$ \hspace{4cm} x=x_1 +\frac{x_2}{2!}+\frac{x_3}{3!} + \cdots + \frac{x_n}{n!} + \cdots, \hspace{2cm} (*) $

**where $x_1$ can be any integer, but for $ n \geq 2$, $x_n \in \{ 0,1,…,n-1 \}.$ Furthermore, if we require that the partial sums be strictly smaller than x, then such a representation is unique.**

**Remark:** One cannot help recalling decimal or binary expansion of numbers. Notice that $\frac{n}{n!}=\frac{1}{(n-1)!}$ (drops back to previous digit), so the bound on $x_n$ is logical.

**Proof:**

Choose the biggest integer $x_1$ *strictly* smaller than $x$. If $x_1+\frac{1}{2!}$ is strictly less than $x$ then choose $x_2=1$, otherwise, choose $x_2=0$. Assume we have picked $x_1, x_2, … , x_n$, then we’ll choose $x_{n+1}$ to be the largest of $\{ 0, 1, 2, … , n \}$ so that

$ \hspace{6cm} x_1+\frac{x_2}{2!}+ … +\frac {x_n}{n!} + \frac{x_{n+1}}{(n+1)!} < x .$

We’ll prove that this inductive choice of $ \{ x_n \}_{n=1}^\infty $ satisfies the expansion $(*)$.

**Claim**: For every ,

$ \hspace{6cm} 0 < x- (x_1+\frac{x_2}{2!}+ \cdots +\frac{x_n}{n!}) \leq \frac{1}{n!} \hspace{2cm} (ES) $

Proof of the claim: If $x_n \neq n-1$, this is an immediate consequence of the choice of $x_n$: by optimality of $x_n$ we have

$ \hspace{5.5cm} x_1+\frac{x_2}{2!}+ \cdots +\frac {x_n}{n!} < x \leq x_1+\frac{x_2}{2!}+ \cdots +\frac {x_n + 1}{n!} $

Subtracting $ x_1+\frac{x_2}{2!}+ \cdots +\frac {x_n}{n!} $ from each term yields (ES).

If $x_n=n-1$, the maximum possible, then we have the identity

$ \hspace{1cm} x_1+ \cdots +\frac {x_n}{n!} + \frac{1}{n!} = x_1+\cdots + \frac{x_{n-1}}{(n-1)!}+\frac {n-1}{n!} + \frac{1}{n!} = x_1+\cdots +\frac {x_{n-1}}{(n-1)!} + \frac{1}{(n-1)!} $

To obtain (ES) it suffices to show that this quantity is bigger than or equal to x. Comparing the first and last expressions, we see that we have reduced case n to (n-1). Thus, either we work backwards to reach n=1, considering cases for $x_{n-1}$ for this current step and so on, or we switch to an inductive proof, to attain (ES). Case n=1 is obviously true.

The uniqueness part’s proof: Assume that some has two different representations:

$\hspace{3cm} x_1 +\frac{x_2}{2!}+\frac{x_3}{3!} + \cdots + \frac{x_n}{n!} + \cdots = y_1 +\frac{y_2}{2!}+\frac{y_3}{3!} + \cdots + \frac{y_n}{n!} + \cdots $

We’ll prove that one of them is a finite sum. Assume that $k$ is the first index where $x_k \neq y_k$, and, without loss of generality, that $x_k > y_k \ $ . Then,

$ \hspace{6cm} \frac{x_k – y_k}{k!} = \displaystyle\sum _{n=k+1}^\infty \frac{y_n – x_n}{n!} \hspace{2cm} (EQ)$

Notice that while $\frac{x_k – y_k}{k!} \geq \frac{1}{k!}$,

$\hspace{2cm} \left|\displaystyle\sum _{k+1}^\infty \frac{y_n – x_n}{n!}\right| \leq\displaystyle\sum _{n=k+1}^\infty \frac{|y_n – x_n|}{n!} \leq\displaystyle\sum_{n=k+1}^\infty \frac{n-1}{n!} =\displaystyle\sum_{n=k+1}^\infty \left(\frac{1}{(n-1)!}-\frac{1}{n!}\right) = \frac{1}{k!} .$

Thus, the only way for (EQ) to hold is to have $x_k=y_k+1$ and for all $n>k$, $x_n= 0$ and $y_n=n-1$ ; an analogous situation to $1.73=1.7299999…$ in decimal base.

Note that out of the two representations only one is strictly increasing to its limit, proving the uniqueness claim.

**The Backstory: **My independent discovery of this expansion was triggered by my search for a compact set in $R$ with no isolated points (every point’s every neighborhood contains other points of the set as well) and *no* rationals. This was a question my favorite analysis professor, Dr. Rezaee, had asked me to think about.

For months my approach had been to start with $[0,1]$ and then try to remove successively more and more subsets. But I had failed to land on the right set. Then there was this morning that I was sitting in a different class when suddenly I recalled an exercise from Tom M. Apostol’s *Mathematical Analysis* book:

**Exercise: **The number $x=x_1 +\frac{x_2}{2!}+\frac{x_3}{3!} + \cdots + \frac{x_n}{n!} + \cdots $ (as in proposition 1) is rational if and only if there exists an $N \in \mathbb{N} $ such that

$\hspace{6cm} n>N \implies x_n=n-1.$

“I know so many irrational numbers!” I said to myself and there I was with a set.

Solution to the exercise: Suppose the condition holds. Then, as shown by uniqueness part’s proof above, the sum is equal to a finite sum, each of whose terms are rational.

