User amitesh datta - MathOverflow

What is the $L^p$-norm of the (uncentered) Hardy-Littlewood maximal function?

2011-05-25T08:52:55Z

The (uncentered) Hardy-Littlewood maximal function $M(f)$ of (a locally integrable) function $f$ on $\mathbb{R}^{n}$ is defined by the rule $M(f)(x)=\sup_{\delta>0,\left|y-x\right|<\delta} \text{Avg}_{B(y,\delta)} \left|f\right|$, where $\text{Avg}_{B(y,\delta)} \left|f\right| = \int_{\left|z\right|<\delta} f(y-z) dz$.

The following results regarding the (uncentered) Hardy-Littlewood maximal function are well-known and can be found in many basic texts on analysis (e.g. Loukas Grafakos' "Classical Fourier Analysis", Chapter 2, pages 78-81):

The Hardy-Littlewood maximal function is a bounded operator from $L^1(\mathbb{R}^n)$ to $L^{1,\infty}(\mathbb{R}^n)$ (i.e., weak $L^1$) of norm at most $3^n$ ($n$ is the dimension of the Euclidean space).
Since the Hardy-Littlewood maximal function is also a bounded operator from $L^{\infty}(\mathbb{R}^n)$ to itself with norm at most $1$ (this is clear), we can apply the Marcinkiewicz interpolation theorem to conclude that for all $1 < p < \infty$, the operator norm of the Hardy-Littlewood maximal function is at most $2\left(\frac{p}{p-1}\right)^{\frac{1}{p}}3^{\frac{n}{p}}$. In fact, there is a slightly better bound: $\frac{p}{p-1}3^{\frac{n}{p}}$.
The bound given above grows exponentially with the dimension $n$ (if $p$ is fixed). It is a fact that it cannot be improved to a bound that does not grow exponentially with the dimension $n$.

My questions:

Is an exact value for the norm of the (uncentered) Hardy-Littlewood maximal function, viewed as a bounded operator from $L^p$ to itself ( $1<p<\infty$ ), known? If so, what is it?

Also, what is the norm of the Hardy-Littlewood maximal function when it is viewed as an operator from $L^1$ to weak $L^1$ (if it is known)?

Are the answers to the analogous questions regarding the centered Hardy-Littlewood maximal function known?

I apologize if this question is too basic. It seems like a fairly simple question but it is not clear (at least to me) how to solve it.

A request for suggestions of advanced topics in representation theory

2012-07-06T15:55:12Z

Please Note: The main points of the question below are in bold in order to minimize the time required to read the question.

Let me begin by stating that I understand representation theory is a vast and deep area with many different subfields. Of course, any learning roadmap request for representation theory would necessarily have many different answers or at least one answer with many different suggestions. I would be more interested in "mainstream topics in representation theory"; one could define this as "the set of topics which every serious representation theorist should know" (although even this is subjective and varies from subfield to subfield). Of course, I am happy for people to suggest topics which they feel are not necessarily "mainstream representation theory"; I would be interested in as many suggestions as possible.

I am interested in representation theory both as a branch of mathematics in its own right and as a set of tools and ideas which one may use to study different (either related or a priori unrelated) areas of mathematics (please feel free to interpret this in a broad sense). My background in representation theory is almost all of (and will soon be exactly) the contents of the book entitled Lie Groups by Daniel Bump. The interdisciplinary nature of representation theory dictates that I have reasonable background in other branches of mathematics; I think that I have such a background but feel free to assume as prerequisites any branch of mathematics when giving suggestions.

I am interested in studying representation theory beyond that which is covered in Daniel Bump's Lie Groups. In other words, I am happy for suggestions for topics that a potential representation theorist should know after reading Bump's book (this is the key point). Of course, I am also interested in hearing suggestions for topics that a potential representation theorist should know even if they are virtually disjoint from Bump's book. I am certainly happy for suggestions to take either the form of a textbook, research monograph, research paper, or some other form that I have not thought about.

I am not really interested in suggestions for topics that are already subsumed in Bump's book; I certainly do not object to such suggestions but they would not really be in response to this request. (You can view/download free and legally the table of contents of Bump's book at the following website: http://www.springer.com/mathematics/algebra/book/978-0-387-21154-1.)

Thank you very much for all suggestions!

Learning roadmap for harmonic analysis

2011-06-03T07:57:16Z

In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas Grafakos' "Classical Fourier Analysis" (I have progressed to chapter 3). My intention is to read this book and then proceed to the second volume (by the same author) "Modern Fourier Analysis". I have also studied general analysis at the level of Walter Rudin's "Real and Complex Analysis" (first 15 chapters). In particular, if additional prerequisites are required for recommended references, it would be helpful if you could state them.

My request is to know how one should proceed after reading these two volumes and whether there are additional sources that one could use that are helpful to get a deeper understanding of the subject. Also, it would be nice to hear suggestions of some important topics in the subject of harmonic analysis that are current interests of research and references one could use to better understand these topics.

