It's not just numbers: Journeys into exploratory and proof-based mathematics
SAT Score Range
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15 sessions
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AH
LH
SW
+26
About
I believe everyone should be given a space to practice their proof-based math skills, since it teaches you to think logically on your feet, explore, and build/challenge arguments.
PREREQUISITES: This series is meant to be accessible to a broad audience. In reality, much of what we consider to be "college-level math" is completely reachable for students with just a basic proficiency with high school algebra. If any of these sessions require any more advanced prerequisites, I will say so in the description. If you don't have the background knowledge, I encourage you to attend anyway, but also feel free to ask for an excused absence.
Each session, we will explore a different topic in mathematics. These topics are likely unlike any math you've seen before. With this series, I hope to help expand your horizons of what math really is, outside of the classroom. Topics include, but are not limited to, set theory, combinatorics, graph theory, geometry, and more.
These sessions will be proof- and discussion-based, so I would like all participants to engage as much as they are comfortable with! I will provide the basic definitions, pose motivating questions, and provide hints as needed, but the exploration is entirely up to you.
I will add more sessions to this series as we go, so if you have any ideas for future sessions, please feel free to suggest them!
Also, if you're interested in this series but the current set time doesn't work for you, feel free to suggest a better time period that works for you, and if there's enough interest, I can see if I can switch times.
Tutored by
✋ ATTENDANCE POLICY
Each session will be mostly independent from the others, so don't worry if you need to miss a few. However, if you do, please give me as much advance notice as possible.
SESSION 1
25
Jul
SESSION 1
Even More Math
Even More Math
Fri 12:30 AM - 1:30 AM UTCJul 25, 12:30 AM - 1:30 AM UTC
Title: Some infinities are just built different
Is infinity a number? If not, what is it? Are all infinities the same? (well, clearly not, given the title)
In this session, we will discuss cardinality, a way to conceptualize of the "size" of a collection of objects. We will apply this concept to various infinite sets to try to understand their sizes.
SESSION 2
29
Jul
SESSION 2
Even More Math
Even More Math
Tue 12:30 AM - 1:30 AM UTCJul 29, 12:30 AM - 1:30 AM UTC
Title: One step at a time
Mathematical induction is a very powerful technique for proving many classes of mathematical statements. Induction is extremely simple: If I'm trying to prove a statement for every natural number, I need to show that it's true for n = 1 and that if it's true for n = k, then it must be true for n = k + 1. If both of these things are true, then the statement is proven.
In this session, we will review the basic definition of induction, go through an example or two, and then explore some interesting problems to try to come up with an induction-based solution.
SESSION 3
30
Jul
SESSION 3
Even More Math
Even More Math
Wed 12:30 AM - 1:30 AM UTCJul 30, 12:30 AM - 1:30 AM UTC
Title: Mathematicians love coloring and connect-the-dots
Graphs are mathematical representations of networks. A very common flavor of question in the theory of graphs is: if I assign a color to each node in this graph, what is the minimum number of colors I need so that no two connected nodes are the same color?
In this session, we'll explore the concept of chromatic numbers of different graphs, trying to develop an intuition for what a graph's chromatic number says about the graph's structure. I will also mention an open problem in this area, which we can dig deeper into in a later session!
SESSION 4
1
Aug
SESSION 4
Enrichment
Enrichment
Fri 12:30 AM - 1:30 AM UTCAug 1, 12:30 AM - 1:30 AM UTC
Title: The state of math education
Based on a few requests from you all, this session, we will have a more open-ended discussion about your experiences in math education. I'm curious to hear your perspectives about what you think worked well and what didn't work well, for you and for others, in your classroom math experience!
SESSION 5
4
Aug
SESSION 5
Even More Math
Even More Math
Mon 2:30 PM - 4:00 PM UTCAug 4, 2:30 PM - 4:00 PM UTC
Title: Revisiting some third-grade arithmetic
Number Theory is the study of the natural numbers and how they interact with the basic arithmetic operations. Common topics of analysis in this field are prime numbers, divisibility, modular arithmetic, and more. Though it seems simple and perhaps inapplicable, Number Theory has proven incredibly useful in forming the foundation of modern cryptography (and thus computer security). In this session, we will explore some of the basics of Number Theory, which will prepare us to learn about RSA, the central cryptographic algorithm, in a future session.
SESSION 6
6
Aug
SESSION 6
Even More Math
Even More Math
Wed 12:30 AM - 2:00 AM UTCAug 6, 12:30 AM - 2:00 AM UTC
Title: Integration, but fancy and French
The Lebesgue Integral is an improvement upon, and generalization of, the Riemann Integral, making almost all conceivable functions integrable (for example, the Dirichlet function; look it up, it's weird).
NOTE: If you don't have any Calculus experience, this session may be a bit hard to follow. I will try to make it as accessible as possible, so please feel free to still join, and don't be discouraged if you don't understand.
