Colin Ni
- About Me
I'm a third-year graduate student at the UCLA Department of Mathematics interested in tensor-triangular geometry and previously number theory. I passed the algebra and geometry/topology quals in Fall 2021 and the basic in Spring 2022.
I play blitz chess and live high-stakes poker, and I worked at Scale AI as a machine learning research engineer.
I did my undergrad at the College of Creative Studies (CCS) at UCSB. Here is a list of the classes I have taken.
- Teaching History
- Winter 2024: Math 156 (instructor: Jona Lelmi)
- I'm teaching section 1A (Thurs 3pm MS 5118)
- Fall 2023: Math 110AH (instructor: Alexander Merkurjev)
- Spring 2023: Math 115AH (instructor: Paul Balmer)
- I taught section 1A (Tues Thurs 2pm Geology 6704)
- Spring 2023: Math 115B (instructor: Artem Chernikov)
- I taught section 1A (Thurs 1pm Geology 4645)
- Winter 2023: Math 32A (instructor: Brian Shin)
- I taught sections 2E and 2F (Tues Thurs 11am Geology 6704)
- Winter 2023: Math 32A (instructor: Richard Wong)
- I taught sections 3E and 3F (Tues Thurs 12pm Boelter 5422)
- Fall 2022: Math 115AH (instructor: Hong Wang)
- Fall 2022: Math 132 (instructor: Hatice Mutlu Akaturk)
- I taught section 1A (Tues 10am MS 6704)
- Summer 2022 (session A): Math 167 (instructor: Eric Radke)
- I taught section 1A (Wed 1pm online)
- Spring 2022: Math 33B (instructor: Wumaier Maimaitiyiming)
- I taught sections 2C (Tues 1pm Lakretz 120) and 2D (Thurs 1pm Geology 6704)
- Winter 2022: Math 32B (instructor: Koffi Enakoutsa)
- I taught sections 1C (Tues 8am MS 5118) and 1D (Thurs 8am MS 5118)
- Winter 2022: Math 32B (instructor: Richard Wong)
- I taught sections 4A (Tues 4pm Geology 4645) and 4B (Thurs 4pm MS 5127)
- Here are the final review problems, which contain variants of most of the final exam problems
- Fall 2021: Math 32A (instructor: Richard Wong)
- I taught sections 5A (Tues 2pm Boelter 5272) and 5B (Thurs 3pm Boelter 5264)
- Here are the final review problems, which contain slight variants of all the final exam problems
- Fun Problems (sorted by difficulty)
- Color the plane using two colors. Are there always two points 1 distance apart with the same color?
- Two players take turns placing coins onto a round table without overlap, and the last person who places a coin wins. Who wins?
- How many cuts does it take to cut a 3x3x3 block of tofu into 27 1x1x1 pieces? You are allowed to move pieces around between cuts.
- Some ants are walking on a log, all at the same speed. When two ants bump into each other, they turn around and walk the other way, and when they reach the end of the log, they fall off. How long will it take before they all fall off (if ever)?
- Two trains (of possibly different speeds) are traveling toward each other. A fly starts at one train, flies to the other, turns around and flies back, etc. How far does the fly travel?
- A hundred coins are on a table in front of you, exactly 10 of which show heads, but you are blindfolded. Can you split the coins, flipping them as you desire, into two groups that contain the same number of heads?
- What is the limit of v_p(n!) as n -> infinity?
- There are a hundred lightbulbs initially all off. Person i flips the switch for every i'th lightbulb, for i ranging from 1-100. Which bulbs are on in the end?
- An 8x8 chess board is coverable by 32 2x1 dominoes. If you remove the bottom-left and top-right corners, is it coverable by 31 2x1 dominoes?
- A crocodile has n teeth, one of which is sore and will make the crocodile bite if pulled. You and your n - 1 friends decide to each pull a tooth, one by one. Would you rather be first, second, ..., last?
- You and your friend are among 32 people participating in a 5-round single elimination tournament. Assuming everyone has a 50/50 chance of winning each game, what is the probability that you will play your friend?
- Show that the set of rational numbers Q has measure 0, i.e. show that you can cover Q by a collection of line segments whose total length is arbitrarily small.
- On an island lives some cannibals. Two cannibals who are awake will not try to eat each other, but if a cannibal eats a person, then they fall asleep and are thus vulnerable to getting eaten. A visitor who is not a cannibal visits the island. Is the visitor safe?
- Show that there exists arbitrarily large gaps between primes. (In fact, Yitang Zhang proved that there are infinitely many pairs of primes with gap at most 70000000.)
- Four coins are on a spinning platform in front of you, but you are blindfolded. On each turn you may flip any number of the coins you want, but after every turn someone gets to rotate the platform any way they want. Can you make them all heads or all tails in a finite number of turns?
- Take n copies of a solid genus g surface, and identify their boundaries. Compute the homology in your head.
- Can you always rotate a four-legged square table to keep it from wobbling?
- Your friend thinks of a polynomial p with natural number coefficients, and your job is to determine p by asking for the value of p at natural numbers. Can you do with finitely many questions, and if so, how many?
- Show that on a finite-dimensional vector space, a linear operator is invariant on all hyperplanes only if it is a scalar multiple of the identity.
- In a tribe lives some villagers. They all have red eyes but are forbidden to speak of it because the island is cursed: if you fall asleep knowing you have red eyes, you die. One day a foreigner lets it slip that someone has red eyes. What becomes of the villagers?
- Suppose the (complex) character table of a finite group contains the row (1, w, w^2, 1), where w is a primitive 3rd root of unity. What can the group be?
