January 2023
Bill Mance: Descriptive complexity in number theory and dynamics
23/01/23 14:38
Tuesday, January 24, 2023 17:00
Location: room C11-3.11
Speaker: Bill Mance (Adam Mickiewicz University in Poznan)
Title: Descriptive complexity in number theory and dynamics
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. Kechris asked several questions involving descriptive complexity of sets of normal numbers. The first of these was resolved in 1994 when Ki and Linton proved that the set of numbers normal in base \(b\) is \(\Pi_3^0\)-complete. Further questions were resolved by Becher and Slaman. Many of the techniques used in these proofs can be used elsewhere. We will discuss recent results where similar techniques were applied to solve a problem of Sharkovsky and Sivak and a question of Kolyada, Misiurewicz, and Snoha. Furthermore, we will discuss a recent result where the set of numbers that are continued fraction normal, but not normal in any base \(b\), was shown to be complete at the expected level of \(D_2(\Pi_3^0)\). An immediate corollary is that this set is uncountable, a result (due to Vandehey) only known previously assuming the generalized Riemann hypothesis.
Location: room C11-3.11
Speaker: Bill Mance (Adam Mickiewicz University in Poznan)
Title: Descriptive complexity in number theory and dynamics
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. Kechris asked several questions involving descriptive complexity of sets of normal numbers. The first of these was resolved in 1994 when Ki and Linton proved that the set of numbers normal in base \(b\) is \(\Pi_3^0\)-complete. Further questions were resolved by Becher and Slaman. Many of the techniques used in these proofs can be used elsewhere. We will discuss recent results where similar techniques were applied to solve a problem of Sharkovsky and Sivak and a question of Kolyada, Misiurewicz, and Snoha. Furthermore, we will discuss a recent result where the set of numbers that are continued fraction normal, but not normal in any base \(b\), was shown to be complete at the expected level of \(D_2(\Pi_3^0)\). An immediate corollary is that this set is uncountable, a result (due to Vandehey) only known previously assuming the generalized Riemann hypothesis.
Łukasz Mazurkiewicz: Ideal analytic sets
17/01/23 09:14
Tuesday, January 17, 2023 17:00
Location: room C11-3.11
Speaker: Łukasz Mazurkiewicz
Title: Ideal analytic sets
Abstract: We will consider examples of analytic sets which are not Borel. We will focus on, so called, complete analytic sets. Firstly, we will consider ideals on naturals (naturally treated as subsets of the Cantor space). Secondly, we will consider the family of Silver trees. We will compare the later example with theorem of Kechris-Louveau-Woodin.
Location: room C11-3.11
Speaker: Łukasz Mazurkiewicz
Title: Ideal analytic sets
Abstract: We will consider examples of analytic sets which are not Borel. We will focus on, so called, complete analytic sets. Firstly, we will consider ideals on naturals (naturally treated as subsets of the Cantor space). Secondly, we will consider the family of Silver trees. We will compare the later example with theorem of Kechris-Louveau-Woodin.
Artsiom Ranchynski: Ultrafilters avoiding measures
09/01/23 15:18
Tuesday, January 10, 2023 17:00
Location: room C11-3.11
Speaker: Artsiom Ranchynski
Title: Ultrafilters avoiding measures
Abstract: A point \(x\) avoids measures if whenever \(\mu\) is a measure such that \(\mu({x})=0\), then \(x\) does not belong to the support of \(\mu\). In this talk I will construct a point avoiding non-atomic measures in the Stone-Cech compactification of naturals. I will discuss the relation of such points to other special points in \(\beta N\).
Location: room C11-3.11
Speaker: Artsiom Ranchynski
Title: Ultrafilters avoiding measures
Abstract: A point \(x\) avoids measures if whenever \(\mu\) is a measure such that \(\mu({x})=0\), then \(x\) does not belong to the support of \(\mu\). In this talk I will construct a point avoiding non-atomic measures in the Stone-Cech compactification of naturals. I will discuss the relation of such points to other special points in \(\beta N\).