]]>

*As always, there’s lots of thoughtful replies in the comments below, including many who disagree with me. Worth a read!*

]]>

At a conference like the HLF—bringing together researchers from across diverse fields—you’re bound to run into a few turf wars.

Mathematician vs. computer scientist.

Mathematician vs. physicist.

Even—in one delicious exchange on Tuesday—mathematician vs. mathematician.

In his morning talk, Sir Andrew Wiles emphasized a fundamental change in his field of number theory over the last half-century: its move from *abelian* to *non-abelian *realms.

Afterwards, Michael Atiyah—fellow mathematician and fellow Sir—rose to comment. After praising a “brilliant talk,” he started to redraw the intellectual boundaries.

“The whole idea of doing non-abelian theory permeates not just number theory,” Atiyah said, “but physics and geometry and vast parts of mathematics. What we’re really looking for is an overall unification in some distant future.”

Wiles mostly agreed, then laughed: “We’ve had this discussion before.”

“When I gave a lecture, probably 25 years ago,” Wiles said, “Michael told me that the future of number theory was to be subsumed by physics.”

Wiles smiled at the memory. “I was a little taken aback by this. It wasn’t what I planned for my future.”

But Wiles got the last laugh. “[Physicist] Cumrun Vafa came up to me afterwards,” Wiles explained, “and said, ‘Don’t worry about it. It’s the other way around.’”

*This is my last post from the Heidelberg Laureate Forum.
Back to your regularly scheduled bad drawings next week!*

]]>

In 1984, the legendary Alexander Grothendieck released one of the greatest mathematical texts of the century: “Esquisse d’un Programme.”

It became a viral hit. Like an epic novel, it painted a sweeping vision of the blossoming field of algebraic geometry. The topic would soon come to dominate research mathematics, and Grothendieck pointed the way.

The young Vladimir Voevodsky was desperate to read it. There was just one problem: it was in French, and Vladimir didn’t speak French. So he did what any ordinary person would do.

He waited for the translation.

Kidding! Of course he didn’t. Vladimir is not an ordinary person; he is a research mathematician. More than brainpower, I find researchers are defined by their singular drive, their obsessive passion. So Vladimir did what *no* ordinary person would do.

He began teaching himself French for the sake of reading a single text.

He succeeded. Before long, he was helping to pursue the path of research that Grothendieck had outlined. And in 2002 his work led to mathematics’ most prestigious prize: the Fields Medal.

Learning a language is hard.

Learning a language for one book is crazy.

But sometimes, crazy is worth it.

]]>

Early in his talk, computer scientist John Hopcroft noted a funny fact about clustering algorithms: they work better on **synthetic** data than **real** data. But this is more than an odd tidbit about software.

It’s an insight into the nature of our world.

When we invent our own synthetic data, we try to mimic real data by mixing true information with random distraction–combining “signal” with “noise.” But in real data, the divide isn’t so clear. What often looks like noise turns out to be the deep structure we haven’t grasped yet.

**The noise is just signals you can’t yet hear.**

Hopcroft’s insight: data doesn’t just have one structure. It has many. If I scanned notebooks from a hundred people, and made a database of all the individual letters, I could sort them lots of ways. Alphabetically. Capital/lowercase. Size. Darkness. Handwriting. Each of these is a different layer of structure.

And to understand data–and the world–you’ve got to reckon with all those layers.

Here’s the approach Hopcroft outlined. First, cluster your data according to its primary, surface-level structure. Then, wipe the slate—scramble this structure—and run your algorithm again, to find the *hidden* structure.

Uncover the signal you thought was noise.

For example, Hopcroft and his colleagues ran their algorithm on **Facebook data from Rice University**. They had sparse information: no names, no profiles, just who was friends with whom—a skeleton network of connections. Based on this, their algorithm quickly sorted the students into **nine clusters**.

But was it merely random, algorithmic gibberish? After all, the computer knew *nothing* about the students, other than the friendship connections. Did the nine clusters have any actual *meaning*?