For the other direction, suppose $x$ is rational then, for relatively prime $p \in \mathbb{Z},q\in \mathbb{N}$ we have

$\hspace{6cm} x=\frac{p}{q}=\displaystyle\sum_{i=1}^{\infty} \frac{x_i}{i!}$

Multiplying the sides by $q!$ yields

$ \hspace{6cm} q!\displaystyle\sum_{q+1}^{\infty} \frac{x_i}{i!} = p(q-1)!-q!\displaystyle\sum_{i=1}^{q} \frac{x_i}{i!}. $

The right hand side is an integer. If for even only one index $ i>q$ the equality $ x_i = i-1$ failed to hold, then we would have

$ \hspace{6cm} 0 < q!\displaystyle\sum_{q+1}^{\infty} \frac{x_i}{i!} <q!\displaystyle\sum_{q+1}^{\infty} \frac{i-1}{i!} =1 $

Which contradicts it being an integer. (The strict positivity is due to strictly increasing assumption on the series.)

**A Perfect Set**

**Proposition: 2 The set**

$\hspace{6cm} S=\left\{\left.\displaystyle\sum_{i =4}^\infty \ \frac{x_i}{i!} \right| \ x_i \in {1,3}\right\}$

**is a “perfect set” (closed, and each point is a limit point) without rationals.**

Proof: The idea is hidden in the arguments we have already made. The point is that when we change one digit by 1, the tail has to go full speed to catch up. Since here we have restricted to choices 1 and 3, when we change a digit then the new number is by a significant distance away from any member of $S$.

Take $y \in R \backslash S$. Let’s restrict to $y$’s of the form

$\hspace{6cm} y=\displaystyle\sum_{i=4}^{\infty} \frac{y_i}{i!} .$

Other cases where $y$ has earlier digits are just as easy, but we want to avoid complications in notation! Since $y$ is not in $S$, there is an index $j$ such that $y_j \notin \{1,3\}.$ For any given $x \in S$, the representations of $x$ and $y$ will differ somewhere earlier than $j$, say at $k$’th component, $4 \leq k\leq j$. Therefore,

$\hspace{2cm} \left|x-y\right|=\left|\displaystyle\sum_{i=k}^{\infty} \frac{x_i-y_i}{i!}\right|\geq \frac{\left|x_k-y_k\right|}{k!}-\displaystyle\sum_{i=k+1}^{\infty} \frac{\left|x_i-y_i\right|}{i!} \geq \frac{1}{k!}-\left|\displaystyle\sum_{i=k+1}^{\infty} \frac{i-2}{i!}\right|. $

It follows that

$\hspace{6cm} |x-y| \geq \frac{1}{(k+1)!}\geq \frac{1}{(j+1)!}$

The index $j$ depends on $y$ only, thus we proved that within radius $1/(j+1)!$ of $y$ there are no points of $S$, or, equivalently, $S^c$ is open, $S$ is closed.

Now, pick any $x\in S$. Given $\epsilon >0$, in order to find another point in $S$, $\epsilon$-close to $x$, move far enough in the representation of $x$, and switch $x_N$ to $1$ if it is $3$, or to 3 if it is 1. Keep all other digits the same. The new number is again is $S$, closer than $\epsilon$ much to $x$ provided that $\frac{2}{N!}<\epsilon$. Thus, every point in $S$ is a limit point.

**Questions:**

Do you know of other perfect sets without rationals (in the usual topology of $\mathbb{R}$)?

What else can be done with the factorials representation of numbers?!

We have proved that $e$ is an irrational number! Do you see where?

]]>The Bad:

- Keeping track of the papers was a nightmare. I had 30 new papers to go through every day and I spent several minutes of my precious class time handing them back to students. I also think the students struggled to keep all of these short quizzes organized. On my end, the fix is to be more organized (always the dream) and to have admitted that my system wasn’t working in order to try something else. To help keep the students more organized with so much paper floating around, I now try to communicate how I organize my resources more openly and that has seemed to help.
- In the last post, I said that grading was a “Good” but it was also a “Bad.” It felt better when (as planned) I could grade the same day as I gave the quiz. However, that wasn’t always possible. The weeks when I got behind on grading were horrible. I stared at the ever increasing pile of quizzes with dread until I got the courage to tackle it. I don’t really have a fix for this part of grading (it is nearly guaranteed that I get behind in grading at least once during any semester) so perhaps this issue is more of a warning label in case you try this yourself: getting behind in grading is rough.

The Ugly:

- I really didn’t enjoy giving something called a quiz every day. I think my students felt unnecessarily pressured to master material quickly. A bit of that pressure is good, but some students felt palpably apprehensive coming to class each day for several weeks at the beginning. They felt better after several reassurances that it wasn’t as high-stakes as the word “quiz” would imply. One possible way to fix this is to stop calling it a quiz, but that would take away all of the pressure, which doesn’t seem ideal either.
- Giving a quiz every day took class time. Sure, the quiz was written to take five minutes, but between passing them out, collecting them, and returning the previous quiz, I lost nearly ten minutes every day. You could fix this by giving these quizzes online before class, but then you lose the opportunity for feedback on the process and some students will use resources other than their brain (regardless of the rules or what is beneficial).

In trying to hold on to the benefits of daily quizzes while addressing some of the issues, I tried something different last spring. Instead of a quiz at the start of each day, I would write the same question that I would have given as a quiz on the board for the students to work on as they came to class and we would discuss paths to solutions together. On Friday, I would give a quiz consisting of questions nearly identical to those we had worked on for the past week. There was incentive to arrive on time since the students got a preview for the quiz, but I didn’t have to keep track of so many papers. The students didn’t have a strong reason to study ahead of time (since no part of this was graded) but we struggled through the problems together, resulting in some really good conversations about the previous material.

This routine came with its own challenges just like any other experiment with teaching style, but overall, I liked the vibe of my classroom with the “warm-up question” structure better than the daily quizzes. Neither of these options is a “one-size-fits-all” solution but both added a lot of richness to my classroom that I couldn’t have predicted.

