First meeting of SMAG

Posted on 2013-05-14 by

Today was the first meeting of the Seminar on Modern Algebraic Geometry. I started the term off by defining the basic objects of (classical) algebraic geometry and building up to the affine ideal-variety correspondence. Notes are available on the SMAG website in case you missed it, and I will try to continue typesetting and posting everyone’s talks. The next talk is this coming Thursday, May 16 in the same place (MC 5136B) at 5:30pm and the speaker will be Yossef.

Thanks to everyone who came out! We really hope that after a couple of weeks of talks from the coordinators, you’ll help us out by volunteering to talk every once in a while. It’s not very hard to do, and it’s great experience. Let’s keep this thing alive!

If you have any feedback about the first meeting, or things you think we should do differently, feel free to leave a comment. :)

Posted in algebraic geometry | Leave a comment

Algebraic geometry seminar

Posted on 2013-05-07 by

EDIT: Please click “AG Seminar” above for information.

This term I’m doing research, taking a course, auditing a course, and helping coordinate another. In this post I’ll discuss the last point.

This was mentioned a while ago on the PMC group, but for anyone who’s not aware, I’m part of a little group of students planning to coordinate a kind of “seminar” on algebraic geometry this term. As a disclaimer, please note that what follows is merely intended to represent my own opinions and feelings, and none of this has really been discussed at length with others in the group. I’m certainly open to ideas or criticisms from others.

We can call this [seminar/interest group/whatever it is] “PMATH 899″ for reference purposes even though it’s not a real course. I choose the number 899 not because I think it’ll be comparable to actual 800-level PMATH courses, but rather to suggest that one of its purposes is to serve as a prelude to PMATH 900, which is being offered this coming Fall.

We don’t exactly have a detailed outline of what we want to do, but we’re trying to encourage as many interested people to participate as possible. We will strive to make the material comprehensible even for those who haven’t taken PMATH 764. Indeed, PMATH 764 spends a lot of time on topics which are rather orthogonal to what we plan to discuss — topics such as intersection theory, for example. Hence, it seems like the plan is to spend the first couple of meetings mostly getting familiar with the machinery of algebraic varieties (affine and projective), coordinate rings, function fields, and so on. There’s no need to grind through any difficult theorems relevant to this material; we can simply take on faith any facts we need. If you’re taking 764 this term, it will probably end up filling in those details anyway. Even if you’re not, there’s not much need to worry since the technicalities of those proofs won’t be of much importance in the context of 899.

The tentative plan is to book a room, and meet Wednesdays 4:30 — 6:30pm, starting next week, so that the first meeting is on the 15th. There will be some amount of material presented, but ample time will be left for discussion amongst the group. The speaker will probably change each week depending on interest and availability. Of course, I intend to contribute as much as I can (or as much as I need to — the \min of the two, haha) in this regard. If it’s more convenient, we could try meeting twice a week.

Here’s a general idea of what kind of material the first few meetings might contain:

  • algebraic varieties, coordinate rings, function fields… (basic machinery from 764)
  • categories, functors, natural transformations… (will be as brief as possible)
  • presheaves, sheaves, morphisms of sheaves…
  • spectra of commutative rings, locally ringed spaces, affine schemes… (working towards the definition of a “scheme”)

If anything above sounds intimidating and you feel like everyone else is already going to know this stuff and breeze through it, I assure you that’s far from the truth. In my case, at least, I’ll have to work diligently to grasp these new concepts, especially since I’ll be one of the people speaking during the first few meetings.

Please get in touch if you’re interested in participating. I think we’ll have no problem making this a success with all the inspiring people around. And of course, feel free to share it and stuff, to spread the word. Further details about the meeting location will be posted on this blog (probably later this week), and likely scrawled on the blackboard of the PMC as well for good measure.

Posted in algebraic geometry | 2 Comments

Connectedness

Posted on 2013-05-06 by

“Good mathematicians see analogies. Great mathematicians see analogies between analogies.” – Stefan Banach

The more I learn about mathematics, the more it feels like everything is connected. It’s crazy to see people define a bunch of really abstract, high-powered objects, only to later show that there is indeed a fairly natural notion of “morphism” between those objects, and moreover, you actually get a category with unbelievably nice properties (for example it turns out to be abelian or something).

