## July 9 – July 29, 2023

- Residential move-in date: July 9
- Move-out date: July 29

**Applications for summer 2023 are now closed.**

The Mathematics Academy is a unique opportunity for students interested in examining mathematical concepts rarely offered at the high school level. The Mathematics Academy is fully residential with no commuter or online options. This rigorous, proof-oriented program will fuse lectures, problem sessions, demonstrations, and exploratory research to engage students in topics such as:

**Discrete mathematics**

- Combinatorics (enumerative, algebraic and geometric)
- Generating functions and partitions
- Graph theory
- Probability
- Combinational game theory

**Algebra and number theory**

- Linear algebra
- Prime and factorization algorithms
- Congruencies and quadratic reciprocity
- Galois theory
- Geometry of numbers

**Geometry and topology**

- Euclidean and non-Euclidean geometries
- Geometric transformations
- Algebraic geometry
- Point-set topology
- Knot theory

*If you attend a School District of Philadelphia public or charter high school, you may be eligible to attend a Penn Summer Academy free of charge with a *

*Penn Summer Scholarship*.

## Features

**Lectures and discussions:** Attend sessions with mathematics faculty and experts which include lectures, recitations, group work, computer simulations, and problem sessions.

**Comprehensive mathematics:** Explore topics such as combinatorics, generating functions and partitions, graph theory, probability, combinational game theory, Galois theory, linear algebra, prime and factorization algorithms, congruencies and quadratic reciprocity, geometry of numbers, Euclidean and non-Euclidean geometries, geometric transformations, algebraic geometry, point-set topology, and knot theory.

**Daily Schedule: **The Summer Mathematics Academy will meet Monday - Friday, from 9:30 a.m. - 5 p.m. There will be a morning lecture period from 9:30 a.m. - 12 p.m., followed by a break for lunch, then an afternoon lecture period from 1:30 - 3:30 p.m. The day will close with recitation from 3:30 - 5 p.m., where students can review concepts with the teaching assistant and work on the problems assigned for that day. Lecture periods will be led by Penn faculty and will have significant active learning components, where students work to solve problems together in groups, with assistance from the faculty speaker, academy director, and teaching assistant. Other activities after 5 p.m. and on the weekends are organized by Penn SAS High School Programs and Summer Discovery.**Academic Content:** The Summer Mathematics Academy will expose students to the diverse field of mathematics over the three-week program. The academy will begin by introducing the theory of mathematical logic and proofs, and then will cover a selection of topics in higher-level mathematics, with different Penn faculty coming in for several days to teach about each new topic. Last year, the topics covered in the academy were difference equations and recurrence relations, graph theory, linear algebra, and the preliminaries of Galois theory.

**Prerequisite:** One year of high school Algebra II/Trigonometry is required for application.

**Program Director: Ellen Urheim**

Ellen Urheim received her undergraduate degree in mathematics and chemistry from Johns Hopkins University and spent several years working as a business strategy consultant before coming to the University of Pennsylvania for her graduate studies. She received her PhD in mathematics while doing research in the field of harmonic analysis, studying objects called oscillatory integral operators which are special ways of transforming functions; a famous example of such an object is the Fourier transform. She is currently a lecturer at the University of Pennsylvania and has taught a range of classes that include linear algebra and an introduction to proofs in analysis.