]]>But one article in particular, “The AI detectives,” captured my attention. Rather than highlighting a specific application of AI, as the other articles do, this piece draws attention to the lack of transparency in certain machine learning algorithms, particularly neural networks. The inner workings of such algorithms remain almost entirely opaque, and they are accordingly termed “black boxes”: though they may generate accurate results, it’s still unclear how and why they make the decisions they do.

Researchers have recently turned their attention to this problem, seeking to understand the way these algorithms operate. “The AI detectives” introduces us to these researchers, and to their approaches to unlocking AI’s black boxes.

One such “AI detective,” Rich Caruna, is using mathematics to impose greater transparency in artificial intelligence. He and his colleagues employed a rigorous statistical approach, based on a generalized additive model, to produce a predictive model for evaluating pneumonia risk. Importantly, this model is intelligible; that is, the factors that the model weighs to make its decisions are known. Intelligibility is crucial in this setting, as previous, more opaque models conflated overall outcomes with inherent risk factors. For example, though asthmatics have a high risk for pneumonia, they typically receive immediate, effective care, which leads to better health outcomes—but which also led early models to flag them, naively, as a low risk group. Caruna et al.’s model is also modular, meaning that any faulty causal links made by the algorithm can be easily removed from its decision-making process. But while it is powerful, this approach is not well-suited to complex signals, like images—and it circumvents the problem of intelligibility in artificial intelligence, rather than addressing it head-on.

Gregoire Montavon and his colleagues, by contrast, have developed a method that uses Taylor decompositions to study the most opaque of machine learning algorithms, Deep Neural Networks. Their approach (which was not mentioned in the *Science *article) has the advantage of explaining the decisions made by Deep Neural Networks in easily interpretable terms. By treating each node of the neural network as a function, Taylor decompositions can be used to propagate the function value backward onto the input variables, such as pixels of an image. What results, in the case of image categorization, is an image with the output label redistributed onto input pixels—a visual map of the input pixels that contributed to the algorithm’s final decision. A fantastic step-by-step explanation of the paper can be found here.

Of course, none of these artificial intelligence techniques would be possible without mathematics. Nevertheless, it is interesting to see the role that math is now playing in furthering artificial intelligence by helping us understand how it works. And as AI is brought to bear on more and more important decisions in society, understanding its inner workings is not just a matter of academic interest: introducing transparency affords more control over the AI decision-making process and prevents bias from masquerading as logic.

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In order to make sure that we are all on the same page, let’s briefly review the difference between topological and smooth manifolds. Recall (see for example previous posts on this blog!) that an $n$-dimensional topological manifold $M$ is defined using the data of local charts, each of which may be identified with an open subset of $\mathbb{R}^n$, together with continuous transition functions between them. If these transition functions are in addition smooth (in the usual sense of smooth maps on $\mathbb{R}^n$), then we say that $M$ has been given a smooth structure and is a smooth manifold. In order to talk about a diffeomorphism between two manifolds, we of course require that the manifolds themselves are smooth.

To those who have not thought about low-dimensional topology (and even for those who have), it is often difficult to get a feel for the difference between topological and smooth manifolds. (I certainly do not have any such intuition.) This is partly due to the fact that in low dimensions (less than or equal to three), every topological manifold admits a unique smooth structure. Thus, in order to think of a topological manifold that (for example) has no smooth structure, or two different smooth structures, we are already forced to consider examples in more dimensions than most of us are comfortable visualizing. Even setting this aside, it is difficult to see how one would go about distinguishing two smooth structures anyway, or how to prove from the definitions that a given topological manifold does or does not admit a smooth structure. Indeed, the first construction of a pair of distinct smooth structures on the same topological manifold (given for $S^7$, by John Milnor in 1956) came as a shock to many mathematicians.

Part of the difficulty in studying smooth structures on manifolds is that many of the introductory invariants in algebraic topology are either formulated purely in terms of the continuous structure (as is the case for homotopy or homology), or turn out to depend only on the continuous structure (as is the case for de Rham cohomology). Thus, slightly fancier tools are needed if one wants to systematically study smooth manifolds. We shall see later that, in dimension four, the primary (and in many cases, the only) strategy for studying smooth topology turns out to be afforded by gauge theory.

Before we continue, let’s recall our discussion of Freedman’s theorem. Associated to any simply-connected, topological four-manifold $M$, we described a unimodular bilinear pairing, called the intersection form, which could be viewed as a symmetric integer matrix with determinant $\pm 1$. Freedman showed that for any unimodular pairing $Q$, one could find a simply-connected, topological four-manifold $M$ with intersection form $Q$. Even more surprisingly, he proved that if $Q$ was even, such an $M$ was unique up to homeomorphism (among simply-connected, topological four-manifolds), while for odd $Q$ there were exactly two possible $M$. We might thus hope for a similar relationship to hold in the smooth category; or, failing this, to understand how the behavior of smooth and topological manifolds differ.

The first question to ask is whether given a unimodular pairing $Q$, it is always possible to construct a *smooth* simply-connected manifold with intersection form $Q$. It turns out that the answer is a resounding *no*. The first result towards this end was established by Simon Donaldson in 1983 using gauge-theoretic methods:

Let $M$ be a smooth, simply-connected four-manifold. Suppose that the intersection form $Q$ of $M$ is positive (or negative) definite. Then $Q$ must be diagonalizable.

