June 23, 2014 was a very nice day for five mathematicians: Simon Donaldson, Maxim Kontsevich, Jacob Lurie, Terence Tao and Richard Taylor. They were informed that they would be receiving the inaugural Breakthrough Prizes in Mathematics, each with a cash award of USD$3,000,000. They plan to share their good fortune.

The international, if still Euro-centric, nature of mathematics is obvious when we list where the five were born, educated and currently work:

Name | Born | Educated | Current appointment(s) |

Donaldson | Cambridge, UK | Cambridge, UK; Oxford, UK | Stony Brook, USA; Imperial College, UK |

Kontsevich | Khimki, Russia | Moscow State, Russia; Univ. Bonn, Germany | IHES, France; Univ. Miami, USA |

Lurie | Washington, DC, USA | Harvard, USA; MIT, USA | Harvard, USA |

Tao | Adelaide, Australia | Flinders Univ., Australia; Princeton, USA | UCLA, USA |

Taylor | Cambridge, UK | Cambridge, UK; Princeton, USA | Harvard, USA |

Adelaide, Australia native Terry Tao is perhaps the most international of the group (and is also arguably the most widely known and most respected). His father grew up in Shanghai; his mother is Cantonese by ethnicity, and they were first-generation immigrants from Hong Kong to Australia. At the tender age of 16, Tao received both a bachelor’s and a master’s degree from Flinders University in South Australia. After receiving his Ph.D. from Princeton at 21, Tao joined UCLA, where at the age of 24 he was promoted to full professor, the youngest mathematician ever appointed to that rank at UCLA.

Terry Tao is a frequent return visitor to Australia and is a very vigorous supporter of Australian mathematics and mathematical education. Indeed, he will headline the 2015 Australian Mathematics Meetings, which are to held at Flinders University, his alma mater, in late September.

The Breakthrough Prizes in Mathematics complement the Breakthrough Prizes in Fundamental Physics, which were inaugurated in 2012, and the Breakthrough Prizes in Life Sciences, which were inaugurated in 2013. In future years, there will be one award in mathematics, one award in physics, and six in life sciences. Each of the eight annual awardees will receive USD$3,000,000, as in the inaugural prizes.

As more good news for Aussi science, the 2015 Breakthrough prize in physics announced on November 9th was shared by Australian Nobel astrophysicist Brian Schmidt.

Funding for the Breakthrough Prizes is provided jointly by Russian Internet billionaire Yuri Milner, who founded the prizes, and Facebook founder Mark Zuckerberg. The intent is to establish awards of stature comparable to or exceeding that of the Nobel prizes that are now granted in physics, chemistry, medicine, literature and peace (but not mathematics). The Fields Medal, often called the Nobel of mathematics, carries a very small cash award and are given every four years. Donaldson, Kontsevich and Tao have all three received Fields Medals.

On 10 November 2014, the five inaugural recipients of the Breakthrough Prize in Mathematics were honored at a symposium at Stanford University. Titles and abstracts of the talks they presented are given below. For a summary of their prize-winning contributions, see our earlier Math Drudge blog.

#### Panel discussion

The talks were preceded by a very interesting panel discussion, hosted by Yuri Milner, the Russian billionaire co-sponsoring the awards, who posed questions to the recipients. Here are some notes, roughly paraphrased from the discussion; attribution might not be correct in all cases:

- Q: Is mathematics a human invention, or is it discovered?
*Taylor*: Yes, I definitely feel that I am discovering mathematical ideas that are already in place. Many of us suspect that a certain fact is true long before we can actually prove it.

- Q: If some day we contact alien civilizations, will they have the same mathematics as us?
*Tao*: I certainly think so.*Taylor*: I agree. While the language and notation may differ, it is hard to see how the basic structure of mathematics would be much different.

- Q: Can mathematics be “unified”? Will mathematical discovery come to an end?
*Lurie*: Mathematics is already “unified” in the sense that almost all mathematics is based on a set of axioms that have been fixed for several decades.*Others*: In any event, there is no sense that mathematics is coming to an end — with every advance, new questions are posed.

- Q: We still don’t know if the ABC conjecture has actually been proved; should we expect that there will be an increasing number of cases where a relatively long time will elapse before we are confident that some new result is actually sound?
*Several*: This may well be true, but for a variety of reasons. For example, studying the proposed proof of the ABC conjecture by Shinchi Mochizuki is hampered by the “strange” style that Mochizuki used. In the case of Perelman’s proof of the Poincare conjecture, two years elapsed, with contributions from several other mathematicians, before the full proof was finally settled.*Kontsevich*: The refinement and cleaning up of earlier, more complicated stories is an important and undervalued contribution in mathematics.

