# The toughest math problems that challenge the world

In 1900, German mathematician David Hilbert proposed a list of 23 math problems that would change the world. Some have been solved. Others remain. DARPA attempted to update the list a few years back. Here are the highlights.Solving what look to be "unsolvable" math problems has been a hot topic of math and science connoisseurs for a long time. It also impacts popular movie culture—remember "Good Will Hunting" and more recently "Gifted."

The idea that amazingly difficult, conceptual, unsolvable math problems could change the world can in part be traced back to 1900 when German mathematician David Hilbert proposed his still influential 23 math problems that would change the world.

According to the School of Mathematics and Statistics at the University of St Andrews in Scotland, Hilbert began his talk with these words (translated into English), which still ring true in the world today:

“Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be towards which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?”

### Problems solved from Hilbert's original list (and who solved them)

Some of the 23 problems proposed at the time were solved, while some remain. This “Math Is Good for You” post offers a glimpse of a few of the problems and who is credited with solving them:

- "Is there a number, which is larger than any finite number, between that of a countable set of numbers and the numbers of the continuum?" To think of a continuum, think of a number line—and
*all*the numbers on it—without any gaps. This problem was answered by Kurt Gödel. - "Can it be proven that the axioms of logic are consistent?" Gödel also answered this problem with his "incompleteness theorem," which states that all consistent axiomatic formulations include some undecidable propositions. For more, see the short history of Euclidean and non-Euclidean geometries.
- "Give two tetrahedra that cannot be decomposed into congruent tetrahedra directly or by adjoining congruent tetrahedra." Max Dehn showed this could be done, but he had to invent his own invariants (something that does not change under a set of transformations).

### DARPA updates Hilbert's list

A few years ago, the researchers at the U.S. government’s Defense Advanced Research Projects Agency (DARPA) began a concerted effort to update Hilbert’s list and develop a new Mathematical Challenges program for the 21st century. DARPA proposed 23 updated questions that, if answered, “would offer a high potential for major mathematical and scientific breakthroughs.”

Anthony Falcone, now president and chief technology officer at Functor Reality, was the DARPA program manager during the last couple years of the Math Challenges program, which the agency ended in 2012. “At the time, I think the program was more a consciousness-raising effort for mathematics. The idea was to get as many good, smart people as possible thinking about new ways to solve problems,” Falcone says. “We certainly moved the needle.”

According to DARPA, six projects were funded through 2012 to address five of the 23 challenges. According to program documents, the levels of success in the five individual challenges varied.

A simplified look at these complex findings includes:

**21 ^{st} Century Fluids:** Developed a software implementation based on the extended von Neumann’s formula for cell growth rate from two dimensions to three dimensions.

- Used the software to extract statistical information from grain systems that reached the asymptotic statistical state. Since the simulation was several orders of magnitude larger than anything done before, the error bars for the extracted statistics are the smallest existing.
- Did an analysis on two-dimensional structures, three-dimensional structures, and two-dimensional cross-sections of three-dimensional structures.
- Studied the random topological structure of grain networks and foams, and established the existence of phase transitions based on changes in the underlying parameter.

**Riemann hypothesis:** Generalized the *p*-adic local monodromy theorem to arbitrary differential modules.

- Developed alternate constructions for Fontaine's rings in
*p*-adic rings.

**Hodge conjecture:** Proved that every projective K3 surface over an algebraically closed field contains infinitely many rational curves.

**Arithmetic Langlands:** Discovered the appropriate analog of complex conjugation over a finite field for “ordinary” elliptic curves and “ordinary” abelian varieties.

- Devised and analyzed a class of pseudorandom sequences.

**Algorithmic origami:** Solved the “carpenter’s rule problem” regarding the straightening of polygonal chains of line segments.

- Developed novel approaches for the accurate modeling of hydrogen bonds using the bond geometry and a well-suited mechanical model.
- Extended proofs of Delaunay realizability (for triangulations) from maximal outerplanar graphs to arbitrary outerplanar graphs.

### DARPA's math challenges still waiting to be solved

Other challenges remain unsolved. Among them are:

- The mathematics of the brain: Develop a mathematical theory to build a functional model of the brain that is mathematically consistent and predictive rather than merely biologically inspired.
- The dynamics of networks: Develop the high-dimensional mathematics needed to accurately model and predict behavior in large-scale distributed networks that evolve over time, occurring in communication, biology, and the social sciences.
- What are the symmetries and action principles for biology? Extend our understanding of symmetries and action principles in biology along the lines of classical thermodynamics, to include important biological concepts such as robustness, modularity, evolvability, and variability.
- Geometric Langlands and quantum physics: How does the Langlands program, which originated in number theory and representation theory, explain the fundamental symmetries of physics? And vice versa?
- Capture and harness stochasticity in nature: Address Mumford's call for new mathematics for the 21st century. Develop methods that capture persistence in stochastic environments.

### Applying new and/or underutilized mathematics to real-world problems

It is this concept of bringing stochasticity, or randomness, into the fundamentals of mathematics, technology, and science that still has the opportunity to shape future developments of all kinds.

“Stochasticity should be the bedrock of math and science because it would fundamentally change the way we deal with problems in engineering, physics, and other research areas,” Falcone says. “It would have the biggest impact on technology.”

Falcone’s Functor Reality firm continues that sort of leading-edge work. The company’s website describes its role as aiming to apply new and/or underutilized mathematics to real-world problems. “It is founded on the belief that revolutionary advances will emerge from the introduction of more sophisticated mathematical techniques into technology,” Falcone says.

There continues to be an interest in solving the truly toughest math problems. Probably the most notable are the seven Millennium Prize Problems put forth by the Clay Mathematics Institute in Peterborough, New Hampshire. “The prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude,” according to the group’s website.

It is notable that one of the seven Millennium Prize Problems—the Riemann hypothesis, formulated in 1859—also appears on DARPA's list as well as in Hilbert's address from August 1900.

That’s one tough problem.

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