About me
My name is Alex Youcis. I am currently a senior a first year graduate student at the University of California, Berkeley.
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About me
My name is Alex Youcis. I am currently a senior a first year graduate student at the University of California, Berkeley.
About Me
Blogroll
 Absolutely Useless
 Abstract Algebra
 Annoying Precision
 Aquazorcarson's Blog
 Bruno's Math Blog
 Chris' Math Blog
 Climbing Mount Bourbaki
 E. Kowalski's Blog
 Geometric Group Theory
 Geometry and the Imagination
 Gower's Weblog
 Hard Arithmetic
 HardyRamanujan Letters
 NgÃ´ Quá»‘c Anh's Blog
 Project Crazy Project
 Rigorous Trivialities
 Secret Blogging Seminar
 SymOmega
 TCS Math
 Unapologetic Mathematician
 What's New
Lecture Notes
Categories
Top Posts
 Munkres Chapter 2 Section 17
 Munkres Chapter two Section 12 & 13: Topological Spaces and Bases
 Munkres Chapter 2 Section 18
 Munkres Chapter 2 Section 1
 Extending Uniformly Continuous Functions
 Munkres Chapter 2 Section 2
 Munkres Chapter 2 Section 19 (Part I)
 Review of Group Theory: The Fourth Isomorphism Theorem (The Correspondence Theorem)
 Schur's Lemma (First and Second Forms)
 The Dimension of R over Q

Algebra Algebraic Combinatorics Algebraic Topology Analysis Answers Category Theory Chapter 2 Characters Character Theory Class Functions Compactness Complex Analysis Connectedness Derivative Differential Geometry Differential Topology DIrect Limit Direct Limits Examples Exterior Algebra Field Theory Finite Dimensional Vector Spaces Full Solutions Functor Galois Theory Geometry Group Group Actions Group Algebra Groups Group Theory Halmos Homological Algebra Homomorphism Homotopy Ideals Induced Character Induced Representation Intuition Inverse Limit Irreducible Characters Irreps Isomorphism Linear Algebra Linear Transformations Manifolds Matrices Module Modules Module Theory Motivation Multivariable Analysis Munkres Normal Subgroups Number Theory Permutations PID Polynomials Prime Ideals Products Product Topology Projections Representation Theory Review of Group Theory Riemann Surfaces Ring Rings Ring Theory Rudin Solutions Sylow Theorems Symmetric Group Tensor Product Topology Total Derivative
Alex,
I am impressed!
Comment by Scholar Pillion  April 19, 2010 
Alex,
I am impressed! 2 times!
Comment by Scholar Pillion  April 19, 2010 
Hey Drexel,
it looks like MHF has been undertaking some heavy maintenance for the last few days. But I just looked up MHF on google and it seems google has detected suspect activity on the website recently, like lots of malware being uploaded onto it. Do you have any idea what is going on, and do you actually get the same message as me ? (Math Help Forum in google). It might just be a generic message from my school’s security policy.
Nice blog though ðŸ™‚
Bacterius.
Comment by Bacterius  June 14, 2010 
Hi Bacterius,
Yeah, I get a similar message saying it’s under maintenance. I’m not entirely sure what it means, but I hope it will be back up soon! It would be horrible if it was gone forever :S
Comment by drexel28  June 15, 2010 
Hi Drexel,
bad news. Here is the message broadcasted by MHF :
“Site currently under maintenance – Security check underway – please check back later…thank you Update 6/14/2010… Somehow a site vulnerability was taken advantage of and some malicious code was injected to the site. We are working on rectifying the issue and hope to have the site online in 1 or 2 days.”
I hope it will be fine …
Comment by Bacterius  June 15, 2010 
Yeah, that’s what I get too. I hope also that it will be better. It’s my favorite forum BY FAR. I can’t even post on artofproblemsolving.com …it just doesn’t feel the same haha
Comment by drexel28  June 15, 2010 
Yeah I know, MHF is just … like … simple, clean and at the end of the day we don’t like changing habits anyway. I felt really bored this weekend due to the lack of math problems to crunch xD. Hopefully it should be back up and running in a couple of days … ðŸ™‚
Comment by Bacterius  June 15, 2010 
Hi Alex,
I chanced upon your blog since I am also just beginning to work on Munkres (although I will already be a junior next year). Since you don’t seem to be too far ahead of me yet, I thought maybe we could occasionally discuss. Let me know.
In fact, I too am just beginning to blog about mathematics; glad to see someone else doing the same. One quick suggestion I have: use the more feature in your posts so that they don’t blow up the front page.
Teddy
Comment by Teddy Ni  June 28, 2010 
Cool.. I have some topology questions. Maybe you undertsnad them cuz i don’t
Comment by Carmen Browne  October 12, 2010 
im a math major too. talk again soon
Comment by Carmen Browne  October 12, 2010 
Hi Alex,
I’ve really enjoyed reading your blog for leisure over the past couple of months. It seems like there is infinite potential for interaction between group theory and linear algebra.
I’m an undergrad, and I’ve been asked to give a small talk at an undergraduate math seminar. I’ve been trying to come up with simple examples which have interesting applications
I have to keep in mind it must be accessible to an undergraduate audience, so I’ve been thinking of studying some finite group structure in different lights, e.g. as the permutation group, the dihedral group, general linear group, and perhaps as a graph, just to illustrate how versatile this underlying idea is. Obviously, I’ve been reading up online, but I was wondering if you could suggest any particular things you might’ve stumbled upon?
Feel free to email me if you have any ideas.
Regards,
Michael
I’ve of course been looking around online
Comment by Michael  January 19, 2011 
Hello Michael!
