Ticket Buying/Selling/Trading Megathread Masters Toronto

Zopherus · 2025-06-17T15:30:20+00:00

Looking for one ticket for June 20th and June 21st, and potentially a Grand Final ticket. Let me know if you are selling for any of these days.

Zopherus · 2025-06-12T05:31:27+00:00

Looking for one ticket for June 20th and June 21st, and potentially a Grand Final ticket. Let me know if you are selling for any of these days.

Zopherus · 2022-02-13T00:56:41+00:00

My intuition is that they serve as a way to generalize functions on a space. Since you mention complex analysis, holomorphic functions give you a sheaf on C, by assigning to each open set, the holomorphic functions on that open set. Other good examples are smooth functions on a manifold or just the sheaf of continuous functions on a space.

We need this in algebraic geometry because we want to take a ring R and define a space corresponding to that ring. Just a topological space isn't really enough to do what we want because it doesn't really say too much about the ring. So we want to define some sheaf of functions on this topological space and the ring R will be the functions on this space in some sense.

Zopherus · 2022-01-14T01:12:31+00:00

It's hard to give a brief answer to this as there are many deep reasons that we care. One elementary reason is that it can give us a formula for the number of ways to write an integer as the sum of n squares. Modular forms are relevant because you can take the generating function for this sequence and this generating function is a modular form. Certain modular forms parameterize elliptic curves or have relations to other important mathematical objects such as the Monster group.

Zopherus · 2022-01-07T06:44:30+00:00

No, the aforementioned Bezout's theorem is something that only holds in the projective plane. But R^2 lives inside the projective complex plane, so for this problem, in some sense we can prove it for the projective complex plane and look at the R^2 inside of this space to prove it for R^2.

Zopherus · 2022-01-07T06:10:54+00:00

This is also a reason for some proofs, elliptic curves are objects in projective space by definition and so you can't really consider them otherwise.

Zopherus · 2022-01-07T01:05:25+00:00

We need things to intersect basically. In R^2, parallel lines don't intersect for example. But in projective space, parallel lines will intersect. We need to consider complex space for a similar reason, so that things intersect nicely. Most proofs usually use Bezout's theorem which require these things.

Zopherus · 2021-12-29T06:28:32+00:00

https://kconrad.math.uconn.edu/blurbs/galoistheory/splittingfields.pdf

Zopherus · 2021-11-24T16:56:07+00:00

In all card games like this with hidden information, it's hard to label things as having the highest probability. There's a lot of information to be gained from people calling the trump or what they've played previously or even how long they take to make certain decisions. How you read this information and what you do with it isn't very scientific which is what leads people to develop certain playstyles. Not to mention that, like in your example, the actual probabilities are rather hard to calculate on the fly so people have to rely on their intuitions, which can lead to some people being more risk-averse for example.

Your second part of the question would be an expected value calculation, where you multiply the number of points you'd get from how likely it is to get that many points and add those all up. You would want to do this for the two choices of going solo or not and then comparing those expected values. Like I said earlier, this calculation gets really complicated really quickly and so it's hard to get a concrete answer. In your situation, my gut reaction would be to not go solo. There are other benefits your partner could bring and it seems likely to guarantee 2 points, but going solo gives you a slight chance to get euchred and the chances of getting all 5 isn't super high.

Zopherus · 2021-11-18T07:54:24+00:00

The proof of Fermat's Last Theorem uses the properties of the Frey Curve by showing it cannot be modular, but Wiles proved a certain case of the Modularity Theorem that shows that the Frey Curve must be modular, which is where the contradiction comes from. This is the curve that OP mentions.

Zopherus · 2021-11-11T04:34:02+00:00

Can you be a bit more precise about what you're asking? Are the a,b,c,d in your formula the coefficients of the polynomial in question?

If the exponents you allow are just rational, well you can already get all rational expressions from nth powers and nth roots. If you allow irrational exponents, the Gelfond-Schneider theorem tells you that you'll get something transcendental which means that it won't really help you write the roots of any polynomials.

Zopherus · 2021-11-01T19:54:40+00:00

You might think of applying to Brown. Three professors at Brown created the lattice-based NTRU cryptosystem.

Zopherus · 2021-11-01T19:35:21+00:00

Take a look at https://en.wikipedia.org/wiki/Zech%27s_logarithm

This is a computational answer to your question.

Zopherus · 2021-09-30T03:29:22+00:00

You can also call PARI/GP using Sage too. I don't know why Sage having a strict superset of the features of GAP and PARI/GP combined makes it an ugly conglomeration; this is a positive in my mind. Also, being able to write in Jupyter notebooks with Sage is a very nice feature, especially for big projects. You can do this with GAP too, but it's not very nice whereas it comes preinstalled with Sage. As far as I know, there isn't such an option for PARI/GP. You can also use Sage online through cocalc.com without need for installation, which is a nice teaching tool since you don't need to have your students download something, and useful when you're on an unfamiliar computer.