Sure. They corresponded precisely to the nine **student dorms**.

Then they weakened that structure, and ran the algorithm again. This time it produced only **four clusters**. The computer, of course, had no idea what these clusters meant—it only knew that friendships at Rice reflected a hidden four-group structure. Those groups?

**Freshman**, **sophomore**, **junior**, and **senior**.

The most tantalizing fact: there were two more layers of structure revealed by the algorithm, but it remains a mystery what they are. In the lives of those undergraduates, among everything we call “noise,” there are still hidden signals.

*You can see the entire talk here.*

]]>

Mathematics is lonely work. Or so the romantic stereotype has it: the lone genius in an empty library. The sage on the mountaintop. Andrew Wiles in the attic.

But most mathematical work is profoundly collaborative.

I caught four young researchers between events, and gave them the prompt: *On one piece of paper, show me the essence of good collaboration*.

Their drawings? Four different flavors of brilliant.

First, from **Ana Djurdjevac**, born in **Serbia** and now studying **partial differential equations**:

In pursuing PDEs, Ana perhaps missed her other calling: as a painter specializing in stark symbolism.

“First, you need different types of people,” she explained. “Men and women. Standing and sitting.”

“Gray and purple,” I added.

“They all share the same space,” she said, pointing to the stage-like center of her drawing.

“Their own Eden,” I said.

“The sun and the moon represent day and night,” Ana said.

“So they work hard?”

“Yes, and each person plays a different role,” she said. Then Ana pointed to her stick figures, from left to right. “He is listening. She is teaching. He is asking a question. And she is angry.”

“Angry?” I asked. “Is that important for collaboration?”

“Oh yes!” Ana said. “Someone must bring the anger.”

Second, from **Lashi Bandara**, from **Australia** and now studying **harmonic analysis**:

Lashi is effusively social, with a fondness for vulgar humor—I had to warn him to keep it PG-rated.

He was disappointed, but resilient. “Can I at least draw a beer?” he asked.

Lashi produced three playful sketches showing the three pillars of good collaboration, Bandara-style.

First, *communication*: sharing a blackboard, and sharing thoughts.

Second, *socializing*—although Lashi mused that his handwriting looks more like “socialism.” “That also works,” he said.

Thirdly, *diversity*—for which Lashi drew two topologically distinct objects, representing topologically distinct people. “So that torus-person cannot be smoothly deformed into a sphere-person?” I said.

“And that’s the essence of diversity!” Lashi agreed.

Third, from **Chenhao Tan**, born in **China** and now studying **social networks**:

As befits his academic interests, Tan laid out two different graph theoretic models of collaboration: *efficient* and *inefficient*.

In an efficient collaboration, everyone is connected, offering compliments and constructive discussion.

But in an inefficient one, the hierarchy is strict. (Graph theorists would call it a *tree*.) Authoritarianism reigns, and there is no goodwill between collaborators.

Personally, if I ran an MBA program, I would totally build Tan’s illuminating charts into the coursework.

And finally, from **Helena Andre Terre**, born in **Spain** and now studying **computational biology**:

Helena’s cartoon is equal parts cynicism and inspiration.

“So the guy with the roller skates is just coasting, isn’t he?” I asked.

“No,” she insisted. “You need him. The collaboration needs someone providing the carrot.”

“So the carrot is like the question being asked?” I said.

“It could be,” Helena replied. “It’s whatever drives the project forward, gives it a goal.”

“And which one are you?” I asked.

Helena, still working on her PhD at Cambridge University, pointed to the one on the bike. Then she smiled. “But, someday…” she said, and her finger moved towards the other figure.

]]>

At most scientific conferences, you find a cross-section of ages: elder statesmen, rising stars, mid-career workhorses, maturing postdocs, and fresh-faced PhD candidates. The HLF brings together the two extremes: the most legendary of the legends, and the most bright-eyed of the youngsters.

What do such disparate groups have to talk about?