Maybe this is just an odd coincidence, but I just started blogging about Representation Theory of Finite Groups which details precisely a beautiful and fruitful interplay between group theory and linear algebra. More specifically, it turns out that we can study finite groups (representation theory becomes much, much more advanced when dealing with nonfinite, noncompact groups. The representation theory of Lie algebras is particularly interesting! It’s something I might pursue on here) via linear algebra by associating group elements of some group G with unitary endomorphisms on some finite dimensional complex inner product space V. It’s amazing how much we can really find out using this methodology (it’s analogous to the association of homotopy and homology groups to topological spaces, if this is familiar to you).
As for the ideas for an undergraduate seminar…hmm. This sounds oddly like something I discussed with a friend of mine a while ago. Consider the permutation group on nletters. We can think of $S_n$ as being isomorphic to a subgroup of by associating to each permutation a permutation matrix (see here https://drexel28.wordpress.com/2010/11/06/permutationsptiipermutationmatrices/) [FYI, that is an example of a representation as I mentioned in the previous paragraph]. It’s interesting how you can then study the permutations themselves via their permutation matrix representation. For example, it turns out that the sign of a permutation is just the determinant of the associated permutation matrix. You can take this one step further I feel, and possibly associate to each permutation matrix a graph for which matrix is that graph’s adjacency matrix. There seems to be something cool going on there. Does this kind of hit the spot? If not, let me know. I’d love to give you a great idea to show people how seemingly disparate areas of mathematics are really very interconnected.
Now, on a nonmath note. I really, really appreciate you reading my blog. It’s nice to know that I’m not just talking into the void.
You’re an undegraduate, huh? Where at? I assume you’re a math major? Do you want to be a mathematician, etc.?
Sorry for the long response. If you’d rather contact me via email my email address is afy25@drexel.edu.
Best,
Alex
Comment by drexel28  January 19, 2011 
whoops, dunno how that last line got there
Comment by Michael  January 19, 2011 
lol you proved all of munkres? you are a god damn patriot brother
Comment by whokebe  February 28, 2011 
God Bless America Son. Someone’s got to do work.
Comment by drexel28  February 28, 2011 
[…] as my favorites and first to be my favorite consecutively for two months. Abstract Nonsense is Alex Youcis‘s blog on theoretical mathematics in which he proves (..that he’s theoretical..) one […]
Pingback by Blog of the Month October 2011  MY DIGITAL NOTEBOOK  October 6, 2011 
I really like U’re blog!!!
It’s very useful for me.
Comment by yoona  October 14, 2011 
Hey, do you know what is going on with mathhelpforum?
Comment by ymar  October 28, 2011 
Yeah, I have no idea what’s up with it.
Comment by Alex Youcis  October 28, 2011 
Hi Alex,
I been looking at your blog and recognizing many of the kinds of math I’m interested in. If you’re ever interested in discussing some of your subjects sometime you can reach me by email at librarylance@gmail.com
Comment by librarylance  December 30, 2011 
Well, what kind of math are you interested in Lance(?)!
Best,
Alex
Comment by Alex Youcis  December 30, 2011 
I sometimes think the most honest answer might be “all kinds of math”, but the best answer comes from a major independent study based on the observation that much of the deepest results of math ultimately arose because of problems in physics, a phenomena that became even more glaring in the late twentieth century. The project was a failure is it turns out that the physicists in fact do not have a clear enough picture of physics to let you guess what math they need or will need, but I’ve been returning to things like Abstract Algebra, Differential Geometry, Representation Theory, and so forth in a quixotic quest to understand exactly what are the structures that seem to control reality. I know this is a bit of a vague answer, but hopefully it’s enough to let you figure out your next question.
Comment by librarylance  December 30, 2011 
Ah, while I agree that is a noble, and deep pursuit, I feel as though I cannot say much on it. Much to the chagrin of my father (a physics enthusiast) I know very, very little of physics–basic mechanics, at best, and so while I ‘know’ that some deep results of math have come out of physics, I have no idea what that physics is. I assume you mean things like quantum groups, and hopf algebras?
Comment by Alex Youcis  December 31, 2011 
Certainly those are some of the recent examples, but when you get down to it a lot of things from the beginning on are physics inspired: geometry, calculus and what have you. In the twentieth century differential geometry and functional analysis have been heavily inspired by modern physical developments (though ironically the mathematicians barely anticipated the physicists in going after functional analysis)
Comment by librarylance  December 31, 2011 
Hello
Google just sent me here: Your blog looks great! I share your enjoyment wrt simple proofs, can’t wait to have a poke around the site. I’m an undergrad physicist doing a lot of theory courses I have a feeling I’ll be back here very often.
All the best
Comment by christine  March 30, 2012 
Dear Christine,
Thank you very much for the compliments! I hope that my blog will be as useful to you as it first appears. I wish you the best of luck with your physics adventures (something I know very little about–very basic mechanics at best). As always, feel free to shoot me an email or just comment on here if you have any questions about anything!
Best,
Alex
Comment by Alex Youcis  April 1, 2012 
hello your blog is nice for munkres book. i am self studying topology with plain calculus ode some discrete math for CS background.
what courses u did before topology can u tell me , cause some set theory proofs are obvious but difficult to write in clear expressions for me in chapter1 munkres
Comment by mehdi  May 8, 2012 
Hey man, topology was probably one of the first courses I took. The only thing really before that was analysis via Rudin.
Have you read Munkres set theory part? It’s pretty good, and self contained.
Best,
Alex
Comment by Alex Youcis  May 9, 2012 