Here, an intersection form $Q$ is said to be *positive* (or *negative*) *definite* if $Q(x, x) > 0$ (or $Q(x, x) < 0$) for all $x$, and is said to be *diagonalizable* if it is equivalent to a diagonal matrix over $\mathbb{Z}$ (which in this case must be plus or minus the identity). Although being definite is certainly a restriction on the lattice, it turns out that most unimodular lattices in high dimensions are in fact either positive or negative definite—for example, there are over a billion distinct definite lattices of rank 32. According to Freedman’s theorem, each one of these arises as the intersection form of a simply-connected topological four-manifold. But by Donaldson’s theorem, almost all of these are *not* smooth manifolds—the only definite unimodular pairings that arise as the intersection forms of smooth four-manifolds are (equivalent to) plus or minus the identity!

Donaldson’s theorem immediately implies the existence of a vast class of topological four-manifolds without any smooth structure. Moreover, it showed that the relationship between smooth four-manifolds and their intersection forms is rather more subtle than in the topological category. Since Donaldson’s theorem, much work has been done on investigating exactly which pairings $Q$ can arise as the intersection forms of smooth four-manifolds. So far, what is known is the following:

Let $M$ be a simply-connected, smooth four-manifold with intersection form $Q$. Then:

1) If $Q$ is definite, then $Q$ must be diagonalizable by Donaldson’s theorem. Conversely, all definite, diagonalizable $Q$ indeed arise as the intersection forms of smooth manifolds (namely, $m\mathbb{C}P^2$ or $m\overline{\mathbb{C}P}$$^2$).

2a) If $Q$ is indefinite and odd, then it is an algebraic fact (due to the classification of unimodular lattices) that $Q$ is necessarily equivalent to a diagonal matrix with $\pm 1$ on the diagonal. All such $Q$ indeed arise as the intersection forms of smooth manifolds (namely, $m\mathbb{C}P^2 \# n\overline{\mathbb{C}P}$$^2$).

2b) If $Q$ is indefinite and even, then it is an algebraic fact (due to the classification of unimodular lattices) that $Q$ is necessarily equivalent to a direct sum $aH \oplus b E_8$, where

\[H = \begin{bmatrix}

0 & 1 \\

1 & 0

\end{bmatrix}\]

and

\[E_8 = \begin{bmatrix}

2 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\

1 & 2 & 1 & 0 & 0 & 0 & 0 & 0\\

0 & 1 & 2 & 1 & 0 & 0 & 0 & 0 \\

0 & 0 & 1 & 2 & 1 & 0 & 0 & 0 \\

0 & 0 & 0 & 1 & 2 & 1 & 0 & 1 \\

0 & 0 & 0 & 0 & 1 & 2 & 1 & 0 \\

0 & 0 & 0 & 0 & 0 & 1 & 2 & 0 \\

0 & 0 & 0 & 0 & 1 & 0 & 0 & 2

\end{bmatrix}\]

If $M$ is smooth, then it is known that $b$ must be even and $|a| > |b|$. If $|a| \geq \frac{3}{2}|b|$, then one can explicitly realize $Q$ as the intersection form of a connected sum $mK3 \# nS^2\times S^2$. It is conjectured that for a smooth four-manifold the inequality $|a| \geq \frac{3}{2}|b|$ must hold in general; the strengthening of the condition $|a| > |b|$ to the condition $|a| \geq \frac{3}{2}|b|$ is referred to as the “11/8-conjecture”.

For those unfamiliar with the classification of unimodular lattices, the exact casework above is unimportant—the point is that unlike in the case of topological manifolds, the question of which lattices arise as the intersection forms of smooth four-manifolds is rather more complicated and involves some peculiar numerology. These results indicate that the theory of smooth four-dimensional manifolds is radically different from the study of topological four-manifolds, a divide that has colored the field to the present day. We should note also that the uniqueness analogue of Freedman’s theorem is significantly less well-understood than the existence part—there is not even a single four-manifold for which an exhaustive list of smooth structures has been proven, and in many examples there are an infinite number of known distinct smooth structures on the same topological four-manifold!

We have now come to the end of our brief history of the divide between continuous and smooth topology in dimension four. In the next post, we will begin introducing some basic ideas from gauge theory itself, but I will give an (extremely vague) overview here. Earlier, we alluded to the difficulty of studying smooth structures on manifolds using classical invariants due to their dependence only on the continuous structure. What was needed was thus a new set of tools which were formulated in such a way so as to explicitly see the smooth structure present on a manifold. The idea of mathematical gauge theory was to take certain partial differential equations from physics and study the moduli space of solutions to these PDEs when defined over the target manifold. These moduli spaces see both the topology of the manifold (in constraining the global solutions) and also, implicitly, the smooth structure (in defining the PDEs). The miracle of this approach is that, for the right PDEs, not only does the moduli space succeed in capturing the smooth structure, but also that it is tractable enough to condense into an invariant with good functorial properties!

]]>Hello and welcome to the September AMS Notices Spotlight. Since the last spotlight, many of you have started a new school year and if you haven’t started yet you are getting ready to start very soon. With that in mind, take a moment before the busyness of the semester sets in and peruse the September AMS Notices. There are many great articles in this month’s notices, including a sampler from the three sectional meetings that are going to occur this fall and several articles about bikes and math. We mentioned in our last spotlight that every issue of the AMS Notices includes a dedicated graduate student section. In the graduate student section, there is usually an interview of someone notable, and at least one article written with graduate students in mind. This month we are highlighting one of these articles.

The spotlight this month falls on the article entitled “What is a Sobolev Orthogonal Polynomial?” written by Francisco Marcellán and Juan J. Moreno-Balcázar. This article provides a brief introduction to the so-called Sobolev orthogonal polynomials, and then discusses the history, various areas that utilize the Sobolev polynomials, certain variations of these polynomials and concludes with a brief outline of the current research into these polynomials. It is a brief article but very understandable and a good read. In order to get the most out of this article, the reader need only have a basic understanding of Lebesgue measure and integration along with a little bit of inner product theory, most which is typically seen in a core analysis sequence at the graduate level. We encourage you to take a moment out of your busy day to take a look at this article, or if not this one, find an article that excites you to look through sometime this month. Good luck with your semester and tune in next month for the next AMS Notices Spotlight!