SESSION 7
7
Aug
SESSION 7
Even More Math
Even More Math
Thu 10:00 PM - 11:30 PM UTCAug 7, 10:00 PM - 11:30 PM UTC
Title: In Manhattan, pi = 2.8284
In this session, we will be exploring the concept of a metric space, defined as a set equipped with some way to measure the distance between two points. Traditional Euclidean space is a metric space in which the ratio between a circle's circumference and its diameter is pi = 3.1415..., but with different ways to measure distance, this number can be very different. Different metric spaces are very useful in different contexts in math and in real life.
SESSION 8
12
Aug
SESSION 8
Even More Math
Even More Math
Tue 12:30 AM - 2:00 AM UTCAug 12, 12:30 AM - 2:00 AM UTC
Title: Woah, is this movie in 4D?
In this session, we'll explore the concept of dimension, which shows up absolutely everywhere in math. There have been numerous ways to understand dimension, each generalizing a different aspect of our intuitive understanding of our 3D universe. We will explore concepts like higher dimensions, infinite-dimensional spaces, non-integer dimension, and more.
SESSION 9
12
Aug
SESSION 9
Even More Math
Even More Math
Tue 10:00 PM - 11:30 PM UTCAug 12, 10:00 PM - 11:30 PM UTC
Title: The mathematics of pancake toppings 🥞
A few days ago, my friends and I were making pancakes, but we ran into a problem: between all of us, there were many combinations of preferences for types of pancakes, each of which required us to put different combinations of toppings into the batter before cooking. I wanted blueberries, someone else wanted chocolate chips, a third person wanted both, and so on.
With so many preferences, it was hard to keep track of how to split up the batter so everyone got their preference. The problem of figuring out how to manage all of this is deeply connected to the mathematical structure of a partially ordered set (or poset, for short). In this session, we'll try to understand this concept and the many places it shows up in math and the real world, including your next brunch!
SESSION 10
15
Aug
SESSION 10
Even More Math
Even More Math
Fri 12:30 AM - 2:00 AM UTCAug 15, 12:30 AM - 2:00 AM UTC
Title: Why must the soccer ball pattern have pentagons on it? ⚽
In this session, we'll dive into the Euler Characteristic, investigating questions such as the one in the title. Through a relatively straightforward counting-based exploration, we will uncover a deep truth about connections between polyhedra, graphs, and topology.
SESSION 11
20
Aug
SESSION 11
Even More Math
Even More Math
Wed 12:30 AM - 2:00 AM UTCAug 20, 12:30 AM - 2:00 AM UTC
Title: Is base-10 really the best?
We've lived our whole lives working with exactly 10 numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. To count past 9, you need to flip the "ones" place to 0 and the "tens" place to 1. In other words, we use base-10 to represent our numbers. There are many properties of base-10 that make it useful, for example the fact that divisibility by 2 and 5 are easy to determine, but there are others that are a pain to do in our heads, for example multiplying a number by 7. In this session, we'll explore these properties and try to build a case for ourselves either arguing that base-10 is, in fact, best, or if humanity should add/remove a few numerals and switch bases entirely.
SESSION 12
21
Aug
SESSION 12
Even More Math
Even More Math
Thu 12:30 AM - 2:00 AM UTCAug 21, 12:30 AM - 2:00 AM UTC
Title: (Rescheduled from earlier) The math that makes your computer secure
In this session, we'll explore the RSA cryptographic algorithm that forms the basis of all modern computer security. To prepare for this session, please review the notes from the previous session introducing basic number theory ideas such as divisibility, modular arithmetic, and finite fields.
Note: I will most likely not be able to host any more sessions after this one until late September. After that, I would love to continue if there is still interest!
SESSION 13
28
Sep
SESSION 13
Even More Math
Even More Math
Sun 4:00 PM - 5:30 PM UTCSep 28, 4:00 PM - 5:30 PM UTC
Title: What is a number?
In math, we frequently work with objects like integers, rational numbers, and real numbers, but we rarely stop to think about what these numbers actually are or how they are defined. In this session, we'll construct these numbers from the ground up, starting with empty sets and building up to the continuum of real numbers and beyond.
Though they have not been scheduled yet, I plan to host more sessions in this series. If you're interested, keep posted by registering!
SESSION 14
5
Oct
SESSION 14
Even More Math
Even More Math
Sun 4:00 PM - 5:30 PM UTCOct 5, 4:00 PM - 5:30 PM UTC
Make-up for "What is a number?" (sorry for dropping the ball on last week's scheduled session 😔)
In math, we frequently work with objects like integers, rational numbers, and real numbers, but we rarely stop to think about what these numbers actually are or how they are defined. In this session, we'll construct these numbers from the ground up, starting with empty sets and building up to the continuum of real numbers and beyond.
SESSION 15
12
Oct
SESSION 15
Even More Math
Even More Math
Sun 4:00 PM - 5:30 PM UTCOct 12, 4:00 PM - 5:30 PM UTC
Title: Counting?? I thought we're not in preschool anymore...
Many problems in math require you to count the number of possibilities for a given system. For example, one might ask: How many ways can four dice sum to 10? In this session, we will discuss the principles of counting that form the basis of Combinatorics, diving into some surprisingly rich ideas in math such as Pascal's Triangle, partitions, and more.
Though they have not been scheduled yet, I plan to host more sessions in this series. If you're interested, keep posted by registering, and feel free to suggest ideas for future sessions!