However, I understand that as one gets deeper into a subject such as harmonic analysis, one would need to understand several related areas in greater depth such as functional analysis, PDE's and several complex variables. Therefore, suggestions of how one can incorporate these subjects into one's learning of harmonic analysis are welcome. (Of course, since this is mainly a request for a roadmap in harmonic analysis, it might be better to keep any recommendations of references in these subjects at least a little related to harmonic analysis.)

In particular, I am interested in various connections between PDE's and harmonic analysis and functional analysis and harmonic analysis. It would be nice to know about references that discuss these connections.

Thank you very much!

Additional Details: Thank you for suggesting Stein's books on harmonic analysis! However, I am not sure how one should read these books. For example, there seems to be overlap between Grafakos and Stein's books but Stein's "Harmonic Analysis" seems very much like a research monograph and although it is, needless to say, an excellent book, I am not very sure what prerequisites one must have to tackle it. In contrast, the other two books by Stein are more elementary but it would be nice to know of the sort of material that can be found in these two books but that cannot be found in Grafakos.

Answer by Amitesh Datta for Solvable PDE Problems

2011-05-25T12:22:14Z

The PDE that I shall suggest is quite common and therefore it is likely that it has already been selected by another student. However, the analysis of this PDE is vast and very interesting.

The motivation is as follows: let $D$ be the unit disk in the plane (i.e., ${x\in \mathbb{R}^2: \left|x\right|\leq 1}$) and let $f$ be a continuous function defined on the boundary of $D$. We wish to find a harmonic function $u$ defined in the interior of $D$ (i.e., ${x\in\mathbb{R}^2:\left|x\right|<1}$) whose boundary values are $f$; i.e., $u$ is a continuous function required to satisfy the Laplace equation $u_{xx}+u_{yy}=0$ and the function $F$ defined on $D$ by the rule $F(x)=u(x)$ if $\left|x\right|<1$ and $F(x)=f(x)$ if $\left|x\right|=1$ is continuous. This is called the Dirichlet problem in the unit disk.

Similarly, let $1\leq p<\infty$ and let $f\in L^p(\mathbb{R})$. We wish to find a harmonic function $u$ defined in the upper half plane such that $u(x,0)=f(x)$ almost everywhere on the real line. This is called the Dirichlet problem in the upper half plane.

There exist approaches to both problems that use general measure theory in a particularly enlightening manner. I will briefly sketch the solutions; if you wish to see a more comprehensive treatment, you may look at Walter Rudin's Real and Complex Analysis (2nd. edition), chapter 11, and Loukas Grafakos' Classical Fourier Analysis, chapter 2, pages 84-87.

Solution to Dirichlet's problem in the unit disk: the general approach is to define $u$ as the Poisson integral of $f$. More precisely, we define $u(re^{i\theta})=\frac{1}{2\pi} \int_{-\pi}^{\pi} P_r(\theta - t)f(t) dt$ for $0\leq r < 1$, where $P_r(t)= \frac{1-r^2}{1-2r\cos(t)+r^2}$ is the Poisson kernel.

Solution to Dirichlet's problem in the upper half plane: the general approach is to first define the Poisson kernel $P_t(x)=c\frac{t}{x^2+t^2}$ (for $t>0$, $x\in\mathbb{R}$, and $c=\frac{1}{\pi}$) and then define $u(x,t)=(P_t * f)(x)$; the convolution of $P_t$ and $f$ on the real line. Since ${P_t}_{t>0}$ is an approximate identity on $\mathbb{R}$, it follows that $u(x,t)$ converges to $f(x)$ in $L^p$ as $t\to 0$. In fact, this convergence is a.e. (the proof is non-trivial; one approach is to use maximal functions) and this implies that we have solved the Dirichlet problem in the upper half plane.

I hope that I have helped and I apologize for the somewhat sketchy proofs! I have certainly noted some non-trivial facts and I recommend you to look at Rudin and Grafakos for the details. Of course, I should add that the solutions that I have presented will be much more meaningful if you are familiar with measure theory and elementary complex analysis.

Modern algebraic geometry vs. classical algebraic geometry

2010-07-13T02:04:00Z

Can anyone offer advice on roughly how much commutative algebra, homological algebra etc. one needs to know to do research in (or to learn) modern algebraic geometry. Would you need to be familiar with something like the contents of Eisenbud's Commutative Algebra: With a View Toward Algebraic Geometry, or is less needed in reality? (I am familiar with more commutative algebra than that which is covered in Atiyah and MacDonald's *Introduction to Commutative Algebra", but less than that which is covered in Eisenbud's textbook.)

Also, is modern algebraic geometry concerned with abstractions such as schemes, sheaves, topological spaces, commutative and noncommutative rings etc., or is it just classical algebraic geometry in an abstract form? Perhaps more specifically, to do research in modern algebraic geometry, do you need to be familiar with classical algebraic geometry, or is it possible to think of algebraic geometry as an "abstract language" and do research based just on this perception?

