# Projects for newbie math lovers This page includes some potential projects for students who loves mathematics yet are new to college mathematics. Once you have chosen a topic, explore the following aspects to gain a full understanding. 1. **Digest the definition.** Read the Wikipedia page of the concept. Remember the definition and think about why/how people come up with this idea. If necessary, find some textbook to learn more details. 2. **Generate some examples.** Read some positive examples and negative examples so that you can distinguish them. Generate more examples and analyze them using the definition. 3. **Establish some properties.** Find some properties of the concept and think about why. 4. **Make your own example.** Apply what you have learned to make your own examples. Try some more challenging examples, or use a computer program to simulate the idea. Suprise the audience! :::info :bulb: When using generative AIs, the following prompt might be useful. * Please tell me what is XXX. + What is the formal definition of XXX? + What is the motivation of XXX? + What is the intuition of XXX? * Please tell me some examples of XXX. + Please give me an example that meets the definition of XXX. + Please give me an example that does not meet the definition of XXX. * Please tell me some properties of XXX. + What is the formalt statement of this property? + What is the intuition of this property? + Please tell me some examples of this property. + How to prove it? + I am doing research on XXX. I would like to work on some cool projects to suprise my audience. Could you tell me some potential hand-on projects? ::: ## Limit of a function Limits formalize the idea of approaching a value, even at points where the function is undefined. They allow mathematicians to define continuity, derivatives, and integrals rigorously. Without limits, the entire foundation of calculus collapses. - [Wikipedia: Limit of a function](https://en.wikipedia.org/wiki/Limit_of_a_function) - Can you use `desmos` to demonstrate your idea? ## Derivative The derivative measures instantaneous change—how a quantity varies at an exact moment. It underlies physics, optimization, differential equations, and modeling real-world processes. Derivatives let us analyze growth, motion, and sensitivity in countless scientific fields. - [Wikipedia: Derivative](https://en.wikipedia.org/wiki/Derivative) - How to find the derivative of $x^n$ by definition? How about the cases when $n$ is a negative number, a fraction, or even an irrational number? ## Riemann sum A Riemann sum approximates area by adding small rectangles. This idea motivates the definite integral and shows how continuous accumulation arises from discrete pieces. It provides an intuitive bridge from geometry to rigorous calculus. - [Wikipedia: Riemann sum](https://en.wikipedia.org/wiki/Riemann_sum) - Can you write a computer program to find the Riemann sum? ## Vector space Vector spaces unify diverse objects—vectors, functions, polynomials—under one algebraic framework. Their structure enables linear combinations, dimension, and basis, forming the foundation of linear algebra, optimization, data science, quantum mechanics, and more. - [Wikipedia: Vector space](https://en.wikipedia.org/wiki/Vector_space) - What is the craziest vector space you have ever heard of? ## Linear map Linear maps preserve addition and scaling, making them the simplest structure-preserving maps. They capture symmetries, transformations, and approximations. Many nonlinear phenomena become simpler when locally approximated by linear maps. - [Wikipedia: Linear map](https://en.wikipedia.org/wiki/Linear_map) - What is the craziest linear map you have ever heard of? ## Determinant The determinant measures how a linear transformation scales area or volume. It encodes geometric information like orientation and algebraic properties like invertibility. Determinants also appear in eigenvalues, differential equations, and change-of-variables formulas. - [Wikipedia: Determinant](https://en.wikipedia.org/wiki/Determinant) - How many terms are there in the determinant formula of a $4\times 4$ matrix? ## Countable set Countability distinguishes between different sizes of infinity. It shows that some infinite sets (like integers) are “smaller” than others (like real numbers). This idea is essential in analysis, probability, and the foundations of mathematics. - [Wikipedia: Countable set](https://en.wikipedia.org/wiki/Countable_set) - Is $\mathbb{Q}$ countale? ## Equivalence relation Equivalence relations group objects that should be considered “the same” under some criterion. They allow mathematicians to form quotient sets, classify mathematical objects, simplify structures, and capture symmetry and redundancy. - [Wikipedia: Equivalence relation](https://en.wikipedia.org/wiki/Equivalence_relation) - What is the craziest equivalence you have ever heard of? ## Partial order Partial orders describe systems where not every pair of elements is comparable, such as tasks with dependencies or subsets of a set. They provide a flexible language for hierarchy, optimization, and combinatorial structures. - [Wikipedia: Partially ordered set](https://en.wikipedia.org/wiki/Partially_ordered_set) - What is the craziest partial order you have ever heard of? ## Infimum and supremum Infimum and supremum capture the best possible lower and upper bounds of sets. They complete the real numbers and allow precise handling of limits, convergence, and optimization. Many theorems in analysis depend on them. - [Wikipedia: Infimum and supremum](https://en.wikipedia.org/wiki/Infimum_and_supremum) - What is the difference between minimum, minimal, and infimum? - What is the difference between maximum, maximal, and supremum? ## Open set and closed set Open and closed sets describe the shape of spaces in terms of neighborhoods and boundaries. They provide a framework for continuity, convergence, compactness, and connectedness, forming the foundation of topology and modern analysis. - [Wikipedia: Open set](https://en.wikipedia.org/wiki/Open_set) - [Wikipedia: Closed set](https://en.wikipedia.org/wiki/Closed_set) - Can a set be open and closed at the same time? ## Group Groups formalize symmetry and reversible operations. They appear in geometry, physics, number theory, cryptography, and algebra. Understanding groups allows us to classify structures, analyze transformations, and solve equations systematically. - [Wikipedia: Group (mathematics)](https://en.wikipedia.org/wiki/Group_(mathematics)) - Do you find some examples of groups in real world? ## Field Fields generalize the arithmetic of rational and real numbers. They allow addition, subtraction, multiplication, and division, making them essential for linear algebra, number theory, coding theory, and algebraic geometry. Fields unify many mathematical systems. - [Wikipedia: Field (mathematics)](https://en.wikipedia.org/wiki/Field_(mathematics)) - - Do you find some examples of fields in real world?