<p>What do you think is the most challenging math class for undergraduate math majors?</p>
<p>Advanced Calculus, which would be the more advanced option of Analysis. From just the schools I have surveyed, there is an Analysis course at the junior level that can prepare you for the senior-level Advanced Calculus course OR you can just go directly to Advanced Calculus.</p>
<p>For me, it was extremely hard because it was a pure theory course and I was a Computational Math major…so no computer assignments bailing me out (approach was B+ on exams and A+ on computer assignments). I had to learn the proofs and there was no way around it.</p>
<p>Advanced Calculus at my school was simply a mix of Linear Algebra and Differential Equations, and the text book we used was titles “Engineering Mathematics”</p>
<p>For me it was Abstract Algebra. To me it seemed to involve a whole different way of thinking. I could feel my brain rewiring itself as I pushed on through it. </p>
<p>I also found Algebraic Topology (homology theory) difficult to get my brain productively wrapped around.</p>
<p>In contrast, I found Advanced Calculus and Real Analysis straightforward, Complex Analysis absolutely gorgeous, and Topology (point-set and homotopy theory) an absolute blast…</p>
<p>I have seen Advanced Calculus refer to freshman/sophomore level multivariable calculus or junior/senior level Real Analysis.</p>
<p>Abstract Algebra is a class that many students struggle with. Analysis, while painstakingly rigorous, isn’t too bad because you can rely on your calculus intuition. Most students hit abstract algebra without much intuition for algebraic structures though.</p>
<p>Analysis and Algebra are the two “hard” classes that are required for a math major virtually everywhere. There are electives that are much harder than that, but being electives you can choose not to take them.</p>
<p>
Are we the same person?</p>
<p>“I have seen Advanced Calculus refer to freshman/sophomore level multivariable calculus or junior/senior level Real Analysis.”</p>
<p>I have seen that also. At my school (Michigan State), we had a senior-level Advanced Calculus course and a senior-level Real Analysis sequence. The Advanced Calculus course was the required one (thank goodness).</p>
<p>Is geometry a difficult elective? (Well at my school this is an elective, I don’t know if it is mandatory at other schools or not.)</p>
<p>What sort of geometry? </p>
<p>Differential geometry/curves and surfaces: feels like a natural extension of multivariable calculus</p>
<p>Differential geometry/abstract Riemannian geometry: hard if it’s your first exposure to abstract manifolds; if you have studied topology and manifolds before, you know what you are getting yourself into</p>
<p>Algebraic geometry: I hear this one is <em>really</em> hard but I have no experience myself</p>
<p>Other geometry topics: analytic geometry, projective geometry, spherical and hyperbolic geometry, combinatorial geometry, etc. These are less standard topics in an undergraduate or graduate curriculum but sometimes taught as electives. They can be a lot of fun unless your professor is sadistic and likes to torture helpless students ;)</p>
<p>I take my hat off to anyone who likes geometry.</p>
<p>I guess it would be differential geometry of curves and surfaces because the only prerquisite is multivariable calculus. I want to find out because I am scheduled for it next semester, but the course description only says foundations of geometry/euclidean and non-euclidean geometry.</p>
<p>Computational Geometry seems to be a course that I see in more job requirements for CS major jobs.</p>
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You will probably develop high school geometry axiomatically and explore what happens when you work with different sets of axioms. Take a look at this book: [Amazon.com:</a> The Foundations of Geometry and the Non-Euclidean Plane eBook: G.E. Martin: Kindle Store](<a href=“http://www.amazon.com/gp/product/B000QTEB4Q/ref=pd_lpo_k2_dp_sr_2?pf_rd_p=486539851&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0716724464&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=0GWDSKHCPMH874YFE560]Amazon.com:”>http://www.amazon.com/gp/product/B000QTEB4Q/ref=pd_lpo_k2_dp_sr_2?pf_rd_p=486539851&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0716724464&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=0GWDSKHCPMH874YFE560)</p>
<p>BCEagle, that’s interesting. I am surprised that it’s such a common course. When I first encountered it, it seemed like such an exotic subject to me. It was so different in philosophy from the other areas of CS: people were spending a lot of time on curious problems, although their solution did not have any obvious practical applications. Not that there aren’t plenty of practical applications; but computer scientists usually tend to look down on the mathematicians who spend all their time on problems of their own fancy w/o regards of applications.</p>
<p>There is no such thing as a universal most challenging math course. Algebra and analysis are usually the courses used to weed out the students at first, but after that, it depends on the student’s abilities and interests, and it can get arbitrarily hard. Those who like algebra might not be so fond of analysis and vice-versa. I’ve never had good intuition for geometry personally. And the pure math students usually have a hard time with even the statistics classes (even the theoretical courses, the applied stuff they wouldn’t touch with a ten foot pole).</p>
<p>Has anyone here taken set theory?</p>
<p>That’s not good on one’s mental health. There’s a reason Cantor went insane. All joking aside, set theory is not usually a course made easy.</p>
<p>Thank you for the responses. Could anyone suggest a way to prepare for real analysis and abstract algebra? I am trying to do some work in proofs now because I have to take an intro to proofs/group theory class and real analysis next year and i do not want to just go in without knowing anything.</p>
<p>Preparing For Real Analysis:</p>
<p>-Take the calculus sequence and know your calculus very well.</p>
<p>Preparing For Abstract Algebra:</p>
<p>-If you haven’t taken Linear Algebra yet, take a <em>proof based</em> linear algebra course.</p>
<p>For Both:</p>
<p>-Some schools have classes called “Transition to Upper Level Math” or “Introduction to Proof Writing.” If these are offered at your school, take them. If not, there are still other things you can do, like participating in problem solving groups if they are offered.</p>
<p>As for “hardest classes,” most people say real analysis (as well as abstract algebra) are the math department’s answer to organic chemistry. In all reality though, there are still harder classes.</p>
<p>Algebraic Geometry is just plain hard. I hate all kinds of geometry though lol. If you take a class in Hilbert Space and their operators, those can be hard too.</p>
<p>calculus is easy, number theory is hard.</p>
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<p>Real Analysis can be a pain, good background in Linear algebra is great but not required.
Also it will always depend on the Professor, he can make your life easy or really miserable if he really wants.</p>