- A mouse is stuck in a circular pond, and a cat is at the edge of the pond, trying to catch the mouse. The cat can run four times as fast as the mouse can swim, but the mouse can outrun the cat. The cat cannot swim. Can the mouse escape the pond?
- Show that a finite group acting transitively on a non-singleton set has a fixed-point-free element.
- Given an n by n matrix of positive numbers and a threshold t, find an efficient algorithm to determine the maximum k such that any k by k contiguious submatrix has sum at most t.
- Show that you can cover any 10 points in the plane with disjoint unit disks.
- Let p > 3 and p + 2 be twin primes. Add up the digits of p(p+2), add up the digits of that resulting number, and so on, until you get a one-digit number. Show that this one-digit number is 8.
- Let B be the square matrix with ij-entry given by the number of shared divisors of i and j. What is the determinant of B?
- Settlers discover an infinite grid and decide to create a city. On day 1, they build a finite number of height 1 buildings. On day n they build a new building of height n such that the heights of the surrounding (eight) buildings sum to n, but they stop building altogether if this is impossible. Does the city ever stop expanding?
- Two players each choose a positive integer. You pay the other player $1 if your number is exactly one less than theirs, but you get paid $2 if your number is more than one less than theirs. What is each player's strategy?
- A safe has 30 lights on it and contains 2^29 dollars. At the cost of 1 dollar, you may flip any of the lights on or off and then try opening the safe; if all 30 lights are correct, then the safe opens, but otherwise the safe indicates whether you got at least half of the lights correct. How much money do you expect to make?
- Show that a square matrix with positive entries has a positive eigenvalue.
- Is 9223837124123 a square mod 3568943567827? What about 9223837124125?
- A lion is trying to catch one of 100 sheep in the plane R^2. Once the sheep decide on a starting position, each turn the lion moves in any direction up to 10 meters and then so does one of the sheep. Can the lion catch a sheep?
- In each of the squares in an infinite grid, draw an arrow pointing N, NE, E, SE, S, SW, S, or NW such that horizontally or vertically adjacent squares have arrows differing by at most 45 degrees. Is there a closed loop?
- Show that for any hexagon inscribed in a nondegenerate conic, the three intersection points of its opposite sides are collinear.
- Construct a bijection from (0, 1) to [0, 1).
- Your friend chooses two distinct real numbers and gives you one at random. Would you bet $1 on whether you can guess whether it was the higher number of the two?
- At what angle should you throw an object off a cliff to maximize how far away it lands? Explain without calculus.
- Can you solve a 4x4 sliding tile puzzle where the 14 and 15 are swapped?:
1 2 3 4
5 6 7 8
9 10 11 12
13 15 14 []
- A hundred prisoners are numbered 1-100, and release forms numbered 1-100 for the prisoners are placed randomly in 100 drawers, one in each drawer. If all prisoners find their release forms, they will all be released. However, each prisoner may look in 50 drawers, and they are forbidden to communicate after the process begins. Can they do better than (1/2)^100?
- Show that if a smooth compact connected oriented manifold without boundary has Euler characteristic zero, then it admits a nowhere vanishing vector field.
- Pick a finite number of vectors in a finite-dimensional vector space over an infinite field. Show that there exists a nonzero functional that vanishes on all of these vectors.
- You and an accomplice steal a necklace and open it at the clasp. The necklace contains jewels of n different colors, and there are an even number of jewels of each color. Show that by cutting the necklace at n spots and taking altnerating segments, you can split the jewels of each color evenly with your accomplice.
- Given a p-faced die for p = 2, 3, 5, ..., 41, how can you generate a uniformly random integer in [1, 42] in the fewest number of rolls (on average, if non-deterministic).
- Take a finite collection of unit disks in the plane, and let S denote the area of their union. Show that there is a subset of the disks whose union has area at least 2S/9.
- Two players ante $1, have $1 behind, and are given a uniformly random number in [0, 1], with the higher number winning at showdown. Either player can bet $1, but player one acts first. What is each player's strategy and EV?
- Can you partition R^2 into circles? What about R^3?
- Show that for every n there is a real extension of the rationals with Galois group Z/n.
- Alice, Bob, and Charles are in a 3-way duel. They take turns shooting or passing their turn until only one person survives, with Alice going first, then Bob, then Charles, then back to Alice, etc. They respectively can hit a target 10%, 30%, and 60% of the time. What is each player's strategy?
- Three blind spiders and a crippled fly walk along the edges of a regular tetrahedron. The spiders can all walk faster than the fly but cannot see it. Can the spiders catch the fly?
- Prisoners 1, 2, 3, ... are each wearing a hat with a real number on it and lined up so that prisoner n can see prisoner n + k for every positive k. The prisoners are asked to guess the number on their hat, being set free if they guess their number correctly. If they go one by one in order so that the prisoners hear the previous guesses, then how many prisoners can be set free? What if they must guess all at once?
- Show that any simple (Jordan) curve in the plane contains four points that form a rectangle.
- Two players each ante $1, have $1 behind, and are dealt one hole hard out of a three-card deck consisting of a J, Q, and K. What is each player's strategy and EV? (The only allowed bet size is $1.)
- Let b be a nonzero root of a monic polynomial f with integer coefficients (i.e. an algebraic integer), and suppose the other roots of f lie in the complex unit disk. Show that b is a root of unity.
- A group of friends is quarelling while trying to plan a trip. Each friend has a list of places they want to visit and refuses to go on the trip unless they get to visit all of them. How can you efficiently compute the maximal (number of friends on the trip) / (number of places visited)?
- Top 5 Favorite...
Theorems
Math books
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- Resources (sorted by technical-ness)
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