A lot, it turns out.

During the opening ceremony, **Jean-Pierre Bourguignon**—the president of the European Research Council—told a story from his own days as a young mathematician.

“In 1973, I was spending the summer at Stanford University,” he explained. Fancy post at a prestigious university—he must have been pretty cocksure? Not exactly. He felt like most young researchers: a little anxious, a little unsure.

“At that time,” he admits, “I had not produced much.”

Then, out of the blue, he got a lunch invitation from an eminent researcher across the bay at UC Berkeley. He was dumbfounded by his good luck. “I started to wonder why on earth this world-famous mathematician would want to talk to me,” Bourguignon said. “I had met him briefly, only once.”

When he got to the lunch date, he found out: “He was simply curious to know what my projects were.”

A casual gesture from the world-class researcher—but a transformative moment for this young scholar. “Maybe,” Bourguignon reflected afterwards, “what I was trying to do had some value after all.”

And what did the eminent scholar get out of it? Lots, apparently! He often made similar invitations to young researchers. Their fresh and daring thoughts nourished his mature, considered ones.

That’s the HLF, in a nutshell: bringing the youngsters and the legends together, for the enrichment of both.

]]>

The joke among bloggers over breakfast: What award *hasn’t* Atiyah won?

(My suggestion: the gold glove? But maybe he’s won that, too.)

A Fields Medalist and Abel Prize recipient, he is a living legend: his index theorem (developed with Singer) revolutionized both mathematics and quantum physics.

In a sweeping talk on Monday morning, he leapt so nimbly from the upper echelons of abstraction to the gritty details of reality that you began to realize that they are one and the same. The purity of mathematics, the practicality of computer science—they’re interwoven.

Here are 21 quotes that capture the experience of hearing this knighted mathematician, a thinker of extraordinary depth and curiosity.

First, he introduced himself as our tour guide through the last century of theory in mathematics and computer science:

**I’m as new as the hall.**

*on lecturing in the so-called New University Building, built in 1929—the same year as Atiyah’s birth*

**I’m going to be your tour guide of the last century. But don’t believe everything the tour guide says. They exaggerate.**

*a self-effacing introduction to his far-ranging talk*

** **

**My first slide is Heidelberg in 1904. I don’t think many of you were here.**

** **

**Among [Hilbert’s challenges] were questions like, ‘Find the mathematical foundations of physics.’ You know, small questions.**

*on the ambitious questions laid out by David Hilbert in 1904*

** **

**There’s a picture of [Alan Turing’s code-breaking machine] the Colossus, with two young ladies doing all the work. There were lots of ladies doing all the work.**

*on Alan Turing’s work in Bletchley Park*

** **

**You can’t get more famous than that: You win a million dollar prize and turn it down. You get even***more*publicity that way.

*on Grigori Perelman turning down the Millennium Prize*

** **

**These are the grandfathers, the prophets, upon whose work computer science is based. They are almost gods.**

*on legends like Turing, Kurt Gödel, and John Von Neumann, whose theoretical mathematics paved the way for computer science*

Then, Atiyah began to explore the tensions between theory and practice.

**I had a banker who came into my office one day and said, ‘My shares went down 30 percent! I’m broke and it’s your fault!’ But mathematicians just give information. It’s not our fault.**

*on the role of mathematics*

** **

**I’m sure up there**[*he points upwards*]—**I don’t mean in heaven, I mean up in the ceiling—there are bugs.**

*on the ubiquity of governmental ‘data gathering’*

** **

**Astronomy: if you’re getting within 10 orders of magnitude, you’re okay.**

*on the different meanings of “solution” in different fields*

** **

**What is a solution? It entirely depends on the customer.**

*connecting the philosophical question ‘What is it to solve a problem?’ to the gritty practicalities of actual work*

