Link Floer homology of links in S^3 can be computed as the homology of a grid chain complex defined using grid diagrams. I will describe a construction of a CW spectrum whose cells correspond to the generators of the grid chain complex, and whose cellular chain complex is the grid chain complex (and therefore, the homology is link Floer homology). This is joint with Ciprian Manolescu.

The Robinson-Schensted (RS) correspondence is a famous bijection between permutations and pairs (P,Q) of standard tableaux of the same shape, called the RS partition. The RS partition and its conjugate record certain permutation statistics called Greene’s theorem statistics.
A box-ball system is a discrete dynamical system which can be thought of as a collection of time states. A permutation on n objects gives a box-ball system state by assigning its one-line notation to n consecutive boxes. After a finite number of steps, a box-ball system will reach a steady state. From any steady state, we can construct a tableau (not necessarily standard) called the soliton decomposition. The shape of the soliton decomposition is called the BBS partition. An exciting discovery (made in 2019 by Lewis, Lyu, Pylyavskyy, and Sen) is that the BBS partition and its conjugate record a localized version of Greene’s theorem statistics.
We will discuss a few new results:
(1) The Q tableau of a permutation completely determines the dynamics of the corresponding box-ball system.
(2) The permutations whose BBS partitions are L-shaped have steady-state time at most 1. This large class of permutations include column reading words and noncrossing involutions.
(3) If the soliton decomposition of a permutation is a standard tableau or if its BBS partition coincides with its RS partition, then its soliton decomposition and its P tableau are equal.
(4) Finally, we study the permutations whose P tableaux and soliton decompositions coincide and refer to them as “good". These “good” permutations are closed under consecutive pattern containment. Furthermore, we conjecture that the “good” Q tableaux are counted by the Motzkin numbers.
This talk is based on REU projects with Ben Drucker, Eli Garcia, Aubrey Rumbolt, Rose Silver (UConn Math REU 2020) and Marisa Cofie, Olivia Fugikawa, Madelyn Stewart, David Zeng (SUMRY 2021).

The Kaczmarz algorithm is an iterative method for solving linear systems of equations of the form Ax=y. Owing to its low memory footprint, the Kaczmarz algorithm has gained popularity for its practicality in applications to large-scale data, acting only on single rows of A at a time. In this talk, we discuss selecting rows of A randomly (Randomized Kaczmarz), selecting rows in a greedy fashion (Motzkin's Method), and selecting rows in a partially greedy fashion (Sampling Kaczmarz-Motzkin algorithm). Despite their variable computational costs, these algorithms have been proven to have the same theoretical upper bound on the convergence rate. Here we present an improvement upon previous known convergence bounds of the Sampling Kaczmarz-Motzkin algorithm, capturing the benefit of partially greedy selection schemes. Time permitting, we also will discuss an extension of the Kaczmarz algorithm to the setting where data takes on the form of a tensor and make connections between the new Tensor Kaczmarz algorithm and previously established algorithms. This presentation contains joint work with Jamie Haddock and Denali Molitor.

We'll discuss the background material for the main result of the following paper: https://arxiv.org/abs/2107.07351. We'll start by talking about quasi-split special unitary groups and the associated Steinberg groups. If we have time, we'll talk about algebraic rings. This will be the first in a sequence of two talks on this paper.