Not only that, but the way certain intuitively familiar (say, geometric) concepts can be recast so naturally in completely different (say, algebraic) terms is amazing. It’s even a bit creepy, like suddenly becoming aware of the existence of so many esoteric foreign languages, most of which nobody even currently speaks, but which all have their own advantages and disadvantages. This is one of the things that lends these “analogies” and tactics of “mathematical translation” their immense power. You may have to stutter your way through unnatural, abstruse constructions in one of these “languages”, only to find a ridiculously elegant formulation of that same concept in another such language. The answer, clearly, is to learn as many languages as you can. Taking this sentence literally is also something I’ve wanted to do for as long as I can remember, but math is keeping me busy at the moment…

I was reading about the Banach-Stone theorem, which is concerned with recovering a compact Hausdorff space from the algebra of continuous functions on the space (!!!). That’s pretty remarkable if you ask me. Also in this vein is the way you can conflate points of affine n-space over an algebraically closed field k with maximal ideals of the corresponding polynomial ring k[x_1, \ldots, x_n]. That is, the point (a_1, \ldots, a_n) can be replaced by the maximal ideal (x_1-a_1, \ldots, x_n - a_n).

This leads to the definition of the spectrum, \mathrm{Spec} \; R, of a ring R as a locally ringed space (you take \mathrm{Spec} \; R, equip it with the Zariski topology, and then define a sheaf of rings on it by using localisation at the prime ideals, more or less). After doing this, you’re pretty much inches away from defining affine schemes, and then general schemes. Grothendieck’s introduction of these objects elevated algebraic geometry to where it is today.

Even if you disregard that rippling shockwave of power, check this out: it turns out people have used very similar kinds of ideas to translate topological properties of a space into purely algebraic properties of its function algebra, for example (roughly speaking) compactness of a space corresponds to its algebra being unital. Now, the C*-algebra you obtain from a space in this sense is always commutative (of course), but with this “topology-to-C*-algebra” dictionary in hand, people decided to replace it with a noncommutative C*-algebra, which then lead to a noncommutative version of topology!

For another example, I was reading (in the Princeton Companion, for whoever’s wondering) earlier today about different ways of writing polynomials: if I write a polynomial down, say, x^4 + 3x^2 + 7x + 9, then this representation is clearly “centered at 0″ in a certain sense. Namely, it makes it clear that if I plug in x=0 I will get 9. However you could also write a polynomial centered at another point; one might say (x-3)^2 + 7(x-3) + 2 is “centered at 3“. Because both of these representations are “biased” toward a certain point, we think of them as “local” in nature. However you could also write down a polynomial as (x-2)(x-1)^2(x-7), or something — notice how this tells you about all the roots and hence is much less biased, so one might think of this representation as “global”. Anyways, the idea is that a guy named Hasse thought long and hard about this, and tried to make an analogy between numbers and polynomials, and that’s how p-adic expansions were invented: they give you information about how the number behaves locally, “at the prime p“. It’s called “Hasse’s local-global principle”. The initial response to Hasse’s work was widespread skepticism. People thought it was useless until eventually it was realized one can sometimes “piece together local information” to prove a global theorem. Then they changed their minds.

All in all, I just find myself becoming even more addicted to mathematics. I would have withdrawals if I ever stopped studying this stuff. I’m definitely going to do every problem in Hartshorne, no matter how long it takes. It’s a pretty formidable task though, since I have to supplement it with multiple other books (mainly for the commutative algebra).

Posted in articles | Leave a comment

Constant presheaf

Posted on 2013-04-26 by

Here’s a problem from the Sheaves section of Hartshorne. I’m having difficulty with the very last part…

Background. Consider the forgetful functor from the category of sheaves on X to the category of presheaves on X. Then if \mathscr{F} is a presheaf on X, there is a universal morphism, say \theta : \mathscr{F} \to \mathscr{F}^+, from \mathscr{F} to this functor. \mathscr{F}^+ is called the sheaf associated to the presheaf \mathscr{F} or alternatively its sheafification.

Let A be an abelian group. Equip it with the discrete topology. The corresponding constant sheaf, denoted \mathscr{A}, is the one which maps each open set U to the group of continuous functions U \to A, where A is equipped with the discrete topology. The restriction maps are the obvious ones.

Problem. Let A be an abelian group, and define the constant presheaf associated to A on the topological space X to be the presheaf U \mapsto A for all U \neq \varnothing, with restriction maps the identity. Show that the constant sheaf \mathscr{A} defined in the text is the sheaf associated to this presheaf.

