Point groups in two dimensions are mathematical concepts used in the study of symmetry in two-dimensional objects or systems. A point group is a collection of symmetry operations (such as rotations and reflections) that leave a geometric figure unchanged when applied. These symmetry operations involve rotating, reflecting, or translating the figure, but in the context of point groups, we mainly focus on operations that keep the center of the object fixed.

A **spherical 3-manifold** is a type of three-dimensional manifold that is topologically equivalent to a quotient of the 3-dimensional sphere \( S^3 \) by a group of isometries (which preserve distances). More formally, a spherical 3-manifold can be described as a space of the form \( S^3 / G \), where \( G \) is a group of finite isometries of the 3-sphere.

A finite ring is a ring that contains a finite number of elements. In abstract algebra, a ring is defined as a set equipped with two binary operations: addition and multiplication, which satisfy certain properties. Specifically, a ring must satisfy the following axioms: 1. **Additive Identity**: There exists an element \(0\) such that \(a + 0 = a\) for all elements \(a\) in the ring.

In ring theory, a branch of abstract algebra, a **reduced ring** is a type of ring in which there are no non-zero nilpotent elements. A nilpotent element \( a \) in a ring \( R \) is defined as an element such that for some positive integer \( n \), \( a^n = 0 \). In simpler terms, if \( a \) is nilpotent, then raising it to some power eventually results in zero.

In mathematics, particularly in the fields of topology, algebra, and lattice theory, a **closure operator** is a function that assigns a subset (the closure) to every subset of a given set, satisfying certain axioms. A closure operator \( C \) on a set \( X \) must satisfy the following three properties: 1. **Extensiveness**: For every subset \( A \subseteq X \), \( A \subseteq C(A) \).

"Irrigation" is a type of puzzle or simulation game that typically involves managing water resources to effectively irrigate crops and maximize agricultural output. The primary goal is usually to create a network of irrigation channels that deliver water to fields while overcoming various challenges such as terrain obstacles, limited resources, and time constraints. In these games, players often have to strategically plan and construct irrigation systems, considering factors like water flow, planting schedules, and crop types.

GADDAG is a word game that serves as a variation of the classic game Scrabble. The name "GADDAG" stands for "Go Ahead & Don't Do A Letter At the Gap." The game is structured around creating words on a game board using letter tiles, with players aiming to score points based on the letters they use and the placement of their words on the board.

Mahbub ul Haq (1928 – 1999) was a prominent Pakistani economist and politician, known for his contributions to economic policy, development, and human welfare. He is particularly recognized for his role in advocating for human development and for being one of the architects of the Human Development Index (HDI), which measures a country's social and economic development based on factors such as life expectancy, education, and per capita income.

Michael Maschler is an Israeli mathematician known for his contributions to game theory, particularly in the areas of cooperative games and bargaining theory. He is also recognized for his work in the field of mathematical economics. Maschler has co-authored several influential papers and works in these domains. His research often focuses on the mathematical foundations of decision-making processes and strategic interactions among rational agents.

The Dictator Game is a widely studied economic experiment that explores concepts of altruism, fairness, and decision-making. It involves two players: one designated as the "dictator" and the other as the "recipient." The basic structure of the game is as follows: 1. **Endowment**: The dictator is given a certain amount of resources (commonly money, but it can be points or goods in different variations of the game).

The Two Knights Endgame is a situation in chess where only two knights are left on the board for one side, typically against a lone king or another minor piece (often a pawn). This endgame is distinct because it usually represents a challenging scenario for the player with the two knights, as they cannot checkmate a lone king without the assistance of a pawn or another piece. In its purest form, the most common scenario involves one player having two knights and the other player having just a king.

Game balance refers to the process of ensuring that all elements of a game—such as characters, abilities, weapons, items, or mechanics—are designed and adjusted in a way that creates a fair, enjoyable, and challenging experience for players. Effective game balance aims to prevent any single aspect of the game from being overwhelmingly powerful or weak, which could lead to frustration or diminish the enjoyment of the game.

Astranis is a company focused on developing small, affordable satellites that provide internet connectivity, particularly for underserved regions. Founded in 2015 and based in San Francisco, Astranis aims to bridge the digital divide by leveraging technology to offer reliable internet access in areas where traditional infrastructure may be lacking or too costly to deploy. Their satellites are designed to be smaller and more cost-effective compared to traditional communication satellites, making it easier and more economical to deploy broadband services.

Friedrich Otto Rudolf Sturm, also known simply as Fritz Sturm, is a renowned figure in the field of mathematics and is known for his contributions to various areas including differential equations, control theory, and numerical analysis.

Richard M. Pollack is a prominent American scientist known for his research in the fields of biology and biophysics. He is particularly recognized for his work on the mechanisms of molecular motors and their role in cellular processes. His research often intersects with topics such as energy conversion, protein structure, and the physical principles underlying biological functions. Pollack has published numerous scientific papers and has contributed to the understanding of how molecular machines operate at the cellular level.

Yuri Burago is a mathematician known for his contributions to the fields of geometry and topology. He has made significant advancements in the study of metric spaces, as well as in the areas of differential geometry and geodesic flows. In addition to his research, Burago is also known for his role in mathematics education, having authored several textbooks that are used in university-level courses.