Mathematics
math·e·mat·ics · mathēma: that which is learned · manthanein: to learn
Mathematics is the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions — where evidence means proof, and a single counterexample can overturn a claim.
- Start from a known theorem and change one condition — what breaks?
- Compute small cases, conjecture a pattern, then prove it.
- State definitions precisely before you use them.
- Expository work counts too: a clear proof of a hard result, written for your peers.
Related areas & concentrations
Mathematics is the language of the sciences. Its neighbors supply real problems — and often inspire new mathematics.
Related research areas
- Physics: modeling motion, forces, and the laws of nature
- Engineering: applying math to design and optimize real systems
- Chemistry: symmetry, rates, and quantitative models of reactions
- Computer science: algorithms, logic, and computation
Sample concentrations
- Introduction to combinatorics and graph theory: counting, networks, and connections
- Coding theory and cryptography: sending messages reliably and securely
- Fibonacci numbers and visual proofs: patterns you can see and prove
- Applied mathematics for engineers: the tools that solve real-world problems
Four ways into a question
Most student math research works in one of these areas. All of them start with playing with small examples until a pattern appears.
Combinatorics & Graphs
Counting arrangements and studying networks of points and connections.
How many ways can you color a map of n regions with three colors so no neighbors match?
Number Theory & Patterns
Properties of whole numbers — primes, divisibility, and famous sequences.
Which Fibonacci numbers are divisible by 5, and can you prove the pattern?
Codes & Cryptography
The math behind correcting errors and keeping information secret.
How many errors can a simple repetition code detect and correct?
Applied Math & Modeling
Using equations and data to describe and predict real-world systems.
How well does a simple model predict how quickly a cup of coffee cools?
Turning an interest into a question
A topic isn't a research question. Narrow to a specific object or problem, explore small cases, and state a precise claim you can try to prove.
Start from a pattern or puzzle
A sequence, a game, a shape, a counting problem — something you can explore by hand.
Compute small cases
Work out the first several examples, by hand or with code, and record everything.
State a precise conjecture
Turn the pattern into an exact claim with clearly defined terms.
Prove it — or find out why not
Try induction, contradiction, or a visual argument. A counterexample is a result too.
Methods & where to look
Pair the core techniques of proof with the tools and references mathematicians actually use.
Common methods
- Direct proof: reasoning step by step from definitions.
- Induction: proving a statement for every whole number.
- Contradiction: showing the opposite leads to something impossible.
- Computational exploration: using code to test many cases quickly.
- Visual proof: arguments built from diagrams and pictures.
Computer checks build confidence, but only a proof settles the question.
Where to find sources
- Sequences: the On-Line Encyclopedia of Integer Sequences (OEIS).
- References: Wolfram MathWorld for definitions and known results.
- Preprints: the math section of arXiv for current research.
- Student journals: Math Horizons and the College Mathematics Journal.
Free tools like Desmos, GeoGebra, and Python are great for exploring patterns.
Writing a clear proof
A proof has to convince a careful reader, not just you. Three habits keep your mathematics rigorous and readable.
Define
State every definition and assumption up front. Ambiguous terms are where proofs go wrong.
Justify
Give a reason for every step. “It's obvious” usually hides the hardest part.
Check
Test your claim on edge cases and try to break it. Look for the counterexample before a reader does.
Cite every theorem you use that you didn't prove yourself, and credit any code or tools.