Solution. Call this presheaf \mathscr{F}. We need to find a morphism \theta : \mathscr{F} \to \mathscr{A} with the property that for any morphism \varphi : \mathscr{F} \to \mathscr{G}, there is a unique morphism \psi : \mathscr{A} \to \mathscr{G} such that \varphi = \psi \circ \theta.

Hence, let U \neq \varnothing be an open set. We need a morphism of abelian groups \theta(U) : \mathscr{F}(U) = A \to \mathscr{A}(U). Recall that \mathscr{A}(U) was the group of all continuous maps of U into A. The natural choice seems to be to send a \in A to the constant function a \in \mathscr{A}(U). Let us verify that \theta really is a morphism of presheaves: suppose we have an inclusion V \subseteq U. Then the statement that the appropriate diagram commutes is the same as saying “the restriction to V, of the constant function a on U, is the same as the constant function a on V”, which is clear. So \theta is a morphism of presheaves.

Before proceeding, we note that if Y is discrete then any continuous map f : X \to Y is locally constant, in the sense that each x \in X admits some neighbourhood on which f is constant. Locally constant maps are constant on each connected component.

Finally, suppose \varphi : \mathscr{F} \to \mathscr{G} is a morphism of presheaves, i.e. for each open set U we have a morphism of abelian groups \varphi(U) : \mathscr{F}(U) = A \to \mathscr{G}(U). Define, for each open U, a morphism \psi(U) : \mathscr{A}(U) \to \mathscr{G}(U) as follows. If U is connected, then since any continuous f : U \to A is constant (say f \equiv a for a \in A), we may define \psi(U)(f) = \varphi(U)(a). If U is not connected, then we should have \mathscr{A}(U) \cong \prod \mathscr{A}(U_i) where \{ U_i \} are the connected components of U, and then it should be clear where to go from here.

However, it seems like to finish, we instead want \mathscr{A}(U) \cong \bigoplus \mathscr{A}(U_i) — we want to decompose any continuous f : U \to A as a finite sum of constant functions, and then the way to define \psi(U)(f) will be clear. My question is: how do we know this is possible? What about, say, the function \mathbb{R} \setminus \mathbb{Z} \to \mathbb{Z} given by x \mapsto \lfloor x \rfloor?

EDIT: Maybe one should look at the maps induced on the stalks by \varphi, since to figure out what \psi(f) should be, it seems sufficient to only have local data about f, which would certainly make our life easy since f is locally constant…

Posted in algebraic geometry | 5 Comments

Sheaves

Posted on 2013-04-24 by

Alright so, with all the algebraic geometry lined up for the Spring and Fall, we might as well get a head start. That’s right — time to define sheaves. I admit these things sound pretty intimidating at first, but I think I see why you might want to talk about them. The idea is that you have some topological space X and you want to attach some algebraic data to the open sets. For example, you might want to attach to each open set U of X the ring C(U) of continuous functions U \to \mathbb{C}, say. Or, if you were working with a variety, you could talk about the regular functions on U instead.

Here is the definition (from Hartshorne). A presheaf \mathcal{F} of abelian groups on X consists of the data

  • for every open subset U \subseteq X, an abelian group \mathcal{F}(U), and
  • for every inclusion V \subseteq U of open subsets of X, a morphism of abelian groups \rho_{UV} : \mathcal{F}(U) \to \mathcal{F}(V)

subject to the conditions

  • \mathcal{F}(\varnothing) = 0, where \varnothing is the empty set,
  • \rho_{UU} is the identity map \mathcal{F}(U) \to \mathcal{F}(U), and
  • if W \subseteq V \subseteq U are three open subsets, then \rho_{UW} = \rho_{VW} \circ \rho_{UV}.

That’s quite a mouthful, especially considering we’re defining something which isn’t even good enough to be a legit sheaf. However, you probably see what’s going on. The first two points may have had you thinking: “\mathcal{F} is a functor”. Once you read the last three points, though, your suspicions were surely confirmed.

Indeed, the above merely defines a presheaf \mathcal{F} to be a (contravariant) functor \mathcal{O}(X) \to \mathbf{Ab} where \mathbf{Ab} is the category of abelian groups. Here, \mathcal{O}(X) is a category “manufactured out of X“. Its objects are the open sets of X, and the morphisms are the inclusions. Simple as that. One can easily define presheaves of other algebraic structures than abelian groups, just by replacing the target category.