** **

**Your computer’s fast, but it can’t actually travel faster than light.**

*on the limits of computation*

** **

**‘Money’ is just a word to describe resources. Don’t think I’m being mercenary. Without resources, nothing happens.**

*on practical considerations in research*

** **

**It’s worse than moving the goal-posts. In this game, you start playing football, and by the end you’re playing ping-pong. And the Chinese always win at ping-pong.**

*on how new advances in computing can alter research*

He ended with a series of stirring meditations on the purity and beauty of mathematics:

**Perfect spheres do not exist in the real world, but they do have reality. They exist in the human imagination—and that’s the most important world there is.**

*on Platonism*

** **

**Plato has infinite time and money and can outrun Moore. In the ideal world, time has no boundaries, and we can travel faster than light.**

*on the potential of theory to escape the constraints of practice*

** **

**You do it every day without realizing it: you break the barriers of space and time.**

*on the power of human imagination*

** **

**We as mathematicians don’t have to be apologetic about saying we like beautiful things. We live on beautiful things.**

*on aesthetics in mathematics*

** **

**You may have heard about it—there are things called spheres. The two-dimensional sphere has been known for a long time. You can play football with it.**

*on the practical uses of low-dimensional geometry*

** **

**We mathematicians believe in brevity, so when I wrote a poem, it was a short poem. Don’t worry.**

*introducing a poem to close the lecture*

** **

**Under the full moon, they dream.**

*from his poem on mathematicians*

]]>

A combinatorist working in India, **Manjil Saikia** is soft-spoken and super-knowledgeable: we chatted about Isaac Asimov and the history of the Fields Medal before getting into his passion project, which (like mine!) is a blog.

It’s called **Gonit Sora**.

That’s Assamese for “**Gateway to Math**.”

As a monolingual American, it’s easy to forget just how easy I have it. My native tongue happens to be the global language. Case in point: I blundered into Germany yesterday not speaking a word of German. No problem! For me, provincialism carries no penalty.

But for Manjil, growing up in northeast India, he had to fight for access to knowledge in a world that catered far better to folks like me. When the internet arrived in his home at age 18, it was a revelation—but even online, he had to leap linguistic hurdles. There were almost no sources on math and science in his native Assamese.

It was English or bust.

As an adult, Manjil (and his blog’s other co-founder) are trying to change that. For the 15 million speakers of Assamese—as many as Swedish and Finnish combined—Gonit Sora really is the gateway to mathematics, the only blog of its kind. It presents interviews, explains ideas, and tries its best to deliver the mathematical experience to Manjil’s home community.

They call math “the universal language.” But that’s not a guarantee. The ideas of mathematics are universal, yes, but they perish unless housed in human minds. It helps to speak to those minds in the language they know—whether it’s English or Arabic, Afrikaans or Assamese.

]]>

One of the hardest things about research in technical fields: Explaining what the heck it is that you do.

The natural sciences have it easy: they study physical, tangible things. Perhaps those things are weird and exotic (bosons, mRNA, kangaroos, etc.) but hey, at least they’re *things*.

Mathematicians and computer scientists face a taller order. They study concepts, processes, algorithms. The “things” they research aren’t really *things *at all: they’re creations of rigorous human thought, abstract structures of logical language.

Not so easy to explain.

So as they sipped on coffee and Coke, waiting for the HLF opening ceremony to begin, I ambushed seven young researchers and goaded them into explaining their work to me. Characterizing your specific research can be simply too hard, so I gave them a slightly broader invitation: *On a single piece of paper, illustrate what your research area is about*.

Here is what they (very gamely!) contributed:

From **Tetiana Klychmuk**, studying **linear algebra** in Ukraine:

From **Opeyemi Aborisade**, studying **cryptography** in Senegal:

From **Mariia Fedorova**, studying **automata** in Ukraine:

From **Collins Amburo Agyingi**, studying **topology** in South Africa:

From **Gilbert Bernstein**, studying **computer graphics** in the USA:

From **Pacome Ambassa**, studying **information security** in South Africa:

And from** Haji Ali**, studying **mobile health systems** in South Africa:

]]>