In this course, we will practice and learn to:
write clearly, concisely, and correctly. This will be a writing-intensive course. This involves:
writing in prose
using proper mathematical style
using proper mathematical notation
using precise language
communicating enough, but no more
develop a healthy respect for the importance of definitions. This involves:
internalizing definitions so that you understand them
stating definitions verbatim
constructing examples and constructing non-examples
comprehend the structure and importance of theorems. This involves:
stating theorems accurately; identifying assumptions and conclusions
constructing examples and constructing non-examples
formulating and testing conjectures
comprehend the structure and importance of mathematical proofs. Proofs are a form of persuasive writing that are logical, comprehensible, and explanatory. This involves:
analyzing and/or critiquing existing proofs and/or non-proofs
creating your own proofs
be able to abstract appropriately. This involves
(application) draw a conclusion by showing a particular case satisfies the assumptions of a theorem
(generalization) develop conjectures based on common properties observed in specific examples
In this class, we will learn about (not necessarily in this order):
Groups and symmetry (with a strong emphasis on the theory of finite groups)
Subgroups
Cosets
Normal subgroups and Factor Groups
Products of Groups
Examples of Groups: Cyclic Groups, Dihedral Groups, Permutation Groups, Groups of Matrices
Isomorphisms, Homomorphisms
Rings and ideals
Have fun and gain confidence in your mathematical reasoning and skills
Improve as a learner, teacher, communicator, and mathematician by working individually and as part of a community to further not only your personal growth, but also our collective knowledge and understanding of the world around us
read and communicate (verbally and in writing) clearly, concisely, and correctly using relevant vocabulary, notation, and colloquial language of modern mathematics
understand and appreciate philosophical and conceptual ideas of modern mathematics, and how they lead to models for, and solutions of, common (and uncommon) problems