I will discuss higher semi-additive character theory, a framework for (n-t)-fold characters that unifies the classical monoidal character and the transchromatic character. Roughly speaking, the goal is to study character maps from R-valued functions on a pi-finite space A to S-valued functions on the (n-t)-fold p-typical free loop space of A, in a way that is functorial not only for restriction maps but also for transfers along pi-finite spaces. A central point is that every infinity-commutative monoid admits a universal character. I will describe key structural properties of this universal character, including chromatic blue shift and higher cyclotomic descent, and I will briefly indicate how it interacts with other higher semi-additive constructions such as the semi-additive Fourier transform. For K(n)-local objects, I will describe an explicit formula for the universal character and explain how it recovers the transchromatic character for Morava E-theory. I will also discuss the natural action of GL_{n-t}(Zp) on the universal character, whose fixed points agree with rationalization when t=0.
This talk is a sequel to Zoltan Vidyanszky's talk a couple weeks ago. I will explain some of the questions descriptive set theorists ask about homomorphisms between relational structures, and I will indicate the subtleties involved in adapting the algebraic approach to homomorphism problems in this context. I will also pose some purely algebraic questions that arise from this project.
Gessel and Zhuang introduced the concept of shuffle-compatibility of statistics on permutations of subsets of \N to describe statistics whose multiset of values on the shuffles of two disjoint permutations is determined exactly by the size and statistic values of the two permutations being shuffled. Shuffle-compatibility implies the existence of an algebraic structure on the equivalence classes induced by the statistic. For example, the descent set statistic is shuffle-compatible, and the algebra on the equivalence classes it induces is isomorphic to the Hopf algebra of quasisymmetric functions.
In this talk, we generalize shuffle-compatibility to objects such as words, parking functions, and set partitions using the Hopf monoids associated with these objects. We cover a variety of well-known statistics on each object. We define various algebraic structures on the equivalence classes formed by these statistics that are, in many cases, quotients of (or isomorphic to) well-known combinatorial Hopf algebras such as FQSym, WQSym, PQSym, and NCSym.
Generalizing shuffle-compatibility using Hopf monoids
Tue, Mar. 10 5pm (Math 350)
Math Club
Harrison Stalvey (CU Boulder)
X
Take notes, outline textbooks, work out a lot of problems, visit instructor office hours—These are among answers to the question, "How do we study?" But what about the question, "How do we learn?" Specifically, what happens in our mind developmentally as we study mathematics? In this talk, we will discuss one particular learning theory that describes the learning of advanced mathematical concepts. Then we will apply the learning theory to make a cognitive model of a construction of the integers, which is a topic covered in MATH 2002, Number Systems, at CU Boulder.
Higher semi-additive character theory