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# Geometry MMA Seminar: Normal numbers and the Borel hierarchy

Oct 16, 2023, 10:30 - 11:30 AM

**Speaker: **Bill Mance, Adam Mickiewicz University in Poznań

**Title: ** Normal numbers and the Borel hierarchy

**Abstract:** Informally, a real number is normal in base $b$ if in its $b$-ary expansion all digits and blocks of digits occur as often as one would expect them to, uniformly at random. Borel introduced normal numbers in 1909 and proved that Lebesgue-almost every real number is normal in all bases $b \geq 2$. Even though this shows that, in some sense, normal numbers are "typical," no example of a number normal in all bases was given until 1939 by Turing. In the last 100 years, the study of normal numbers has spread over a wide breadth of seemingly unrelated disciplines. Normality is closely related to number theory, ergodic theory, theoretical computer science, probability theory, fractal geometry, descriptive set theory, and other areas of math. We will provide a gentle introduction to the Borel hierarchy and discuss recent classifications of sets of normal numbers in this hierarchy as well as some surprising corollaries.

**Time:** Monday, October 16, 2023 – 10:30am-11:30am

**Place:** Exploratory Hall, Room 4106