While I suspect that, as with other branches of mathematics, "abstraction was invented to analyze the concrete", with all the emphasis currently given to the understanding of abstract tools, for someone who is not very familiar with the subject (such as myself), it seems that algebraic geometry is a "mixture" of general topology and abstract algebra. Is this right? If not, succinctly my question is: how great an influence does classical algebraic geometry have on modern algebraic geometry today?

Feit-Thompson Theorem: The Odd Order Paper

2010-07-14T08:50:25Z

For reference, the Feit-Thompson Theorem states that every finite group of odd order is necessarily solvable. Equivalently, the theorem states that there exist no non-abelian finite simple groups of odd order.

I am well aware of the complexity and length of the proof. However, would it be possible to provide a rough outline of the ideas and techniques in the proof? More specifically, the sub-questions of this question are:

Are the techniques in this proof purely group-theoretic or are techniques from other areas of mathematics borrowed? (Such as, for example, other branches of algebra.) In the same vein, how great an influence do the techniques (if any) from number theory and combinatorics have on the proof? (Here "combinatorics" is of course not very specific. I should emphasize that I mean "tools from combinatorics that are pure and solely derived from techniques within the area of combinatorics and that do not require "deep" group theory to derive". Similarly for "number theory".)
What sorts of "character-free" techniques and ideas exist in the proof? Does a character-free proof of this result exist? (Since I suspect the answer to the latter is in the negative, I am primarily interested in an answer to the former.)
What are the underlying "intuitions" behind the proof? That is, how does one come up with such a proof, or at least, certain parts of it? This is a rough question of course; "coming up" with things in mathematics is very difficult to describe. However, since the argument is so long, I suspect some sort of inspiration must have driven the proof.
I have observed in group theory that many arguments naturally divide into "cases" and often the individual cases are easy to tackle and the arguments naturally "flow". Of course, here I speak of arguments whose lengths are no more than a few pages. Does the proof of the Feit-Thompson theorem share the same "structure" as smaller proofs, or is the proof structurally unique?
How often do explicit "elementwise computations" arise in the proof?
Is there any hope that one day someone might discover a considerably shorter proof of the Feit-Thompson Theorem? For example, would the existence of a proof of this theorem less than 50 or so pages be likely? (A proof making strong use of the classification of finite simple groups, or any other non-trivial consequence of the Feit-Thompson Theorem, does not count.) If not, why is it so difficult in group theory to provide more concise arguments?

While I have Gorenstein's excellent book entitled Finite Groups at hand, I did not go far enough (when I was reading it) to actually get into the "real meat" of the discussion of the Feit-Thompson theorem; that is, to actually get a sense of the mathematics used to prove the theorem. Nor do I intend to do so in the near future. (Don't get me wrong, I would be really interested to see this proof, but it seems too much unless you intend to research finite group theory or a related area.)

Thank you very much for any answers. I am aware that some aspects of this question are imprecise; I have tried my best to be as clear as possible in some cases, but there might still be possible sources of ambiguity and I apologize if they are. (If there are, I would appreciate it if you could try to look for the "obvious interpretation".) Also, I have a relatively strong background in finite group theory (but not a "research-level" background in the area) so feel free to use more complex group-theoretic terminology and ideas if necessary, but if possible, try to give an exposition of the proof that is as elementary as possible. Thanks again!

Non-trivial consequences of Baer's theorem and Lucchini's theorem in subnormality theory

2010-09-19T11:04:43Z

There are a couple of beautiful results in finite group theory that look trivial, at least on a first glance, but require non-trivial facts to prove. I am basically interested in whether these results actually have relatively easier proofs to the ones I will outline below. More specifically, I am interested in whether these results may be proven without the aid of the powerful subnormality theory.

The first, and perhaps most beautiful result, is due to Horosevskii:

Let $\sigma\in Aut(G)$ where Aut($G$) denotes the automorphism group of a non-trivial finite group $G$. Then the order of $\sigma$ in Aut($G$) in less than $\left|G\right|$.

The second result, though perhaps more specialized, actually is an important ingredient in the purely group-theoretic proof of Burnside's $p^aq^b$-theorem:

Let $t$ be an involution in a finite group $G$, and assume that $t\not\in {\bf{O}}_2(G)$, where ${\bf{O}}_2(G)$ denotes the 2-core of $G$. Then there exists an element $x\in G$ of odd prime order such that $txt=x^{-1}$.

(I think that this result is due to Matsuyama but please do not take my word for this because I am not completely certain. For definiteness, I will refer to this result as the "result on involutions".)

Now the only proof I know of the second result (on involutions) relies on Baer's theorem in subnormality theory. For reference, Baer's theorem states that:

Let $H$ be a subgroup of a finite group $G$. Then $H\subseteq {\bf{F}}(G)$ if and only if $\left\langle H,H^x \right\rangle$ is nilpotent for all $x\in G$.

