## The Shapes of Computations

* Or rather, what can the shapes of proofs tell us about them? *

April CACM source |

Juris Hartmanis did much to lay the landscape of computational complexity beginning in the 1960s. His seminal paper with Richard Stearns, “On the Computational Complexity of Algorithms,” was published 50 years ago this month, as observed by Lance Fortnow in his blog with Bill Gasarch. It is a great achievement to open a new world, but all the more mysterious that after 50 years so much of its landscape remains unknown.

Today we ask what might determine the unseen topography and how much some recent large-data discoveries may help to map it.

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## A Tighter Grip on Circuit Depth

* The polynomial hierarchy is infinite for a random oracle *

Benjamin Rossman, Rocco Servedio, and Li-Yang Tan have made a breakthrough in proving lower bounds on constant-depth circuits. It came from a bi-coastal collaboration of Rossman visiting the Berkeley Simons Institute from Japan and Tan visiting from Berkeley to Servedio at Columbia University in New York. Their new paper solves several 20- and 30-year old open problems.

Today we congratulate them on their achievement and describe part of how their new result works.

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## Transcomputational Problems

* Problems beyond brute force search *

Cropped from Wikipedia source |

Hans-Joachim Bremermann was a mathematician and biophysicist. He is famous for a limit on computation, Bremermann’s limit, which is the maximum computational speed of a self-contained system in the material universe.

Today Ken and I wish to talk about the limit and why it is not a limit.

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## The Anti-Pigeonhole Conjecture

* A conjecture about faculty behavior *

“Dr. Kibzwang” source |

Colin Potts is Vice Provost for Undergraduate Education at Georgia Tech. His job includes being a member of the President’s Cabinet—our president, not the *real* one—and he is charged with academic policies and changes to such policies. He is also a College of Computing colleague and fellow chess fan.

Today I want to state a conjecture about the behavior of faculty that arose when Tech tried to change a policy.

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## Case Against Cases

* How to avoid too many cases at least some of the time *

Theon of Alexandria was history’s main editor of Euclid’s *Elements*.

Today I want to talk about case analysis proofs.

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## Kinds of Continuity

* Congratulations to John Nash and Louis Nirenberg on the 2015 Abel Prize *

Combined from src1, src2. |

John Nash and Louis Nirenberg have jointly won the 2015 Abel Prize for their work on partial differential equations (PDEs). They did not write any joint papers, but Nirenberg evidently got Nash excited about David Hilbert’s 19th problem during Nash’s frequent visits to New York University’s Courant Institute in the mid-1950s. Nash in return stimulated Nirenberg by his verbal approach of barraging a problem with off-kilter ideas. The Norwegian Academy of Sciences and Letters recognized their ‘great influence on each other’ in its prize announcement.

Today we congratulate both men on their joint achievement.

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## A Quantum Two-Finger Exercise

* More mileage than expected from a little example *

Cropped from World Science Festival source |

Sean Carroll is a cosmologist in the Department of Physics at Caltech. He also maintains a blog, “Preposterous Universe,” and writes books promoting the public understanding of science. I have recently been enjoying his 2010 book *From Eternity to Here: The Quest for the Ultimate Theory of Time*.

Today—yes, Carroll would agree that there is a today—I would like to share an interpretation of a little quantum computing example that occurred to me while reading his book.

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