- Q: Will we eventually be able to verify every new paper, say with an advanced computer-based system?
*Tao*: I hope so. Perhaps at some point we will write our papers not in LaTeX but instead directly in some formal mathematics system.

- Q: Is the collective intelligence of mathematicians moving forward?
*Tao*: The best example of this at the present time is the PolyMath project, where we now have numerous instances of collaborations with tens of mathematicians participating. Yet there are biology papers with over 300 authors and physics papers with over 2,000 authors.*Lurie*: However, most papers continue to be done by just one or a few authors.

- Q: In 100 or 1000 years, will computers surpass humans in mathematics, as they have in chess?
*Tao*: Computers will certainly increase in power, but I expect that much of mathematics will continue to be done with humans working with computers.*Taylor*: A better question is whether a computer will receive a Fields medal.*Kontsevich*: I think they will.*Lurie*: 1000 years is too long a time to project reliably.

- Q: What are the three greatest mathematicians?
*Taylor*: I would say Gauss, Euler and Hilbert.*Tao*: I would add Newton and Fermat.*Simon*: I would add Riemann and Poincaire.

- Q (posed from the audience by DHB): What computer-based tool would be most useful in advancing research mathematics?
*Tao*: We really need a search facility that searches in a mathematical context. Too often the results returned are not useful.*Lurie*: I would like to see a computer proof verification system with an improved user interface, something that doesn’t require 100 times as much time as to write down the proof. Can we expect, say in 25 years, widespread adoption of computer verified proofs?

- Q (posed from the audience): What career advice would you offer to a budding mathematician?
*Tao*: Ask lots of dumb questions. Don’t be afraid to question whether something is well understood.

#### Titles and abstracts of seminar talks

Here are the talks that were presented at the symposium by the Breakthrough Prize recipients:

- Terence Tao: “Polymath Projects: Massively Collaborative Online Mathematics.” Introduced by Timothy Gowers in 2009, “Polymath projects” are online projects in which many mathematicians (both professional and amateur) collaborate on a single research program. While some problems have turned out to not be well suited to the “Polymath” approach, there have been notable successes, most recently in building upon Yitang Zhang’s breakthrough towards the twin prime conjecture. We discuss some past Polymath projects, and speculate on the future utility of this research paradigm for mathematics.
- Jacob Lurie: “Analogy and Abstraction in Mathematics.” A great deal of mathematics is inspired by analogies: that is, relationships (often unexpected) between phenomena that arise in seemingly different contexts. When analogies are sufficiently strong, we often introduce definitions or axioms that attempt to capture the common features of the underlying phenomena. In many cases, these axioms (and the structures they describe) can take on a life of their own and become objects of mathematical study in their own right. In this talk, I’ll discuss some examples of this paradigm, drawn from classical abstract algebra and modern algebraic topology.
- Richard Taylor: “The Langlands Program.” The Langlands conjectures provide a remarkable framework linking algebra and geometry. They have been extremely influential in number theory, and in recent years have made contact with physics. In the last 40 years we have made extraordinary progress on these conjectures, but even more remains to be done. Progress on the Langlands conjectures lay behind both Wiles’ celebrated proof of Fermat’s Last Theorem and the advances made on the Birch-Swinnerton-Dyer conjecture. In this talk, I will try to illustrate the Langlands conjectures by looking at some concrete examples.
- Simon Donaldson: “Geometry in Manifolds of Exceptional Holonomy.” We will first discuss “exceptional” structures on spaces of dimension 7 and 8 and connections with the quaternion and octonion number systems discovered in the 19th century. Then we will try to give an idea of some questions of great current interest in this field (connected to M-theory in physics), in which it seems reasonable to hope for progress over the next few years. These questions involve the exceptional structures themselves, and also geometric objects — special minimal submanifolds and gauge fields — that can be defined in them. The fundamental difficulties, which are formidable, lie in the areas of analysis and partial differential equations.
- Maxim Kontsevich: “Calabi-Yau Motives.” I’ll speculate on a possible interaction between two very different topics/lines of thought in algebraic geometry: (1) Calami-Yau varieties; i.e. smooth projective varieties with vanishing Ricci curvature (equivalently, the canonical class is trivial), playing a central role in birational geometry, (2) Grothendieck’s yoga of motives; i.e. “natural” building blocks for cohomology of all algebraic varieties.