(Here, ${\bf{F}}(G)$ denotes the Fitting subgroup of $G$ and $H^x$ denotes the conjugate of $H$ by $x$; that is, $H^x = {x^{-1}hx|h\in H}$.)

The proof (at least the one I know) of the result on involutions uses Baer's theorem, but actually, it really only uses a very special case of Baer's theorem: the subgroup $H$ in the statement of Baer's theorem is chosen to be ${1,t}$ in the proof, where $t$ is the involution quoted in the result. (The proof also uses a very easy fact about dihedral groups.)

This leads me to wonder whether there exists a "Baer-free" proof of the result on involutions. More specifically, my question is:

Question 1: Does there exist a proof of the result on involutions independent of subnormality theory?

It would be interesting if such a proof existed since Baer's theorem relies on the non-trivial Wielandt "zipper lemma" (which I will quote at the end of my question) - and the zipper lemma really looks unrelated to the result on involutions.

Now Horosevskii's theorem relies on another non-trivial consequence of subnormality theory: Lucchini's theorem. For reference, Lucchini's theorem states that:

Let $A$ be a cyclic proper subgroup of a finite group $G$, and let $K=\mbox{core}_G(A)$. Then $\left|A:K\right|<\left|G:A\right|$, and in particular, if $\left|A\right|\geq \left|G:A\right|$, then $K>1$.

What is amazing, at least to me, is that a fact that one might consider fundamental about automorphisms (Horosevskii's theorem) relies on a result (Lucchini's theorem) that at least on a first glance seems very specialized. My second question is nearly identical to my first:

Question 2: Does there exist a proof of Horosevskii's theorem independent of subnormality theory?

(What follows is not really subsumed in my question, but it may be useful for answering my question.)

Succinctly, the proof of Baer's theorem (at least the proof I know) relies on the amazing Wielandt "zipper lemma" which states:

Suppose that $S\subseteq G$, where $G$ is a finite group, and assume that $S$ is subnormal in $H$ for every proper subgroup $H$ of $G$ that contains $S$. If $S$ is not subnormal in $G$, then there is a unique maximal subgroup of $G$ that conains $S$.

Likewise, the proof of Lucchini's theorem relies on Zenkov's theorem on intersections of abelian subgroups, the proof of which in turn relies on Baer's theorem. Zenkov's theorem states:

Let $A$ and $B$ be abelian subgroups of a finite group $G$ and let $M$ be a minimal member of the set ${A\cap B^g|g\in G}$. Then $M\subseteq {\bf{F}}(G)$.

(Note that "$B^g$" denotes the conjugate of $B$ by $g$; that is, $B^g={g^{-1}bg|b\in B}$.)

(Note: I am well aware that there may be other theorems due to Baer, Lucchini and Horosevskii that are referred to as "Baer's theorem", "Lucchini's theorem" or "Horosevskii's theorem". However, I hope that by stating the results above, no confusion arises. For proofs of the theorems quoted, please see the book Finite Group Theory by I. Martin Isaacs. More specifically, please see Theorem 2.9., Chapter 2, Section 2A, page 50 for the statement and proof of the Wielandt "Zipper Lemma", Theorem 2.12., Chapter 2, Section 2B, page 55 for the statement and proof of Baer's theorem, Theorem 2.18., Chapter 2, Section 2D, page 61 for the statement and proof of Zenkov's theorem, and Theorem 2.20., Chapter 2, Section 2D, page 63 for the statement and proof of Lucchini's Theorem.)

Thanks!

Answer by Amitesh Datta for The role of completeness in Hilbert Spaces

2010-09-15T10:49:54Z

We should not forget the famous Riesz-Fischer Theorem:

Let $H$ be a Hilbert space and let ${u_{\alpha}:\alpha\in A}$ be an orthonormal set in $H$. Suppose $\phi$ is in the $l^2$-space of $(A,\mu)$ where $\mu$ is the counting measure on $A$. Then $\phi=\widehat{x}$ for some $x\in H$ where $\widehat{x}:A\to \mathbb{C}$ is defined by $\widehat{x}(\alpha)=(x,u_{\alpha})$, the inner product of $x$ with $u_{\alpha}$, for each $\alpha\in A$.

(I quote Theorem 4.17, page 89, in the second edition of Walter Rudin's Real and Complex Analysis.) In fact, I do not think it would be an exaggeration to say that the Riesz-Fischer Theorem is nothing but a reformulation of the completeness of $H$ - that is how crucial the assumption of completeness is to the proof.

Answer by Amitesh Datta for Undergraduate roadmap to algebraic geometry?

