Mathematics

This lecture is part of a course on the geometry of numbers.

• Lecture 1
• Lecture 2
• Lecture 3
• Lecture 4
• Lecture 5
• Lecture 6
• Lecture 7
• Lecture 8
• Lecture 9
• Lecture 10
• Lecture 11
• Lecture 12
• Lecture 13

For more information about these lectures, please see the course website (external).

This lecture is part of a course on the geometry of numbers.

• Lecture 1
• Lecture 2
• Lecture 3
• Lecture 4
• Lecture 5
• Lecture 6
• Lecture 7
• Lecture 8
• Lecture 9
• Lecture 10
• Lecture 11
• Lecture 12
• Lecture 13

For more information about these lectures, please see the course website (external).

This lecture is part of a course on the geometry of numbers.

• Lecture 1
• Lecture 2
• Lecture 3
• Lecture 4
• Lecture 5
• Lecture 6
• Lecture 7
• Lecture 8
• Lecture 9
• Lecture 10
• Lecture 11
• Lecture 12
• Lecture 13

For more information about these lectures, please see the course website (external).

In The Hitchhiker’s Guide to the Galaxy, by Douglas Adams, the number 42 was revealed to be the “Answer to the Ultimate Question of Life, the Universe, and Everything”. But he didn’t say what the question was! I will reveal that here. In fact it is a simple geometry question, which then turns out to be related to the mathematics underlying string theory.

Speaker Biography

John Baez is a leader in the area of mathematical physics at the interface between quantum field theory and category theory, and has broad interests in mathematics, and science more generally. He created one of the earliest blogs "This week's finds in Mathematical Physics" (before the term blog existed!)

Baez did his PhD at MIT, and was a Gibbs Instructor at Yale before moving to University of California, Riverside in 1988.

About the series

Starting in 2021, PIMS has inaugurated a high-level network-wide colloquium series. Distinguished speakers will give talks across the full PIMS network with one talk per month during the academic term. The 2021 speaker series is part of the PIMS 25th Anniversary showcase.

In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation $T$, one takes averages of a given integrable function over the intervals $\{x, T(x), T^2(x), \hdots, T^n(x)\}$ in the forward orbit of the point $x$. In joint work with Jenna Zomback, we prove a “backward” ergodic theorem for a countable-to-one pmp $T$, where the averages are taken over subtrees of the graph of $T$ that are rooted at $x$ and lie behind $x$ (in the direction of $T^{-1}$). Surprisingly, this theorem yields (forward) ergodic theorems for countable groups, in particular, one for pmp actions of free groups of finite rank where the averages are taken along subtrees of the standard Cayley graph rooted at the identity. For free group actions, this strengthens the best known result in this vein due to Bufetov (2000). After reviewing the subject history and discussing the statements of our theorems in the first half of the talk, we will highlight some ingredients of proofs in the second half.