Description: This presentation discusses research on a tiling game in which two players alternate placing dominoes over a chessboard pattern of a given size (possibly infinite). Play continues until no more tiles can be placed. Player 1, blocker, wins if less than a certain percentage of a board is tiled, while player 2, tiler, wins if that percentage or more of the board is tiled. When the percentage is 100, there are simple strategies for winning. When the percentage is less than 100, the minimum percentage of the board that can be tiled by the end of play must be determined in order for the winning percentage for tiler to not be trivial. In discovering this, many properties of the tiled space under the rules of the game can be found. This presentation focuses on these properties and their relationship to the game.

Description: This article discusses Bär's conformal lower bound for the spectrum of generalized Dirac operators. Abstract: We use the orthogonal splitting of a certain Clifford module as a direct sum of generalized spinors and twistors to give a short and natural proof to Bär's conformal lower bound for the spectrum of a generalized Dirac operator on a compact Riemannian manifold.

Description: This article discusses projecting on polynomial solutions of second order partial differential operators. Abstract: We extend Axler and Ramey's method of constructing the harmonic part of a homogeneous polynomial to polynomial solutions of homogeneous elliptic real second order partial differential operators with constant coefficients. The results yield a constructive solution to a Dirichlet problem with polynomial boundary data.

Description: In this presentation, the author will briefly introduce the subject of Descriptive Set Theory and the motivation for its study. The author will discuss the idea of a projective set and also define the mathematical notion of a "tree" as an example of a projective set. The author will conclude with a brief mention of a significant result that can be proved using the notion of a tree.

Description: This presentation discusses research on predictable patterns in continued fractions and decimal expansions. Abstract: A continued fraction is a representation of a number by series of fractions inside fractions, such that the numerator of every fraction is a one. Decimal expansion is another way to express a number. Many times in mathematics when we have two different ways to write the same expression, we look for connections between the two notations. When trying to express a number, we encounter an interesting anomaly between continued fractions and decimal expansions. A randomly chosen number, with a predictable decimal pattern, will have an unpredictable continued fraction. Furthermore, a randomly chosen continued fraction, with predictable partial quotients, will have an unpredictable decimal expansion. However, there are those few exceptions, one of which the author is studying in depth. It is a binary sequence closely tied with the Fibonacci numbers, a series of numbers that often occur in nature. Through the author's research, the author hopes to find an understanding to this aforementioned anomaly and bridge continued fractions and decimals. In this presentation, the author will show where predictability lies and furthermore where there is chaos still to be settled.