2010-08-11T23:31:41Z

There is an excellent book on algebraic geometry entitled Algebraic Geometry: A First Course by Joe Harris. This book, however, emphasizes the classical roots of the subject but if you have not yet seen too much of algebraic geometry, it is worthwhile getting this book and reading a few lectures. (The book is split into "lectures" rather than "chapters".) There are many beautiful constructions in classical algebraic geometry that can be understood without too much background (and which lay the foundations for some aspects of modern algebraic geometry) and this can perhaps give you a rough indication of the geometric intuitions in algebraic geometry. And in my opinion, the book does an excellent job of conveying the beauty and elegance of algebraic geometry.

The prerequisites for reading this book (according to Harris) are: linear algebra, multilinear algebra and modern algebra. However, since this is a "Graduate Texts in Mathematics" book, there are some places where it is very helpful (but not essential to the point that you cannot read the book otherwise) to have a basic knowledge of commutative algebra, complex analysis and point-set topology. (E.g., basic facts about topological spaces, local rings, basic constructions in commutative algebra, holomorphic functions etc.) Atiyah and Macdonald's an An Introduction to Commutative Algebra should furnish more than enough preparation. (You can also concurrently read commutative algebra if that is your preference.)

Since you are an undergraduate student, you should not worry too much about learning "background material" just yet before at least seeing what classical algebraic geometry is about. If at some point you decide to specialize in the subject, you will need to learn the "modern tools" such as, for example, schemes, sheaves and sheaf cohomology. The "classic book" for this is Robin Hartshorne's Algebraic Geometry but since that does require a solid background in commutative algebra (or at least the mathematical maturity to accept facts without proofs), you might want to try other books. (But this is, I hasten to add, an excellent book if you do have the background to understand it.)

As Bcnrd (on MathOverflow) recommended to me, Qing Liu's Algebraic Geometry and Arithmetic Curves seems to be an excellent book on the subject. Most of the background material in commutative algebra is developed from scratch, and the first six chapters furnish a good introduction to the "modern tools". The last three chapters focus more on the arithmetic side of algebraic geometry, but you can always omit that if you so desire. (But if you are interested in number theory, definitely take a look at that!)

Succinctly, I recommend: Take a look at Atiyah and Macdonald and at least read the first few chapters. (The book is roughly 120 pages so covering the first few chapters is not too hard. Though be warned: Some people say that Atiyah and Macdonald is "dense", but I personally found it a very readable book and I think the majority find that so as well.) Then you should have the right background to read Harris and I hope that that will show you how fascinating the subject of algebraic geometry is. Good luck!

Answer by Amitesh Datta for Finite nonabelian groups of odd order

2010-07-14T04:41:09Z

You might be interested in the result that if n is odd, |G| = n for a finite group G, and if every subgroup of G is normal, then G is abelian. (This does not hold if the hypothesis that n is odd is ommitted as the quaternion group of order 8 demonstrates.)

A group whose every subgroup is normal is called a Dedekind group. A non-abelian Dedekind group is called a Hamiltonian group. With this terminology the result simply states that a Dedekind group of odd order is abelian.

The proof is not immediately obvious. It relies on a classification result that states that every Hamiltonian group is a direct product of the quaternion group of order 8, an elemetary abelian 2-group, and a periodic abelian group of odd order. Once this classification result is established, however, the result can be seen easily.

Is an English translation of Grothendieck's EGA available?

2010-07-13T01:46:09Z

I have always wondered whether there is an English translation of Grothendieck's EGA (Elements de Geometrie Algebrique) available. Does anyone know whether there is and if so where I can find it? If not, are there English texts that cover similar material to the EGA that you would recommend? (My knowledge of French is very rudimentary, and while I can roughly make meaning out of some (non-mathematical) passages, it seems (from what I have heard) that some mastery of French is necessary to leisurely read the EGA.)

Answer by Amitesh Datta for An advanced exposition of Galois theory

2010-07-13T01:18:36Z

I have not actually read this book entirely but Hideyuki Matsumura's Commutative Algebra is a relatively advanced text on the subject. I did read the first few chapters of this book, but having done so I prefer David Eisenbud's Commutative Algebra; in any case, there are certain important concepts which Matsumura discusses towards the end of his book which may be worthwhile to read. (Matsumura does occassionally allude to geometric connections in his book, but Eisenbud alludes to them far more often and in far greater depth.)

All that said, this text due to Matsumura, especially part 2 (the last four chapters) does have some more "advanced field theory" in the context of commutative algebra and algebraic geometry. On the other hand, Matsumura's other book (which I believe was published later) on Commutative Ring Theory also has some more advanced field theory towards the end of the book, and perhaps could be more useful since it is actually designed as a textbook in the subject. (Whereas, I believe, Matsumura's Commutative Algebra was not written with this as the primary goal.)

I should add, however, that it takes time to become accustomed to Matsumura's exposition. He does state several facts without proofs (and some of them are quite fundamental to the rest of the text) but if you are fairly accustomed to homological algebra and commutative algebra in general, you should have little or no difficulty working out the proofs yourself. Also, the prerequisites for both texts is "graduate-level algebra", the fundamentals of homological algebra (i.e., all homological algebra up to, and including, the development of the torsion and extension functors), and familiarity with the exterior algebra. The appendices in Matsumura's Commutative Ring Theory do give a description of the necessary background but are perhaps too condensed if you are not already familiar with the material.

