Cluster 1: Logic, Cryptography and Number Theory: Reason and Riddles
What do solving a riddle, playing sudoku and investing in the stock market have in common? The answer, of course, is mathematics. Mathematics is the language and science of learning and problem solving, and in cluster 1 students look at mathematics from (at least) two perspectives.
Logic and set theory form the foundation of all mathematics, and in the logic class students will explore foundational questions like “What is and how do you deal mathematically with infinity?”, "Can we be certain or possibly even prove that mathematics is free of contradictions?" and "Won't there some day be a clever computer who will correctly answer all mathematical questions with the push of a button?". Probability and game theory are two of the most applied fields in mathematics. In the number theory and cryptography course, students will study number theory, and how basic properties of the integers are used to create `unbreakable' codes, like the ones banks use to keep on-line banking safe. In both courses, students will see how pure theory is translated into practical and important applications.
Prerequisite: One year of high school mathematics.
All students in this cluster will be enrolled in the following
courses.
Logic, Riddles, & Paradoxes
Instructor: Frank Bäuerle, Ph.D. (Mathematics Department)
The basic goal in this course is for students to wrack their brains, have fun while they are doing it, and at the same time gain insight into the inner-workings of mathematics. The students' work is loosely grouped into the following four categories.
To infinity and beyond: Students will be exposed to some of the most surprising, interesting, and intriguing aspects of the age-old riddle of infinity. Examples are: There is always room at Hotel Infinity; Countability of the integers and rationals; Can you list the real numbers?; Cantor's Diagonalization Method; The Continuum Hypothesis.
Paradoxes: Students will be confronted with statements that are strange, weird, and sometimes even false. Examples are: Russel's "Salon Paradox" (a paradox on self-referentiality). Epimenides' "Liar's Paradox" (a paradox on self-referentiality). Zeno's "Achilles vs. the Tortoise Paradox" (a paradox on infinity, or how infinitely much stuff doesn't account for much).
Logical Systems: Students will study and practice the basic workings of logical systems. Examples are: Truth tables of logical connectors (and, or, implies, etc.); Tautologie; Natural deduction system for proofs; Sufficient and necessary conditions on sorting your socks into pairs.
Riddles and brain teasers: Students will wrack their brains solving these puzzles from ancient to modern times. A typical example is the following Pirate Problem: A pirate ship captures a treasure of 1000 golden coins. The treasure has to be split among the 5 pirates: 1, 2, 3, 4, and 5 in order of rank. The pirates have the following important characteristics: infinitely smart, bloodthirsty, greedy. Starting with pirate 5 they can make a proposal how to split up the treasure. This proposal can either be accepted or the pirate is thrown overboard. A proposal is accepted if a majority of the pirates agree on it (a tie means the proposal is rejected). What proposal should pirate 5 make? What happens if there are 6, 7 or more pirates? Generalize!
Number Theory and Cryptography
Instructor: Yonatan Katznelson, Ph.D. (Applied Mathematics Department)
Both the theory of numbers and cryptography have been around for thousands of years, and there have always been connections between them. In the past 30 years the interaction between number theory and cryptography has grown tremendously, fueled by the need to transmit information safely on computer networks, like the internet. In this course, we'll explore the integers and their properties and see how number theory is used in the construction of crypto-systems. One of the topics we'll study in some detail is the distribution of prime numbers among all the integers, and why prime numbers are so important to cryptography. Until we meet this summer, here is a simple encrypted message for you to work on: EHZDUHWKHLGHVRIPDUFK.
Hint: The code is famous, and the message is appropriate.
Throughout the course students will be challenged to absorb new concepts technical skills, and apply them to a variety of problems, some familiar and some perhaps less familiar. Ultimately, the goal is for students to gain insight into the ways mathematical analysis is applied to `real world' problems.
Transferable Skills: Tools for Success
It may or may not surprise you that being a university researcher
requires a whole host of skills outside of the specific scientific
knowledge required of your chosen discipline or specialty. It requires
communication skills such as the ability to present your work in
writing and orally. It requires competencies in the realm of information
technology including the ability to find and judge (the validity
of) information and use a variety of hardware and software tools
(e.g. spreadsheets, databases, statistics software, other data manipulation
tools). It requires all of those skills required to effectively
conduct research such as data collection, analysis and interpretation,
critical thinking and problem solving as well as the ability to
conduct laboratory and/or field work. And, of course, a baseline
competency in English, science, mathematics and computers is critical.
The governing mission of the UCSC COSMOS Transferable Skills course
is to promote students’ future academic (and professional)
success through the exploration and development of transferable
skills: i.e. those competencies that students develop while in school
which facilitate academic achievement, the eventual transition into
the work-force and which are applicable in many other life situations.
Go to course information for:
- Logic, Cryptography and Number Theory: Reason and Riddles*
- Engineering
the Future: Autonomous Robots and Nanotechnology*
- Under
the Sea: Exploring Marine Organisms and Their World*
- Everyday
Chemistry: From Perfumes to Pollution*
- Video Games: The Design of Fun - From Concept to Code*
- Chemistry
and Mathematics: From Life to Thought*
- Points in Space: Astronomy and Linear Algebra*
- Marine Mammals and Oceanography: From Prey to Predators
- Particle and Astrophysics: Investigations of the Minuscule to the Massive
