TITLE: What were some major mathematical breakthroughs in 2016?
QUESTION [184 upvotes]: As the year is slowly coming to an end, I was wondering which great advances have there been in mathematics in the past 12 months. As researchers usually work in only a limited number of fields in mathematics, one often does not hear a lot of news about advances in other branches of mathematics. A person who works in complex analysis might not be aware of some astounding advances made in probability theory, for example. Since I am curious about other fields as well, even though I do not spend a lot of time reading about them, I wanted to hear about some major findings in distinct fields of mathematics. 
I know that the question posed by me does not allow a unique answer since it is asked in broad way. However, there are probably many interesting advances in all sorts of branches of mathematics that have been made this year, which I might have missed on and I would like to hear about them. Furthermore, I think it is sensible to get a nice overview about what has been achieved this year without digging through thousands of different journal articles.

REPLY [9 votes]: Logical induction is a probability theory for deductively limited systems. Formulating such a theory is a long open question, with previous attempts by for example Haim Gaifman.
Logical induction not only solves how deductively limited systems should assign credence to unproved arithmetical statements. It also allows deductively limited systems to reason about themselves, elegantly avoiding the associated Godelian/Lobian paradoxes.
"We present a computable algorithm that assigns probabilities to every logical statement in a given formal language, and refines those probabilities over time. For instance, if the language is Peano arithmetic, it assigns probabilities to all arithmetical statements, including claims about the twin prime conjecture, the outputs of long-running computations, and its own probabilities. We show that our algorithm, an instance of what we call a logical inductor, satisfies a number of intuitive desiderata, including: (1) it learns to predict patterns of truth and falsehood in logical statements, often long before having the resources to evaluate the statements, so long as the patterns can be written down in polynomial time; (2) it learns to use appropriate statistical summaries to predict sequences of statements whose truth values appear pseudorandom; and (3) it learns to have accurate beliefs about its own current beliefs, in a manner that avoids the standard paradoxes of self-reference. For example, if a given computer program only ever produces outputs in a certain range, a logical inductor learns this fact in a timely manner; and if late digits in the decimal expansion of $\phi$ are difficult to predict, then a logical inductor learns to assign $\approx 10\%$ probability to “the $n$th digit of $\phi$ is a 7” for large $n$. Logical inductors also learn to trust their future beliefs more than their current beliefs, and their beliefs are coherent in the limit (whenever $\phi\to\psi$, $P_\infty(\phi)\leq P_\infty(\psi)$, and so on); and logical inductors strictly dominate the universal semimeasure in the limit. 
These properties and many others all follow from a single logical induction criterion, which is motivated by a series of stock trading analogies. Roughly speaking, each logical sentence $\phi$ is associated with a stock that is worth \$1 per share if $\phi$ is true and nothing otherwise, and we interpret the belief-state of a logically uncertain reasoner as a set of market prices, where $P_n(\phi) = 50\% $ means that on day $n$, shares of $\phi$ may be bought or sold from the reasoner for 50¢. The logical induction criterion says (very roughly) that there should not be any polynomial-time computable trading strategy with finite risk tolerance that earns unbounded profits in that market over time. This criterion bears strong resemblance to the “no Dutch book” criteria that support both expected utility theory (von Neumann and Morgenstern 1944) and Bayesian probability theory (Ramsey 1931; de Finetti 1937)."