**Click here to apply to PI4 for Summer 2018.**

The program has four components. Students early in the graduate program usually do the Linear Algebra Working Group, Computational Bootcamp, and Prepare/Train Group. Students later in the graduate program usually do the Computational Bootcamp and an Internship.

1. **Linear Algebra Working Group**

**May 16-18** (239 Altgeld Hall), led by Stefan Klajbor Goderich

2. **Computational Mathematics Bootcamp**

Part I: **May 21-26** (239 Altgeld Hall) with focus on Data Science, led by David LeBauer.

Part II: **May 30-June 1** (239 Altgeld Hall) with focus on Mathematica Fundamentals, led by A. J. Hildebrand

3. **Prepare and Train Group – Algorithms for Analytic Combinatorics**

Dates: **June 4-July 13** (with holiday on July 4)

Location: 159 Altgeld Hall (room booked in mornings)

Instructor: Stephen Melczer (U. of Pennsylvania)

The program is a “Research Experience for Graduate Students” style endeavor: after a series of introductory lectures, the students form small groups (2-5 people) to work on open-ended interconnected problems.

*Overview.* The goal is to guide students through the transition from working on “canned” problems to tackling open-ended problems and formulating the problems themselves. We expect the group work to involve a mixture of computational experiments (to generate conjectures) and theory (to prove them).

One of the draws of combinatorics is its ability to draw on, motivate, and even push forward diverse areas of mathematics and computer science. The focus of this program will be to study the methods of analytic combinatorics – a field drawing inspiration from complex analysis, differential geometry, and algebraic geometry – from the perspective of computer algebra.

Students will begin by learning the underlying theory and implementing algorithms which have been previously described at a theoretical level. Later, students will engage with open problems of both a theoretical and computational nature, and examine new applications of this fast-growing theory. A wide range of problems of varying difficulty will be available for students with different backgrounds.

Topics include the new theory of analytic combinatorics in several variables, effective enumeration results for power series coefficients of algebraic functions, and decompositions of multivariate rational functions. Potential applications touch on areas of queuing theory, representation theory, theoretical computer science, transcendence theory, probability theory, and (of course) combinatorics.

A more detailed statement of problems can be downloaded here.

4. **Internships**

Various dates. Hosts to be arranged.