For U an open set, we call \mathcal{F}(U) the section of the presheaf \mathcal{F} over U. The maps \rho_{UV} are termed restriction maps and we sometimes write \left. s \right|_V rather than \rho_{UV}(s) if s \in \mathcal{F}(U).

While presheaves may seem like a reasonably good tool for accomplishing the task of “attaching algebraic data to open sets” that we first mentioned, they turn out to be a bit too general. We need to ask a bit more of these functors to ensure they behave properly in the contexts we wish to employ them.

There are two more assumptions we need to impose on presheaves before they qualify as sheaves:

  • If an open set U is covered by open sets \{ V_i \} and s \in \mathcal{F}(U) is such that \left. s \right|_{V_i} = 0 for all i, then indeed s=0.
  • If an open set U is covered by open sets \{ V_i \} and we have a bunch of “pieces” s_i \in \mathcal{F}(V_i) for each i which “agree” on the overlaps in the sense that \left. s_i \right|_{V_i \cap V_j} = \left. s_j \right|_{V_i \cap V_j} then there is a way of “gluing them together”: there is some s \in \mathcal{F}(U) such that \left. s \right|_{V_i} = s_i for each i.

Next time we will discuss some pathological examples which assure us that indeed, both of these additional axioms are required. After that we’ll talk about morphisms of sheaves. As you probably guessed, those will be natural transformations…

Posted in algebraic geometry | 1 Comment

What is a module?

Posted on 2013-04-17 by

Here is my first effort to explain, in basic terms, the concept of a “module”. I tried to make it accessible to those not studying pure math, while still remaining interesting to those who are. I have no idea how well this can actually work in practice (or if it even did work); hopefully people will just skip over any terminology they don’t understand instead of giving up on the whole article. Better yet, ask me questions in the comments!

In linear algebra courses, we learn about vector spaces: these are algebraic structures where you can add two vectors (v+w), or scale a vector by some amount (\lambda v). In applied treatments, these “scalars” \lambda are usually tacitly assumed to come from the real or complex numbers (\mathbb{R} or \mathbb{C}); it is seldom mentioned that the “scalar multiplication” of a vector space is really an action of a field \mathbb{F} on an abelian group V.

If you’re puzzled by this last phrase, don’t worry. The word action merely indicates a rule for assigning to each element of \mathbb{F} a transformation (indeed, a group homomorphism) of V, in a particularly nice way: to each \lambda \in \mathbb{F} we associate the “multiplication by \lambda” map V \to V given by the rule

v \mapsto \lambda v.

To indicate the field of scalars explicitly, we might call V an \mathbb{F}-vector space or a vector space over \mathbb{F}. To summarize, an \mathbb{F}-vector space is a set V, equipped with a rule for adding two elements of V, as well as a rule for scaling elements of V “by” elements of \mathbb{F}.

A ring is, loosely speaking, a structure in which we can add and multiply elements, which satisfies most of the usual arithmetic laws. Fields are a very special kind of ring: the multiplication of a field is commutative (ab =ba), and every nonzero element a has a reciprocal 1/a. Since they have so many wonderful properties, they are much more “rigid”: on an elementary level, their nature is far less complicated than that of rings in general (don’t get me wrong, there’s still a lot we don’t know about fields). However, there are rings we deal with every day which (for simple reasons e.g. the lack of reciprocals) are not fields, for example, the ring of integers:

\mathbb{Z} = \{ \ldots, -2, -1, 0, 1, 2, \ldots \}.

Something I’ve been meaning to learn more about for a while are “vector spaces where the scalars are allowed to come from any ring R at all, not necessarily a field“. In mathematics we call these R-modules, or modules over R (or just modules when the ring R of scalars is clear). Hence, a vector space is just a module over a field. They’re immensely useful in the study of rings themselves, and most people usually glimpse them for the first time in a course on commutative algebra (perhaps when they begin as a grad student). Unfortunately, it will be another 8 months before I have a chance to take an actual course in commutative algebra (PMATH 446), and I’m too impatient for that.

One thing we notice about vector spaces is that their structure theory is trivial; it’s about as nice as it could possibly be. Namely, \mathbb{F}-vector spaces are in some sense “completely determined” by their dimension. You’re probably familiar (at least in the case where the dimension is finite) with the fact that, by choosing a basis, any such space can be viewed simply as \mathbb{F}^n.