]]>**Lemma: **Given any ring $A$, a prime ideal $ \mathfrak{p} \subset A$, and a finite collection of ideals $I_j,$ where $j \in \{1, 2, … , n\}$, then if $I$ is the intersection of the ideals, then $I \subset \mathfrak{p}$ implies that $I_j \subset \mathfrak{p}$ for some $j \in \{1, 2, … , n\}$.

A couple of things before we give the proof—the first being that if you have experience in ring theory, half the fun is to try this yourself! It’s a pretty fun thing. Note that the lemma holds trivially true for principal ideal domains by the definition of prime ideals. Moreover, you can use this lemma in scheme theory to show that if you take the finite union of “closed subschemes” in a scheme using a more “scheme/ring theoretic” definition of a closed subscheme is topologically the finite union of the subsets. (Chapter 8 of Ravi Vakil’s excellent, free lecture notes give more information on this topic!)

**Proof of Lemma:** We prove this lemma by induction, noting that the base case is trivial. For the inductive step, by clever use of parenthesis, we may also reduce to the case that there are two ideals, i.e. $n = 2$. Swapping if necessary, assume that $I_1 \not\subset \mathfrak{p}$. Then there exists some $f \in I_1$ such that $f \notin \mathfrak{p}$. Let $g$ denote an arbitrary element of $I_2$. Then $fg \in I_1 \cap I_2 \subset \mathfrak{p}$ by assumption, so either $f \in \mathfrak{p}$ or $g \in \mathfrak{p}$ by the definition of prime ideal. But we assumed that the former didn’t hold, so we conclude $g \in \mathfrak{p}$, that is, any arbitrary element of $I_2$ is in $\mathfrak{p}$. Thus $I_2 \subset \mathfrak{p}$.

Pretty cool right?! For a fun bonus problem, you can check if this holds if $n$ is allowed to take on infinite values. For a really fun bonus problem, you can attempt to prove the Birch and Swinnerton-Dyer conjecture. Best of luck on both!

]]>

Sir Simon K. Donaldson FRS is a British geometer. His honors include the Fields Medal in 1986 “primarily for his work on topology of 4-manifolds, especially for showing that there is a differential structure on Euclidean 4-space which is different from the usual structure,” the Shaw Prize in 2009 with C. H. Taubes “for their contributions to geometry in 3 and 4 dimensions,” knighthood in 2012 “for services to mathematics,” and the Breakthrough Prize in Mathematics in 2014 “for the new revolutionary invariants of 4-dimensional manifolds and for the study of the relation between stability in algebraic geometry and in global differential geometry, both for bundles and for Fano varieties.”

Donaldson received his BA in Mathematics from Cambridge in 1979 and his PhD from Oxford in 1983. His thesis “The Yang-Mills Equations on Kähler Manifolds” was supervised by Sir Michael Atiyah and Nigel Hitchin. Donaldson proved his legendary diagonalizability theorem while still a graduate student.

Donaldson has held academic jobs at Imperial College London, Oxford University, and the Simons Center for Geometry and Physics. His students include Dominic Joyce, Simon Salamon, Paul Seidel, and Richard Thomas.

This month marks his 60th birthday. A conference on symplectic geometry will be held at the Issaac Newton Institute, Cambridge to celebrate this occasion and various advances in symplectic geometry.

Below we mention some major contributions of Donaldson to the ancient science of geometry; in particular, to gauge theory and to Kähler-Einstein metrics. This is by no means a comprehensive account of the life and work of Simon Donaldson. I have tried my best to describe some of his contributions to mathematics without totally embarrassing and discrediting myself; for this to be somewhat successful, it was important to restrict attention to theorems I was more familiar with. Necessarily, this has introduced some personal bias into the post, which I hope the reader will forgive. I have also attempted to avoid overlap with other posts on this blog about gauge theory. In any case, since Donaldson and Atiyah are my biggest inspirations, I hope that the occasion of Donaldson’s 60th birthday is sufficient reason to discuss these topics again!

Donaldson’s diagonalizability theorem is an application of physics to the geometry of 4-manifolds. Previous posts on this blog have discussed Freedman’s theorem, which states that for a closed, simply-connected topological 4-manifold $M$, the intersection form $H^2(M,\mathbb{Z}) \times H^2(M,\mathbb{Z}) \to \mathbb{Z}$ is (up to a binary choice) a complete invariant – every possible intersection form can be realized by some $M$, and at most two such manifolds share the same intersection form. However, smooth 4-manifolds are totally different.

**Theorem **[Donaldson]. Let $M$ be a simply-connected, smooth 4-manifold. If the intersection form of $M$ is definite, then it is diagonalizable. That is, if the intersection form is positive (resp. negative) definite, then it is equivalent to $Id$ (resp. $-Id$) over $\mathbb{Z}$.

Combined with Freedman’s theorem, this proves the existence of non-smoothable topological 4-manifolds (such as the $E_8$ manifold). See [5] for a full account.

