# Why “Competitive Marketplaces” are an Awkward Fit for Education

# Who Subsumes the Subsumers?

*a final dispatch from the fourth annual Heidelberg Laureate Forum*

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 You Learn a New Language to Read a Single Book

*a dispatch from the fourth annual Heidelberg Laureate Forum*

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.

# Sometimes the Noise is Signals, Too

*a dispatch from the fourth annual Heidelberg Laureate Forum*

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.

# Collaboration (in Cartoons)

*a dispatch from the fourth annual Heidelberg Laureate Forum*

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. Continue reading

# The Essence of the HLF, in One Story

*a dispatch from the fourth annual Heidelberg Laureate Forum*

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.