Intriguingly, when we move from the setting of vector spaces to the more broad world of modules, the “more complicated” personae of general rings (compared to fields) mangles the situation significantly. A lot of our linear algebra, which we were able to develop with elementary methods, is vehemently defenestrated. In particular, it is no longer even true that a basis always exists for a module (in fact this is a pretty rare situation, and such modules are called free). This means our much-applauded concept of dimension doesn’t, in general, even make sense for modules. Nor is the “completely decomposable” nature of vector spaces shared by modules: it’s actually possible to construct huge modules M which don’t even have a single ”submodule” (other than the obvious ones \{ 0 \} and M).

For our very first example of some of the ideas modules generalize, let’s talk about abelian groups: these are simply sets equipped with an associative, commutative binary operation +, an identity element, and inverses. Given any abelian group, we can define a rule for “scaling” elements of G by integers, that is, elements of \mathbb{Z}. Namely, for n \geq 0 we define ng = g + g + \ldots + g (n times), simply using the group operation +, and otherwise if n < 0 we define ng = (-n)(-g) (that is, with recourse to the n \geq 0 case). This is a perfectly natural way of turning any abelian group into a \mathbb{Z}-module. On the other hand, it is obvious that if you start with a \mathbb{Z}-module you can just forget about the scalar multiplication altogether, and you’re left with an abelian group. So \mathbb{Z}-modules are the same thing as abelian groups.

If you think back, you’ll probably recall that a large part of linear algebra had to do with linear operators (more concretely, matrices) and doing things with them, like finding their eigenvalues and eigenvectors, characteristic polynomials, determinants, traces, and so on. A lot of work was put into discussing when “diagonalisation” is possible, and how to achieve it. Since we’re not always lucky enough to be able to do this, you probably learned about canonical forms: the “next best thing” to diagonalisation where we usually try to get some kind of “block diagonal” form. Namely, Jordan canonical form, rational canonical form, and all that. So why should we even care about modules? Furthermore, why were linear operators (square matrices) so much subtler objects to deal with than vector spaces themselves?

The following cool idea provides what I believe is an epistemologically satisfactory answer. First, recall that the set of all polynomials with coefficients in \mathbb{F} forms a ring under the usual operations of addition and multiplication, known as the polynomial ring \mathbb{F}[x]. Suppose V is an \mathbb{F}-vector space. I claim that a linear operator T : V \to V is the same thing as an \mathbb{F}[x]-module structure on V. Notice that V is already an \mathbb{F}-vector space, and any action must respect the ring structure of \mathbb{F}[x], so my previous claim reduces merely to saying that a linear operator T : V \to V is the same as a rule for scaling an element v \in V by the element x \in \mathbb{F}[x]. What is the obvious thing to do? Well, simply define xv = T(v) for all v \in V, right? Then right away, this gives us a scalar multiplication of \mathbb{F}[x] on V: namely,

(a_n x^n + \ldots + a_1 x + a_0)v = a_n T^n(v) + \ldots + a_1 T(v) + a_0 v.

Here, of course, T^n refers to the composition of T with itself n times, that is, the map (T \circ \ldots \circ T) : V \to V. For this reason many authors will refer to this as an \mathbb{F}[T]-module structure on V, since the indeterminate x is literally acting as T. Okay, so every linear map gives rise to an \mathbb{F}[x]-module structure on V. What about the other way? That is, if we have some \mathbb{F}[x]-module structure on V, can we get a linear map T : V \to V from it? We definitely can: define T by merely setting T(v) = xv where xv denotes the scalar multiplication of v by x, provided to us by the \mathbb{F}[x]-module structure! Then, simply by the conditions we impose on how a “scalar multiplication rule” must behave, it follows that T is linear.

If we think about it for a second, we realize that the submodules of the module obtained from the linear map T : V \to V are precisely the subspaces of V which are invariant under T, namely, the subspaces W \subseteq V such that T(w) \in W for all w \in W, or stated another way, T(W) \subseteq W. When we diagonalise a matrix by finding a basis consisting of eigenvectors, what we’re effectively doing is understanding how the associated linear map’s domain is made up of a bunch of one-dimensional invariant subspaces (the eigenspaces). Since we know this is not possible in general, we deduce that these modules will not, in general, admit a decomposition into one-dimensional submodules. It’s interesting to think about how properties of the matrix, like its characteristic polynomial for example, are encoded in the algebraic properties of the resulting \mathbb{F}[x]-module…

Canonical form theory for square matrices over a field falls out as an easy consequence of structure theory for certain kinds of modules (to be precise, the “finitely-generated modules over principal ideal domains”). Recall that we previously mentioned the interchangeability of the concepts of “abelian group” and “\mathbb{Z}-module”. Since \mathbb{Z} is one of the first examples of a “principal ideal domain”, the celebrated structure theorem for finitely generated abelian groups (and its special case for finite abelian groups) is also a special case of this theorem on modules! So, aside from the formality, one could almost argue that you were essentially doing some primitive, well-cloaked module theory in Linear Algebra 2.

