User:GregariousMadness

User Info

This user does not understand mean people. Please be nice.

ZThis user is a member of
Generation Z.

This user is a member of the LGBTQ+ community.

gradThis user is a graduate school student.

This user is neurodivergent.

This user is cisgender and supports the rights of nonbinary and transgender people.

^_^This user watches anime.

This user is a member of
WikiProject Mathematics.

This user has been on Wikipedia for 4 years, 8 months and 27 days.

This user assumes good faith.

This user loves mathematics

Welcome to my user page! I'm GregariousMadness, nice to meet you! ๑(◕‿◕)๑

About me

I'm a graduate student at Duke University focusing on deep learning and mathematical foundations of artificial intelligence. My research interests lie at the intersection of topology, differential geometry, and neural network architecture design. I also love competition math, having done it since middle school, and am currently trying to make the article Olympiad mathematics.

I also like going through the list of requested Mathematics articles at Wikipedia:Requested_articles/Mathematics and making new articles when I can. I've created Taniyama's problems, Thurston's 24 questions, Darboux cyclide, Yang-Baxter operator, Ultradistribution, Brokard's theorem (projective geometry), and Kobayashi's theorem thanks to that list.

Wikipedia experience

I'm still a new editor, but I try my best! I'm a gregarious person to a fault, and that may leak into my edits at times. Please be patient with me, and I'll be patient with you!

Articles I created

• 15.ai
• Taniyama's problems
• Darboux cyclide
• Thurston's 24 questions
• Yang-Baxter operator
• Ultradistribution
• Kobayashi's theorem
• Chicago movement
• Brokard's theorem (projective geometry)
• Draft:DeepLearning.AI (submitted to AfC)
• Draft:Erdős's problems

Articles I'm helping clean up

• Autoencoders
• Audio deepfake
• Liouville's theorem

My favorite math proofs

Proof of the infinitude of the primes using topology

Consider the set $N_{a,b}=\{a+nb;n\in \mathbb {Z} \}$ for $a,b\in \mathbb {Z}$ and $b>0$. Each set $N_{a,b}$ is a two-way infinite arithmetic progression. Now call a set $O\subseteq \mathbb {Z}$ open if either O is empty, or if to every $a\in O$ there exists some $b>0$ with $N_{a,b}\subseteq O$. Clearly, the union of open sets is open again. If $O_{1},O_{2}$ are open, and $a\in O_{1}\cap O_{2}$ with $N_{a,b_{1}}\subseteq O_{1}$ and $N_{a,b_{2}}\subseteq O_{2}$, then $a\in N_{a,b_{1}b_{2}}\subseteq O_{1}\cap O_{2}$. So, any finite intersection of open sets is again open. Thus, this family of open sets induces a topology on $\mathbb {Z}$. Any number $n\neq 1,-1$ has a prime divisor p, and so is contained in $N_{0,p}$ Thus $\mathbb {Z} \backslash \{1,-1\}=\bigcup _{p\in \mathbb {P} }N_{0,p}$.

Suppose for the sake of contradiction that $\mathbb {P}$ is finite; then $\bigcup _{p\in \mathbb {P} }N_{0,p}$ would be a finite union of closed sets, and hence closed. Thus, $\{1,-1\}$ would be an open set, a contradiction since any nonempty open set is infinite.

Proof of the fundamental theorem of algebra using Liouville's theorem

Suppose for the sake of contradiction that there is a nonconstant polynomial $p$ with no complex root. Note that $|p(z)|\to \infty$ as $z\to \infty$. Take a sufficiently large ball $B(0,R)$; for some constant $M$ there exists a sufficiently large $R$ such that $1/|p(z)|<1$ for all $z\not \in B(0,R)$.

Because $p$ has no roots, the function $q(z)=1/p(z)$ is entire and holomorphic inside $B(0,R)$, and thus it is also continuous on its closure ${\overline {B}}(0,R)$. By the extreme value theorem, a continuous function on a closed and bounded set obtains its extreme values, implying that $1/|p(z)|\leq C$ for some constant $C$ and $z\in {\overline {B}}(0,R)$.

Thus, the function $q(z)$ is bounded in $\mathbb {C}$, and by Liouville's theorem, is constant, which contradicts our assumption that $p$ is nonconstant.