**Yang-Mills Theory and 4-Manifolds **

The proof of Donaldson’s theorem relies on Yang-Mills theory, a classical field theory detailed by Yang and Mills in their 1954 Nobel Prize-winning physics paper [11]. Fix a manifold $M$ and a Lie group $G$. If $P$ is a principal $G$-bundle over $M$, we may consider the space of all $G$-connections on $P$, which we denote by $\mathcal{A}$. In local coordinates, these may be thought of as $\mathfrak{g}$-valued one-forms on $M$. A *gauge transformation* is an automorphism of $P$; the group of gauge transformations acts on the space of connections and is written $\mathcal{G}$. The Yang-Mills action functional is defined on the space $\mathcal{A}/\mathcal{G}$ of connections modulo gauge by

$$YM(A) = \int_M |F_A|^2 d\mu.$$

Here $F_A$ is the curvature of the connection $A$ and may be thought of as a $\mathfrak{g}$-valued two-form on $M$. It can be shown that the critical points of the Yang-Mills functional are given by solutions to the *Yang-Mills equations *

$$d^\ast_AF_A = 0.$$

The Yang-Mills equations generalize Maxwell’s equations. For the Abelian group $U(1)$, the curvature of a connection $A$ is just $F_A = dA$. Here $A$ is called the *electromagnetic potential *and $F_A$ the *electromagnetic field. *A computation in local coordinates shows that $A$ is not gauge invariant but $F_A$ is. This corresponds to the physicist’s idea that the electromagnetic potential cannot be assigned a direct physical interpretation whereas the electromagnetic field can. The Yang-Mills equations for $U(1)$ are just Maxwell’s equations.

Maxwell’s equations yield | Hodge theory |

Yang-Mills equations yield | Donaldson theory |

In four dimensions something special happens – the Yang-Mills equations have *instanton *solutions. For a smooth 4-manifold $M$, the Hodge star operator $\ast : \Omega^2 M \to \Omega^2 M$ is an isomorphism with eigenvalues $\pm 1$. Elements of the corresponding eigenbundles $$\Omega^2M = \Omega^2_+M \oplus \Omega^2_-M$$ are called *self-dual* and *anti-self dual (ASD)* forms, respectively. The curvature of a connection similarly breaks up into self-dual and anti-self dual parts $F_A = F^+_A + F^-_A$. (This is also related to the fact that $Spin(4) = SU(2) \times SU(2) = Spin(3) \times Spin(3)$.) When $\| F^+_A\|^2=0$ (resp. $\| F^-_A\|^2=0$) the connection $A$ is called *self-dual *(resp. *ASD*). The space of ASD instantons modulo gauge, the *moduli space, *is the star of Donaldson’s proof.

There is a lot going on here. Whenever ones takes quotients like $\mathcal{A}/\mathcal{G}$ one must worry about “stability”. Also the group $\mathcal{G}$ is infinite-dimensional, so the moduli space can be a bit wonky and a lot of hard functional analysis has to be done. It is this potent synthesis of analysis, topology, geometry, and physics that Atiyah reverently referenced in this post’s opening quotation.

Much of contemporary topology and geometry is literally unthinkable without the legacy of Donaldson theory, which has led to research trends like Hitchin’s moduli of Higgs bundles and Drinfeld’s geometric Langlands correspondence.

**Donaldson-Thomas Invariants**

If $M$ is a 3-manifold, one can define the *Chern-Simons functional *on $\mathcal{A}$ by

$$cs(A) = \int_M\text{Tr}(A \wedge F_A + \frac{2}{3}A \wedge A \wedge A).$$

It can be checked that the critical points of $cs$ are *flat connections; *e.g., connections $A$ such that the de Rham complex $$\Omega^0 \to\Omega^1 \to \Omega^2 \to \Omega^3$$ coupled to the connection on the adjoint bundle induced by $A$ is an elliptic complex.

The Casson invariant is a gadget of low-dimensional topology. In Taubes’ 1990 paper “Casson’s Invariant and Gauge Theory” he reintepreted the Casson invariant as an Euler characteristic of an infinite-dimensional space that counts the flat connections [9].

In the 1998 paper [3] of Donaldson and Thomas “Gauge Theory in Higher Dimensions” (of which there is a 2011 sequel by Donaldson and Segal [4]) they define a holomorphic Casson invariant. These invariants now go by the named of “Donaldson-Thomas invariants”. This is an active area of research.

Their construction hinges on defining a holomorphic kind of Chern-Simons functional. This requires manifolds with holomorphic volume forms – Calabi-Yau manifolds. In analogy to the real Chern-Simons map, the critical points of the “holomorphic Chern-Simons map” are connections on a complex bundle $E$ for which the Dolbeaut complex

$$\Omega^{0,0}\to \Omega^{0,1} \to \Omega^{0,2} \to \Omega^{0,3}$$

coupled to the induced connection on $End(E)$ is elliptic. Such connections are not flat, but rather correspond to holomorphic structures on $E$. On a Calabi-Yau threefold there are a finite number of critical values. So Donaldson-Thomas invariants count the number of holomorphic bundles over a Calabi-Yau threefold of fixed topological type. In more algebraic language, they count the Hilbert scheme $I_n(M,\beta)$ of ideal sheaves with holomorphic Euler characteristic $n$ and homology class $\beta$ on some three-dimensional Calabi-Yau complex variety $M$.

Calabi-Yau threefolds matter deeply to the string theorist – to her, the universe has 10 dimensions and is locally modeled on $\mathbb{R}^4 \times M$ where $M$ is a compact Calabi-Yau threefold. In string theory the Donaldson-Thomas count corresponds to “the number of BPS states” [7].

If $M$ is a complex manifold, then its complex structure induces a field $J$ of automorphisms of the tangent bundle such that $J^2 = -Id$. For a metric $g$ on $M$ there is an associated form defined by $\omega_g(X,Y) = g(JX,Y).$ When $\omega$ is closed (that is, $d\omega = 0$), $g$ is called a *Kähler metric. * When the metric is a constant multiple of the Ricci tensor, the metric is called *Einstein *because if $g = \lambda Ric(g)$ then the manifold satisfies Einstein’s equations. A *Kähler-Einstein metric *is a metric both Kähler and Einstein. Writing these things down is a big deal.