Answer by Amitesh Datta for Does $Aut(Aut(...Aut(G)...))$ stabilize?

2010-07-12T10:06:23Z

For reference, note that the Wielandt Automorphism Tower Theorem states:

Let G be a finite group, and assume that Z(G) = 1. Write G₁ = G, and for i > 1, G_i = Aut(G_{i - 1}). Then up to isomorphism, there are only finitely many different groups among the G_i.

In fact, I quote the above result from Martin Isaacs' Finite Group Theory; more precisely, the result can be found on page 278 in Chapter 9 of this publication. Isaacs discusses this result in the context of subnormality in some depth and proves it as well.

Answer by Amitesh Datta for Undergraduate Level Math Books

2010-07-11T12:28:32Z

I would also recommend the book entitled Analysis on Manifolds by James Munkres. I think that this is a good undergraduate textbook in mathematics for any student wishing to pursue multivariable calculus in greater depth. My only complaint is that Munkres often chooses to include details which can be seen easily after a little bit of thought. Perhaps this can be viewed as an effort to show the student how to "properly do analysis": doing analysis, just like doing any other branch of mathematics, requires you to carefully apply definitions and theorems, and it is important for the student to appreciate this early in his/her mathematical learning.

That said, the book is an excellent text overall for "advanced calculus". The student will need to be familiar with single variable analysis and perhaps some linear algebra. (Even a rudimentary knowledge of linear algebra will do since Munkres develops most of the necessary theory from scratch.)

Roughly speaking, the book splits into two parts. The first part covers most of the results students see when doing multivariable calculus that are stated "without proof" in their texts. For example, "the equality of the mixed partials", "double integrals can be done in any order", "a bounded function is Riemann integral if and only if it is continuous almost everywhere", "the change of variables theorem" etc., are (very) imprecise forms of some of the results Munkres establishes.

In the second part of the book, manifolds and their theory are introduced. Thus, for example, a rudimentary introduction to tensors is given, and this is supplemented by the basic theory of differential forms, the De Rham groups (of the punctured plane), Stokes' theorem etc.

I think that the exposition could be tightened: if you actually pick up the book and really make an effort to read it, it is quite possible to finish the first half of the book in the space of a week (that is, approximately 200 pages in a week) simply because certain topics are explained in more detail (at least in my opinion) than necessary. (One example is Munkres' proof of the linearity, monotonicity, additivity etc. of the Riemann integral. This is proved in three contexts separately: the case of the integral over a rectangle, that over a bounded set, and that of improper integrals, when essentially the proofs can be left as relatively easy exercises in some cases.)

As the above comments suggest, I think that this is an excellent book for undergraduate students, but perhaps less so for graduate students. (Spivak's Calculus on Manifolds is good for both undergraduate and graduate students, in my opinion, but some people may suggest that it is too hard for undergraduates.) And after reading this book, you should have more than enough preparation to read more advanced texts such as William Boothby's An Introduction to Differentiable Manifolds and Riemannian Geometry.

Answer by Amitesh Datta for Approximation with continuous functions

2010-07-11T09:01:03Z

I quote a theorem due to Lusin:

Let $X$ be a locally compact Hausdorff space and let $\mu$ be a regular Borel measure on $X$ such that $\mu(K)<\infty$ for every compact $K\subseteq X$. Suppose $f$ is a complex measurable function on $X$, $\mu(A)<\infty$, $f(x)=0$ if $x\in X\setminus A$, and $\epsilon>0$. Then there exists a continuous complex function $g$ on $X$ with compact support such that

$\mu({x:f(x)\neq g(x)})<\epsilon$.

Furthermore, the function $g$ can be chosen such that

$sup_{x\in X}|g(x)|\leq sup_{x\in X}|f(x)|.

As an immediate corollary (which is more relevant to your question), observe that:

If the hypotheses of Lusin's theorem are satisfied and if $|f|\leq 1$, then there is a sequence ${g_n}$ of continuous complex functions with compact support such that $|g_n|\leq 1$ for all $n$ and

$f(x)=\lim_{n \to \infty}g_n(x)$

almost everywhere with respect to $\mu$.

Note that the proof of Lusin's Theorem requires Urysohn's lemma for locally compact Hausdorff spaces, or at least a variation of it. (A very similar argument to that used to prove Urysohn's lemma for Normal Hausdorff spaces establishes the fact that any locally compact Hausdorff space is completely regular.) For more details on these results and their proofs, see Chapter 2 of the second edition of Walter Rudin's Real and Complex Analysis. (The results can be more precisely located on pages 56 and 57.)

