**Instructors:**Janko Gravner, Edward (William) Tavernetti**Prerequisites:**Precalculus, or Equivalent**Typical Field Trips:**Exploratorium, California Academy of Science- This is a
option only.**FIRST CHOICE**

## Introduction

This cluster is designed to introduce students with a strong interest in mathematics to several advanced topics. These topics would ordinarily be studied at the advanced undergraduate level, but all lend themselves to an introductory course at the high school level. No prior experience in any of these topics is expected, but enthusiasm for and interest in mathematics is essential. Students in this cluster will gain insight into what to expect from studying mathematics, or applied mathematics, in university. There will be an emphasis on doing mathematics and getting hands-on experience!

### Core Course - Probability

We will learn methods from the mathematical theory of probability that can solve many practical problems, such as this one: suppose you and three friends are each dealt 13 cards from a shuffled deck, and you observe that you were dealt a single ace; what is then the probably that your three friends also each have an ace? On the way, we will also learn many useful counting methods.

Probability is one one the most applicable areas of mathematics, and is used in statistics, finance, gambling, physics, artificial intelligence/machine learning, computer science, game theory, and philosophy to draw inferences about the expected frequency of events.

### Supplementary Course** - Introduction to **Mathematics and Applied Mathematics** **

We will also consider a broad study of mathematics and applied mathematics, covering many topics in brief by examining a core set of representative ideas. These topics will be drawn from logic and set theory, geometry, numbers, infinity, probability and statistics, calculus, applications and modeling. The overall goal of this portion of the cluster focus is to help students gain wider experience and appreciation for the various ideas of mathematics and applied mathematics.

For example, just in our study of numbers we will cover the number sets: naturals, integers, rationals, reals, and complex number systems. Subtle ideas from counting, and their applications will astound students! Large and very large numbers, as well as different sizes of infinities will challenge our imagination. Important numbers and where they come from, as well as some other interesting numbers and their stories will be considered. Deep connections to geometry will be made. Computer representation of numbers and numerical ideas will be explored, and we will better understand how computers work, their capabilities and limitations. The interconnections between areas of mathematics run deep, and as we move from one topic to the next, this rich structure will become more apparent.