**The Kobayashi-Hitchin ****Correspondence**

Stability is a concept from algebraic geometry, originating with David Mumford’s *Geometric Invariant Theory *(GIT). GIT is essentially the study of quotients. Stability conditions are conditions ensuring the quotient is not something ridiculous that everybody hates.

For example, in the above discussion of gauge theory we did not want *all* connections, because the action of the gauge group did not really affect the things we cared about. Thus, we quotiented out by this symmetry. The space left standing is the “space of moduli” or in more common parlance the “moduli space”.

People like to think about moduli spaces of bundles. So there is some notion of a stable bundle. Kobayashi and Hitchin (independently) conjectured that the moduli spaces of stable vector bundles and Einstein-Hermitian vector bundles over a complex manifold were equivalent. This was proved by Donaldson for algebraic manifolds [2],[6], and for Kähler manifolds by Uhlenbeck and Yau [10].

**The Calabi Conjecture for Fano Manifolds **

The “Calabi Conjecture for Fano Manifolds”, often called the Yau-Tian-Donaldson conjecture, is an influential Kobayashi-Hitchin type conjecture that was proved by Chen, Donaldson, and Sun in 2012.

Poincaré’s celebrated uniformization theorem says that there is a unique metric with constant scalar curvature in each Kähler class on a Riemann surface. Calabi conjectured a far-reaching generalization of this theorem, as follows. Suppose the first Chern class of $M$ is a constant multiple of its Kähler form $c_1(M) = \lambda \omega \in H^2(M,\mathbb{Z}).$ Then Calabi conjectured there is a Kähler-Einstein metric cohomologous to $\omega.$

The cases $\lambda < 0$, $\lambda = 0$, and $\lambda > 0$ are quite different. The case $\lambda < 0$ was solved by Aubin and Yau. The case $\lambda = 0$ was solved by Yau in the 1970s. These are the famous “Calabi-Yau manifolds” – Ricci flat Kähler-Einstein manifolds (I know, before I said that meant there was a holomorphic volume form, but $c_1(M)=0$ kind of does that for you).

Manifolds with $\lambda > 0$ are called *Fano manifolds. *In the spirit of Kobayshi-Hitchin, Yau-Tian-Donaldson conjectured that a Fano manifold will admit a Kähler-Einstein metric if and only if it is “stable” for some suitable notion of stability. The correct notion of stability is called “K-stability.” In a series of 2012 papers Chen, Donaldson, and Sun proved the following [1]:

**Theorem **[Chen-Donaldson-Sun]. A Fano manifold admits a Kähler-Einstein metric if and only if it is K-stable.

References

[1] X. X. Chen, S. K. Donaldson, S. Sun, *Kähler-Einstein Metrics and Stability. *arXiv:1210.7494. 2012.

[2] S. K. Donaldson, *Anti-self-dual Yang-Mills connections on complex algrebraic surfaces and stable vector bundles*, Proc. London Math. Soc. 3 (1985), 1-26.

[3] S. K. Donaldson and R. Thomas, *Gauge theory in higher dimensions*, The Geometric Universe, Oxford UP, 1998, pp. 31-47.

[4] S. K. Donaldson and E. Segal,* Gauge theory in higher dimensions, II*, Surveys in Differential Geometry 16 (2011), 1–41. arXiv:0902.3239.

[5] S. K. Donaldson and P. B. Kronheimer, *The geometry of four-manifolds*, Oxford U.P., 1990.

[6] S. K. Donaldson, *Infinite determinants, stable bundles and curvature*, Duke Math. Jour. 54 (1987), 231-47.

[7] D. Joyce and Y. Song, *A theory of generalized Donaldson-Thomas invariants*, arxiv/0810.5645.

[8] G. Tian, On *Kähler-Einstein metrics on certain Kahler manifolds* with c_1(M) > 0. Invent, math. 89 (1987) 225-246.

[9] C. H. Taubes, *Casson’s invariant and gauge theory*, Jour. Differential Geometry 31 (1990), 363-430.

[10] K. K. Uhlenbeck and S-T. Yau, *On the existence of hermitian Yang-Mills connections on stable bundles over compact Kahler manifolds*, Commun. Pure Applied Math. 39 (1986), 257-93.

[11] C. N. Yang and R. Mills, *Conservation is Isotopic Spin and Isotopic Gauge Invariance, *Phys. Rev. 96, 191 (1954).

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In this post, I want to address what to do when a group gets stuck. While using groupwork in my classes, my first instinct when I notice a group that seems to be stuck on a problem is often to step in and tell them how to do that problem. When I do step in, however, it seems to cut off the discussion that was going on in the group and instead shifts the discussion from being student-centered to teacher-centered.

One of the things I do, therefore, is to not jump in right away. Instead, sometimes it is better to wait until the group asks for help or until I sense that they have moved from productive struggle into unproductive struggle. There are several techniques that can be used when interacting with a group. One common approach is to “answer a question with a question”, which helps to set a norm that students are to do their own mathematical thinking. If students are so stuck that they do not have any well-formed questions, it is often useful to ask them to summarize what they have tried so far. It is not uncommon that when group members try to explain, they will find a new path on their own, and you can leave them to explore these new ideas by themselves. It is perfectly okay to tell a group, “why don’t you give that a try, and I’ll be back later”; often, this is better than micromanaging the group.

Another option when one sees multiple groups that are stuck on similar things is to call a “huddle”. In a huddle, one student from each group comes up to talk with the teacher. An easy way to do this is to call for all the students of a specific group role (if you have assigned student roles such as facilitator, recorder, etc.; I’ll discuss this more in future posts). One then can give the students in the huddle a hint or piece of information to take back to their groups.