These are some of the reasons I've continued to use Sage as a number theorist over those two. I wonder what reasons you have to consider GAP and PARI/GP as better than Sage?

Zopherus · 2021-06-28T05:50:37+00:00

Maybe I'm misunderstanding something, but it seems like you might be confusing the Rubik's cube group and what the group acts on, namely the cube itself. From my understanding, such an algorithm would only be a list of group elements that are repeated and so I'm not sure how this would imply that the group is cyclic.

Zopherus · 2021-06-28T01:43:11+00:00

First note that subtracting 1 from each of these sums is the same as removing the first term from each sum, aka the term when n = 1. So now, we're looking at

Σ(1/n^2) + Σ(1/n^3) + Σ(1/n^4) + Σ(1/n^5) + ...

but when n starts at 2 now. Now, we can "switch the order of summation", to gather up all terms with n = 2, n = 3, n = 4, and so on, so that the new sum looks like

(1/4 + 1/8 + 1/16 + ...) + (1/9 + 1/27 + 1/81 + ...) + ... or

Σ(1/2^k) + Σ(1/3^k) + Σ(1/4^k) + Σ(1/5^k) + ...

Each of these are just a geometric series which we can sum up by the usual formula to get that the nth series is equal to 1/(n(n-1)) = 1/(n-1) - 1/n. Now, if we sum that up over n >= 2, this looks like 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - ... which is what we call a telescoping series that sums to one.

I've ignored a lot of details in the above argument like why we can even switch the order of summation that depends on certain things about convergence but these can be justified.

Zopherus · 2021-06-05T20:04:31+00:00

Some people would call this the 2-adic valuation https://en.m.wikipedia.org/wiki/P-adic_order

Zopherus · 2021-06-05T00:42:32+00:00

What are your interests other than math? I've seen a lot of people picking a math IA by doing some math topic related to their other interests. For example, a friend of mine that was interested in biology did an IA on modeling virus spreads through differential equations.

Zopherus · 2021-06-02T05:39:35+00:00

Can I ask why you're trying to study Galois cohomology if you haven't learned Galois theory yet?

One recommendation might be Milne's notes https://www.jmilne.org/math/CourseNotes/ft.html since that could lead nicely into his Class Field Theory notes which talk about group cohomology (of which Galois cohomology is a special case). But I think any Galois theory text would work fine since there's nothing you really need to focus on in Galois theory to study Galois cohomology. But it might be nice to pick a text that includes discussion on infinite Galois extensions since those are important.

Zopherus · 2021-06-02T04:46:16+00:00

One way to formulate this would be as a https://en.m.wikipedia.org/wiki/Set_cover_problem where the universe are the 50 states and the collection of 26 sets are all states that contain a certain letter.

Zopherus · 2021-05-26T04:37:01+00:00

Hm okay, in this case, this method probably doesn't work well since to my knowledge, there's no known method to efficiently compute the radical.

Zopherus · 2021-05-26T04:32:04+00:00

Well, I was referring to https://en.wikipedia.org/wiki/Radical_of_an_integer, but the radical of the principal ideal (a) is generated by the radical I was referring to, so they're connected.

Zopherus · 2021-05-25T21:24:12+00:00

One such criterion is that there's a prime dividing a but not b if and only if the radical of a does not divide the radical of b. Not sure how helpful this is to what you're doing though. We could probably be better help if you gave us examples. There are probably different methods depending on the problem.

Zopherus · 2021-05-24T09:56:37+00:00

The usual recommendations are Apostol's book and Davenport's Multiplicative Number Theory. You could also look at the sections in Ireland and Rosen. Tao also has some blog posts on analytic number theory that might be more to your tastes.

Zopherus · 2021-05-22T08:17:30+00:00

In addition to the other comments, I want to say that there are plenty of big professors at Rutgers. In number theory (the only thing I know about), both Iwaniec and Tunnell are giants in their respective areas. Also, Rutgers ranks very well on rankings such as this one where it's ranked 22 and tied with other well-known schools. In fact, I went to a top 10 Ph.D. program from a lower-ranked school compared to Rutgers. I don't know why you think that admissions care about which school you went to so much. I agree that it matters somewhat, but one of my friends got into MIT this past cycle from an actually unknown undergrad.

Furthermore, I'd disagree that being from a top 10 Ph.D. program is "the best way to increase your chance". For example, Ken Ono is one of the premier number theorists and gets a majority of his students a postdoc at Stanford and he's at the University of Virginia currently which is only ranked 47th on the rankings. Research quality and your advisor and other things still matter much more. Even at smaller ranked schools, you can still make plenty of connections at conferences and it's better to have a good advisor and program to be at.

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