I'll discuss recent progress on Goldbach's conjecture via the pigeonhole principle

The Game of Cycles is a two-player combinatorial game in which players take turns assigning directions to the edges of a graph, subject to restrictions that prevent the creation of sources and sinks. We introduce the mixed Game of Cycles, which allows some edges to be directed before play begins and permits disconnected graphs. This generalization provides a natural framework for studying game positions using Grundy values and decompositions into simpler games.

We develop structural tools that allow us to determine the Grundy value of any mixed path. In particular, we establish a sharp upper bound and characterize the unique family of mixed paths attaining it. We also introduce a pendant decomposition that yields a simple formula for the Grundy value of any caterpillar graph, depending only on the parity of the number of edges in its spine.

These results illustrate how graph structure can reveal surprisingly simple winning strategies in combinatorial games.

In recent decades, considerable statistical interest has focused on shooting streaks in basketball and the validity of the "hot hand" phenomenon—the belief that a player's probability of making a shot increases following a sequence of successful attempts. While most prior research evaluates individual performance, this talk investigates shooting streaks at the team level, focusing on three-point shooting data from the NBA, with a particular case study on the Boston Celtics. Using historical data analysis and probabilistic modeling, we evaluate whether entire teams exhibit "hot hand" behavior or if team-wide streaks align with random variation.

What do spanning trees, graph flows, graph coloring, and network reliability have in common? They can all be studied by a single polynomial! The Tutte polynomial is a remarkably powerful graph invariant that encodes a wide variety of combinatorial information. This talk will explore its construction and some of its evaluations through examples to examine seemingly unrelated information about a graph.

Major League Baseball has just concluded its innaugural season using the Automated Ball-Strike (ABS) Challenge System.
Similar to manager's replay challenges, teams are only allowed two unsuccessful challenges during a
game. Because ABS challenges are initiated by individual players, teams benefit from simple and clear criteria for when a challenges is advisable. We will use data from the 2026 season to consider the differential strategies of batters and catchers initiating challenges, building a simple theoretical model and making reasonable parameter estimates.

"Power analyis" is conducted when one wants to study the relationships between sample size, effect size, and power of some statistical test. In cases where formulas relating these quantities do exist, they require that one be able to estimate the distribution of some test statistic. We can perform the same analyses with much greater flexibility, and no data or distributional assumptions, through simulation. In this talk we demonstrate this approach for "zero inflated" hospital admissions data. Central to the approach is the use of a Cholesky decomposition trick to correlate individual patient variables. This talk does not assume any specialized statistics background.

In special relativity, velocities do not add: an object with velocity u relative to an object with velocity v has combined velocity (u + v) / (1 + uv/c^2). Instead, "rapidity" adds; the rapidity of velocity v is arctanh(v/c), a number that ranges from -infinity to infinity as v ranges from -c to c. How would someone go about devising such a transformation that carries a binary operation to ordinary addition? We'll show a general principle, derived from representation theory but only needing basic calculus, that can transform any group operation on numbers into ordinary addition.

In this talk, we will give a brief history of Ramsey numbers, including recent breakthroughs. Then we will introduce the r-Kneser Ramsey number, which is the minimum n such that every red/blue edge-coloring of the Kneser graph KG(n,r) contains a red clique on s vertices or a blue clique on t vertices. We will give bounds on the r-Kneser Ramsey number as well as some small exact values. This is joint work with Emily Heath, Alex Parker, Coy Schwieder, and Shira Zerbib.