To close off this quick initial glimpse into module theory, I will mention one more place modules crop up: a branch of mathematics called the representation theory of finite groups. Loosely speaking, a representation of a group G is a way of viewing the group as some set of matrices acting on a vector space.

The Yoneda lemma from category theory tells us that contemplating how one algebraic structure acts on others can yield profound revelations about the object itself: for a concrete example, Cayley’s theorem in group theory says that every group “is” just a permutation group of some set, and this lies at the heart of why we study representations of groups, modules over rings, and so on.

Formally, a representation is a group homomorphism G \to \mathrm{GL}(V) where \mathrm{GL}(V) is the “automorphism group” of V, or in undoubtedly more friendly language, the set of invertible linear operators T : V \to V. It turns out that, in much the same way we whipped out a module over a polynomial ring in one variable to capture the “essence” of a linear operator on V, we can construct a pretty natural ring from the group G, known as the group ring or group algebra, denoted \mathbb{F}[G]. Basically, you consider the set of all finite “formal sums” of elements in G with coefficients from \mathbb{F} and define a multiplication on it by using the group operation of G. Then it turns out that representations \varphi : G \to \mathrm{GL}(V) of G and \mathbb{F}[G]-modules are (just like abelian groups and \mathbb{Z}-modules) completely interchangeable concepts. Of course, the analogy becomes a bit more complicated if you move to, say, the representation theory of topological (for example, Lie) groups, since then you need to introduce a kind of “analytic version” of the group algebra.

Anyway, I’ve barely scratched the surface of all the interesting questions you can ask. Thanks for reading, and again, feel free to leave questions or comments.

Posted in abstract algebra | 10 Comments

A remark

Course notes, for example the ones I post on this blog, should not be your sole reference for learning course material.

Perhaps I already said something along these lines, but I just want to elaborate a little. This coming September will denote the end of my fourth full year as an undergraduate, so I feel like I have enough experience to now state, with certainty, my stance on note-taking in pure mathematics courses.

As is probably made obvious by a quick glance at the Notes section, I have no problem keeping up with typesetting during a lecture when the instructor’s board work is somewhat linearly organized. However, for better or for worse, styles vary among instructors and this is not always the case. Some material, especially in a course like graph theory where proofs are almost always accompanied by sequences of diagrams which evolve as the proof unfolds, just does not lend itself easily to being typeset in real time.

I have slowly come to the realization that it is frequently impossible to produce notes which properly capture every single remark or quick explanation provided in class. Of course, a sufficiently strong student could (while reading course notes) fill in these intermediate steps and justifications themselves.

This is unfortunate for a person like me, who is admittedly a bit obsessive-compulsive to include every such detail the instructor writes down, even when I find them unnecessary for understanding (because who knows, maybe for someone else, that small detail would make the difference between understanding and wasting 10 minutes of their time in confusion). You may have noticed this about the PMATH 365 notes, and it may well be one of the prime causes of their general untidiness.

Which is esseeentially what prompted me to write this post. I am growing slightly tired of having my ability to follow a lecture be interfered with by my efforts to encode all of these details in a non-ugly way — especially in a course where probably everything we do is coming straight out of a textbook anyway. The situation is more complicated with courses where instructors present results that may not be as ubiquitous in the literature, for example graduate topics courses.

Regardless, as my Plan page indicates, this term denotes the logical (but not formal) end of my career as an undergraduate. All the courses I plan to take for the next year are either cross-listed as grad courses, or actually are bona fide topics courses (interesting choice of diction, I guess I may be inheriting some habits…) ahem, anyway, the point is that in spite of the remarks above, my notes will probably become more brief, effective as of next term. Henceforth, when elaborate proofs are presented in lectures and seminars, I may elect to merely provide a reference to the literature, rather than typesetting it all in detail — especially since the presentations in books and papers are almost guaranteed to be more elegant and readable.

TL;DR: I’m almost a grad student. I’m going to become a lazier notetaker and a better listener.

Aside | Posted on 2013-04-03 by | Leave a comment