Speaker: Asaf Hadari (UHM)

Title: In search of a representation theory of mapping class groups.

Abstract:

Mapping class groups are nearly ubiquitous in low dimensional topology. They’ve been studied for over a century. Various results discovered during the past few decades it has become quite clear that there is much to gain by studying them via their linear representations.

Somewhat surprisingly, many such representations are known. Unfortunately, until recently there was almost no representation theory, that is – no underlying structure that allows you to say anything about the class of representations as a whole. It is precisely such an understanding that is necessary for studying mapping class groups.

In this talk I’ll talk about the major source of representations of mapping class groups, and talk about new results in their emerging representation theory.

“Iterated ultrapowers for the masses”, part 2

Comparing Near-linearity Notions in Open Induction

There have been works in number theory on characterizing the class of Beatty sequences (integer parts of natural multiples of a fixed nonnegative real slope). The same is true for the inhomogeneous case when a fixed intercept is added before taking the integer part. We consider some notions of multiplicative or additive near-linearity and elaborate on the extent to which they charecterize various such sequences. We show some implications from standard number theory carry over to Open Induction and some do not. [In a second talk we could relate this to the weak fragment allowing the standard integers as a direct summand of a model. That second talk would include two more multiplicative vs. additive topics, details to follow.]

Some additive vs. multiplicative issues in subrecursivity, maximality, and near-linearity

We deal with three topics around addition without or with multiplication.

We first present algorithms to compute a certain real, generating its Beatty sequence or base 2 expansion. The former calculates in integers with addition, in conjunction with the counting operator. The latter calculates in integers with addition and multiplication. Motivation comes from subrecursive reals.

Next, let F be an ordered field, D a maximal discrete subring of F, and G a maximal discrete additive subgroup of F. We point out that although there are examples where F has elements of infinite distance to D, it can never realize any gaps of G. If F is countable, then G can be constructed Delta^0_2 relative to F.

Finally we finish and extend the talk of last week by considering some nonstandard models M of weak arithmetic which have the integers as an additive direct summand. We present functions f and g from M to M whose value at a sum minus sum of values is always 0 or 1 yet for some x,y,u,v ≥ 1in M, we have f(xy) < xf(y) and g(uv) > ug(v) + u – 1.

**Assessing the Reverse Mathematical Strength of Gratzer-Schmidt Theorem**

Gratzer-Schmidt theorem in lattice theory states that all complete and compactly generated lattices are isomorphic to the congruence lattice of an algebra. There has been an effort to assess the strength of this theorem in the reverse mathematical setting. I will discuss my recent progress on this topic and its potential implications.

Title: Analysis of partisan gerrymandering tools in advance of the US 2020 census

Abstract: Over the last decade, mapmakers have precisely gerrymandered political districts for the benefit of their party. In response, political scientists and mathematicians have more extensively investigated tools to quantify and understand the mathematical structure of redistricting problems. Two primary tools for determining whether a particular redistricting plan is fair are partisan-gerrymandering metrics and stochastic sampling algorithms. In this work we explore the Declination, a new metric intended to detect partisan gerrymandering. Within out analyses, we show that Declination cannot detect all forms of packing and cracking, and we compare the Declination to the Efficiency Gap. We show that these two metrics can behave quite differently, and give explicit examples where that occurs.