Title of the course: Analytic Combinatorics, Symbolic Dynamics, and Application
Instructor’s name: Dr. Mark Daniel Ward, Dr. Ricardo Gómez Aíza
Institution: Purdue University, National Autonomous University
Dates: 14-27 July 2025
Prerequisites: Understanding of Taylor series, and understanding of complex analysis
Level: All graduate and advanced undergraduate students welcome
Abstract:
Two instructors for the course: Dr. Ward will coordinate 1 session per day; Dr. Gómez will coordinate 1 session per day; and we will coordinate a problem session each day.
The Workshop on Analytic Combinatorics, Symbolic Dynamics, and Applications will be aimed at graduate students and advanced undergraduate students.  Each day of the workshop will have a session on Analytic Combinatorics, a session on Symbolic Dynamics, and a problem session that focuses on applications.  Generating functions and special functions (for instance, zeta functions, the complex gamma function, hypergeometric functions, etc.) will be unifying themes of this workshop.
The Analytic Combinatorics topics will include:
1. Ordinary generating functions of unlabeled combinatorial structures.
2. Labeled structures and exponential generating functions.
3. Bivariate generating functions and random structures.
4. Asymptotic analysis of coefficients of generating functions.
The Symbolic Dynamics topics will include:
1. Shift spaces and subshifts of finite type.
2. Entropy, periodic orbits, and dynamic zeta functions (generating functions that encode the periodic orbits of dynamical systems).
3. Countable state Markov shifts.
4. Classification problems.
Selected topics and open problems about the interplay of the two main previous subjects, and their applications, will include:
1. Open problems on asymptotic behavior of coefficients of generating functions.
2. Open problems on recurrence relations of coefficients of generating functions.
3. Zero entropy, minimal systems and the specification property on symbolic dynamics.
4. Classification problems in symbolic dynamics; asymptotic behavior of zeta functions.
5. Further applications.
Language: English
Textbooks:
P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge University Press. 2009.
D. Lind and B. Marcus. Introduction to Symbolic Dynamics and Coding. Cambridge University Press. 1995.