum walk examples, then go back to the
fancier result (Theorem 5.3) which is needed for quantum walks.
Papers on Quantum Walk:
READING for FEB 26 and March 17,19: Read through Chapter 7, referring to
the Appendices when necessary.
I plan to postpone Chapter 5 until
we need it and instead dive into Chapter 7, which discusses the general
approach that sets up ACSV computations in terms of integrals over
certain cycles. The cycles are chosen via Morse theory and then evaluated
asymptotically via the techniques in Chapters 4 and 5. In class on FEB 24-26
mostly what we will get to is the background: the algebraic topology that
forms a foundation (Appendix B, drawing on Appendix A which you probably
already know) and Appendix C on intersection cycles and residue forms
which you probably don't know. In class on March 17-19 (recall that I'm
out of town next week, after which we have Spring Break), we will go through
Chapter 7 more or less.
TO DO NOW: check whether you can do all the exercises in A.1 and A.2.1.
I will assume yes, so let me know if not. Skim the rest of of Appendix A,
feel you really don't know, so I can go over them. Ditto Appendix B.
You don't need to go beyond this, since I'm assuming you don't know what's
in Appendices C and D.
Maple worksheet for Nikita's no three in a
row generating function
static version (pdf) of same worksheet
READING for Feb 10-12-17: For February 10 there's no specific reading,
but you might want to skim Chapter 6. I will be spending the day on
an informal introduction to multivariate domeains of convergence
for power series and Laurent series.
For February 12-17, read Chapter 4. We will spend February 12 on 4.1 and
some part of 4.2.
Please let me know your preferences as to what to discuss
among the following topics:
Exercises 4.1 - 4.8
Real life examples of Fourier-Laplace integrals
Proofs of anything in particular in Chapter 4
Any concepts in Chapter 6 (not proofs: this is just a first look)
READING for Feb 03: Chapter 3, concentrating on sections 3 and 4.
From last time: I posted
Christol's paper surveying results and conjectures on diagonals
of rational functions, D-finiteness and global boundedness.
I also posted Mark Ward's paper solving Wilf's problem
on the generating function with the arctangent, under MATERIALS, below.
READING for Jan 27: Remainder of Chapter 2 - sections 3,4,5.
PLANNED DISCUSSION Jan 27: natural operations on ordinary generating
functions; algebraic functions and the kernel method;
exponential generating functions and their natural operations;
D-finite functions.
PROBLEMS to discuss Jan 27:
Exercises 2.1, 2.2, 2.4, 25 from the textbook.
During class we will make a plan for topics and problems to discuss
from the remainder of Chapter 2.
READING for Jan 22:
Chapter 1 (mostly for interest, there's not much formal math there)
Sections 1 and 2 of Chapter 2
PROBLEMS to discuss Jan 22:
Asymptotics exercises posted below: if these don't seem too easy, it's
a sign you should work them.
Asymptotics exercises from the textbook 1.2, 1.3, 1.4: these are
classical examples -- worth knowing, but not straightforward.
Arctangent problem: work on this problem if you're curious about it.
Over the course of the semester, you should choose some nontrivial problems
to work on but you don't need to work on every one.