What if it seems that specific students are not participating on a consistent basis in their groups? This is actually a very common problem. There are many different techniques that have been developed to equalize participation. In my next post, I will discuss what to do when there are particular students that are not participating in a group.

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As mathematicians, there are myriad concepts we think about every day that most people do not. Sometimes this is because they are concepts not known about by non-mathematicians and sometimes it is because they are not pertinent to them. But infinity? Everyone knows about infinity, but no one really thinks about it. Or at least that was my hunch. And, based upon my small [read: VERY small] sample size, I was right. I started texting my close friends outside of the math department, asking them how often they think about infinity. Some said rarely, some said never. This blew my mind. Not because it surprised me, but simply because I couldn’t imagine what life was like without this thought in the back of my head at all times – that there’s always more.

This led me to ask myself how I think about infinity. As a child, infinity meant that I could name a number after myself. I remember telling my parents “If numbers go on forever, then I can name one ‘Deborah-illion’ and it must exist.” As I thought about my current understanding of infinity, I realized I had not progressed much, despite my mathematical studies, because my first thought was “there’s always one more.” But then I realized that this implied my concept of infinity was a countable infinity. So I started to think about uncountable things, and I realized that I think about that as always being able to “stuff more in between” – like how I can choose two real numbers and there are infinitely many more real numbers between them. And then I was overwhelmed with how inadequate my perception of infinity is.

You see, I believe that we as mathematicians fall into this false sense of security thinking that we understand infinity – and justifiably so. We can prove when certain properties hold for finite-dimensional vector spaces but not for infinite-dimensional vector spaces. We can think about limits as some variable tends to infinity and even understand this well enough to explain it to our students. We talk about it over coffee and write it in our proofs. We have a cute little symbol that we have mastered writing (but let’s be honest, that took some of us a while) and a Latex code that we all know by heart. We have taken something so big (is big even the right word?) and stuffed it into a small, tangible little package so that we can carry it around in our finite brains and feel like we know something.

But I think that’s beautiful. Because it’s yet another example of “the more you know, the more you don’t know.” It reminds me that this whole math thing is way out of my reach, but that if I keep reaching and searching for the little things I can understand – the lemma on page 23 or the definition at the beginning of lecture – there is hope that eventually I will be able to use those to prove another lemma, and maybe that will lead to a theorem. It reminds me that there’s always something left to prove, always something I can understand a little better, and that this thing which I have chosen to dedicate my life to will be there as long as I choose to pursue it. Recently I realized how well this concept applies to other areas of life – for me, it’s my faith and the peace that comes along with knowing that while I’ll never understand all of it, I can cling to what is within my reach and hold on. Maybe for you it’s something else. Either way, I think this is a conversation we need to have. So go to your departmental tea and start a conversation about infinity with your colleagues. Talk to your family and your friends and your barista. Ask them how they think about infinity. I would love to hear what you find out.

]]>I was lucky to be an English teacher back home, before coming to US as a graduate student. The English teaching community is much more self-aware of teaching methodologies. For example, to become an English teacher, even if you are a native speaker, you must attend Teachers’ Training Courses (TTC) and get a certificate. This forced me to read about both theoretical and practical aspects of teaching. In what follows I will share some practical advice I have benefited immensely from in my own classrooms.

Let’s say you have been assigned a Calculus I course to teach for this coming Fall. What are your goals, your objectives for this course? “To teach students calculus,” is not a SMART goal. A SMART goal must be (according to https://www.mindtools.com/pages/article/smart-goals.htm):

**S**pecific (simple, sensible, significant)**M**easurable (meaningful, motivating)**A**chievable (agreed, attainable)**R**elevant (reasonable, realistic and resourced, results-based)**T**ime bound (time-based, time limited, time/cost limited, timely, time-sensitive)

“To teach calculus,” is not a measurable target. “My students will be able to differentiate quotients of elementary functions,” is specific, measurable, achievable, relevant (to the whole structure of the course), and time-based (once we get that it will be covered on quiz 3). “I want to get a B in this course,” is in fact quite a SMART goal for a student you tutor.

Having SMART goals agreed to by your student(s) is especially important when individuals pay you directly—you need some proof of what you helped them attain in case you need to.

Having such documents in your teaching portfolio will give a potential employer an impression of a solid organized teacher.

SMART goals will help with writing fair and on-point exams and quizzes. One complaint I received from my students in the teaching survey last year was that my “exams were not consistent.” And they were right: one time I would test concepts, the other time I would rely on computations.

Finally, having set SMART goals helps decide what material to focus on and what material can receive a lighter weight. This semester, for instance, I realized that spending a whole 1.5 hour session on delineating Riemann Sums does not really fit into the overall objectives of a Business Calculus course—at least the objectives I had set for that course. Instead, I was able to allocate that time to Gini Index and Consumers’ Surplus.

To help write SMART goals, start with “students will be able to + action verbs.” Avoid “Students will appreciate…” Appreciation is not easily measurable. Action verbs from Bloom’s Taxonomy is a great way of synching your expectations with the students’ skills. Choose the appropriate column from the taxonomy according to the complexity level of the course, and use verbs from that column in setting goals. These verbs can also be used in phrasing exam questions in a clear and non-confusing way.

Setting SMART goals makes your life easier—you have a mission, you are focused, the students are focused, and your success is measurable, quantifiable. The time that it saves you later is significant (when writing exams, for example) and the confidence you and your students have in the efficiency of the teaching is boosted. Let me finish by this excerpt from my Business Calculus course:

Endnotes:

- SMART goals can be applied to any context other than teaching. I have found that when I set my goals more thoughtfully I am more likely achieve them. I write them on a piece of paper, and draw little squares next to them. I check each one that I accomplish!
- I may follow this article by one on “Components of Class Activities.”

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