Answer by Amitesh Datta for Undergraduate Level Math Books

2010-07-11T08:23:51Z

I would recommend Walter Rudin's classic text entitled Real and Complex Analysis for the mathematically mature undergraduate student. (Of course, this should really only be read after one has familiarity with most of Rudin's earlier textbook: Principles of Mathematical Analysis.)

The beautiful thing about this book is that nearly every example or result stated by Rudin is used in "the big picture". As you progress through the book, you really start to appreciate the magic Rudin has used to weave together analysis in a manner that is rarely done in other textbooks.

While Rudin states in his preface that the textbook is intended as a first year graduate course on the subject of analysis, I believe that it is quite possible for a mathematically mature undergraduate to follow it. There is no assumption in the book that the reader has any familiarity with linear algebra, abstract algebra, general topology etc. beyond that which was covered in the first seven chapters of his earlier book, but realistically one would like to have at least mastered the basics of these areas before attempting to delve deeper into the analysis covered in this book. (There is a chapter on Banach algebras, but all the necessary abstract algebra here is developed from scratch.)

The textbook is indeed challenging with plenty of exercises, but it is not something with which a student with the correct prerequisites should have tremendous difficulty. Furthermore, if a student can successfully read most of the book, he is well-equipped to go deeper into most branches of modern analysis, and should find the other classic texts in the subject, for instance Royden's Real Analysis, Bartle's The Elements of Integration and Lebesgue Measure etc., very easy to follow.

The Amazon page for this book can be found here.

Answer by Amitesh Datta for Why are p-elementary groups so crucial in finite group theory?

2010-07-07T02:20:48Z

Perhaps this is not a direct answer to your question, but one of my favorite examples of $p$-groups entering into the theory of general finite groups (except of course for Sylow theory itself!) occurs in a theorem due to Frobenius. This theorem states:

Let $P$ be a Sylow $p$-subgroup of a finite group $G$. Then the following are equivalent:

(i) $G$ has a normal $p$-complement.

(ii) $N_G(U)$ has a normal $p$-complement for all $p$-subgroups $U\subseteq G$ with $U>1$.

(iii) $N_G(U)/C_G(U)$ is a $p$-group for all $p$-subgroups $U\subseteq G$.

(iv) There is no fusion in $P$.

(Here $C_G(U)$ and $N_G(U)$ denote the centralizer and normalizer of the subgroup $U\subseteq G$, respectively. The quotient $N_G(U)/C_G(U)$ makes sense, of course, since $C_G(U)$ is normal in $N_G(U)$. Also, to say that there is no fusion in $P$ is to say that any pair of elements in $P$ that are $G$-conjugate are already $P$-conjugate.)

A remarkable theorem of Thompson asserts that, if $p\neq 2$ in Frobenius' theorem, to verify that $G$ has a normal $p$-complement is tantamount to checking that $N_G(U)$ has a normal $p$-complement for just two particular $p$-subgroups $U\subseteq G$. The two subgroups $U\subseteq G$ whose normalizers need to be checked are $Z(P)$ and $J(P)$ where $P$ is a fixed Sylow $p$-subgroup of $G$ and $Z(P)$ and $J(P)$ are the center and Thompson subgroup of $P$, respectively.

Answer by Amitesh Datta for Examples of undergraduate mathematics separation from what mathematicians should know

2010-07-06T08:59:03Z

If you wish to become a solid mathematician, there are certain topics with which you will want to have familiarity, even if you do not intend to delve deeply into them. For instance, most research mathematicians will have had a decent training in point-set topology and will have learnt at least a few of the major theorems and techniques in the area (Urysohn's lemma, Tychonoff's theorem, Urysohn's metrization theorem, partitions of unity etc.).

Ultimately, the extent to which you remember these results and techniques does not determine your quality as a research mathematician. However, open sets and their theory are ubiquituous in nearly every branch of pure mathematics, and having a feel for certain topological concepts is certainly desirable. (Perhaps not essential, however, depending on which branch of mathematics you pursue.)

Again, this is not to say that someone can be dismissed for not knowing point-set topology: there are plenty of ways one can do meaningful research without having a training in point-set topology on the magnitude of Munkres' Topology: A First Course or Kelley's General Topology, for example, and there are at least a few professional mathematicians who demonstrate this.

As for Sylow theory, the topic, of course, falls into the area of finite group theory. (I hasten to add, however, that Sylow theory does have applications to the theory of locally finite (but possibly infinite) groups.) While finite group theory is an exciting subject full of rich structure and powerful theorems, I suspect that there are not too many branches of pure mathematics where the methods of this theory are instrumental to doing meaningful research. One notable exception (to some extent) would be algebraic number theory. For instance, the principal ideal theorem of class field theory can be proven using the techniques of transfer (in group theory). Also, it would be fair to add that most algebraists have a solid training in finite group theory, even if their research interests lie in other aspects of algebra.

Succinctly, I think that it is fair to say that being comfortable with the various techniques of group theory and topology, whether or not you pursue either of these subjects, can be helpful in many areas of mathematics. The subject matter in its exact form may not repeat itself in other areas, but the techniques, ideas and intuitions may do so.

Comment by Amitesh Datta

2012-07-07T06:01:17Z

Thank you very much! I will take a look at this reference.

Comment by Amitesh Datta

2012-07-07T05:59:09Z

@Emerton Dear Matthew, Thank you very much for these suggestions; I really appreciate it. Could you please suggest a reference for the geometric perspective? If it is not too general a question to ask (but it probably is), then I would also be interested in the possible steps that one could take to get toward research in the representation theory of Lie groups and Lie algebras and related areas (say, after Knapp's book); more precisely, are there other important/essential topics that one should learn before/while reading research papers? Thank you very much and best regards,

Comment by Amitesh Datta

2012-07-06T16:19:51Z

@JimHumphreys Thank you very much for pointing this out; I have converted the question to community-wiki.

Comment by Amitesh Datta

2012-07-06T16:16:30Z

@JustinCampbell I think that the coverage of Bump's book is broader than simply the representation theory of $\mathfrak{sl}_{2}$. Of course, chapter 12 is entitled "Representations of $\mathfrak{sl}_{2}(\mathbb{C})$" but there are 50 chapters in Bump's book and many of them discuss a wide range of different topics. For example, general semisimple compact Lie groups are studied in chapter 23. Also, standard topics such as root systems, heighest-weight theory, the Iwasawa and Bruhat decompositions, symmetric spaces etc. are discussed in chapters 1 - 33 (i.e., the first two parts of the book).

Comment by Amitesh Datta

2011-06-04T12:12:23Z

@Andrew L I definitely agree that Switzer is not an introductory text. (The author states so himself in the beginning.) I believe it is intended for people who have at least had a standard first course in algebraic topology. On the other hand, if one does have the prerequisites, it is an excellent second or third book.

Comment by Amitesh Datta

2011-06-03T11:10:34Z

Dear Spencer, thanks!

Comment by Amitesh Datta

2011-06-03T11:09:08Z

Dear Alain, thanks!

Comment by Amitesh Datta

2011-06-03T10:59:24Z

Dear Peter, thanks! I was just wondering, in case you knew, do these notes cover the entire course or do they only cover a part of the course? I was thinking of taking this course at some point in the future and it would be helpful to know what is covered; unfortunately, I can't seem to find anything on the ANU maths webpage about it.

Comment by Amitesh Datta

2011-06-03T09:51:54Z

Dear Peter, the link to the ANU course notes appears to be broken.

Comment by Amitesh Datta

2011-06-03T08:01:18Z

math.stackexchange.com is a more appropriate place to ask this question. Please see the faq on what sorts of questions one should ask here and which sites are more suitable for questions that are not appropriate for this site.

Comment by Amitesh Datta

2011-06-03T07:32:38Z

I recently borrowed this book from a library. It looks like an excellent addition to the literature; however, as Andrea says, it covers similar ground to Atiyah and Macdonald. The author himself recommends one to read Eisenbud or Matsumura afterwards if one wishes to further his knowledge in commutative algebra.

Comment by Amitesh Datta

2011-06-03T07:29:12Z

Atiyah and Macdonald is an excellent book. However, the main problem (and it speaks volumes of the text that this is the main problem) is that there is no second volume. In particular, there are many important topics in commutative algebra that one needs (in Hartshorne, for example) that are not covered in Atiyah and Macdonald. Of course, Eisenbud is another option but it is far thicker than Matsumura; each of these books is great and it really is a matter of personal taste which book is preferred.

Comment by Amitesh Datta

2011-06-03T07:18:36Z

I hasten to emphasize, however, that there are difficult exercises in Hartshorne; but it certainly is not true that *most" (or even a reasonable portion) of the exercises are "hard". Furthermore, Hartshorne does explain some proofs (at least in the part that I have read thus far) in at least as much detail as Liu. The moral is (at least in my view) that one should read Hartshorne and, if one is stuck, one should refer to other texts (on algebraic geometry). I have heard that EGA is a good supplementary source.

Comment by Amitesh Datta

2011-06-03T07:09:30Z

I have to disagree that Hartshorne "is as difficult as people say". I am not saying it is not difficult; however, I have seen people saying on the internet that "it takes hours to understand some exercises in Hartshorne". After reading comments such as this, I decided not to learn algebraic geometry from Hartshorne. (I used Liu.) However, a few weeks ago I picked up Hartshorne, interested to see why it is considered difficult, and I have to say that it is not as difficult as some people say. There are hard exercises but it would not be an exaggeration to say that most are do-able ...

Comment by Amitesh Datta

2011-05-25T11:50:17Z

It would help if you could note some of the PDE problems that have already been chosen by other students (if possible). However, I would imagine that if you added some useful insight of your own regarding how to solve a given PDE, then it might not matter so much if another student has already selected it. For example, there can be several ways in which you can motivate the selection of a given solution to a fixed PDE.