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In the talk titled "The (Non-Perfect) Mathematics of Trust," delivered by Vaidehi Joshi at the Rocky Mountain Ruby 2017 event, the main focus revolves around the concept of trust through the lens of mathematics, specifically graph theory, and its implications in technology and real-world scenarios.

Joshi begins her presentation by introducing herself and providing a humorous context about her sunny Portland home before diving into the topic. She discusses her background in writing rather than computer science and explains her recent project of teaching herself computer science concepts. This journey led her to explore the historical narrative of trust in the context of mathematics.

The key points in her talk include: 
- **Historical Origin of Graph Theory:** Joshi traces the story back to 1735 in Königsberg (modern-day Kaliningrad), where residents attempted to cross seven bridges without crossing any bridge twice. This problem intrigued mathematician Euler, leading to the establishment of graph theory, which abstracts problems into nodes (points) and edges (connections).
- **Eulerian Paths:** The conditions for defining an Eulerian path are detailed. Joshi explains that a path can be drawn through every node without repetition under specific degree conditions, thus establishing foundational principles for graph theory.
- **Applications of Graph Theory:** Joshi cites various practical applications in modern computing such as the knight's tour in chess, Rubik's Cube solutions, Google’s PageRank algorithm, and biogenomics. Each example demonstrates how graph theory helps solve complex problems by establishing efficient pathways or relationships.
- **Social Networks and Behavioral Influence:** The exploration extends to sociological studies, notably by Nicholas Christakis on how social networks can affect behaviors like health and lifestyle based on interconnectedness.
- **Trust in the Sharing Economy:** The concept of trust emerges as a crucial factor in today’s sharing economy, particularly in platforms like Airbnb, where user interactions are facilitated by trust metrics based on networks.
- **Biases in Trust Algorithms:** Joshi warns of the mathematical difficulties in discerning biases in algorithms that govern trust within frameworks of prominent platforms such as Facebook, highlighting the necessity for human oversight in interpreting fairness.
- **Königsberg’s Historical Irony:** Concluding her talk, she revisits the fate of Königsberg, which transformed through history, ultimately leading to the realization that constructing an Eulerian path finally became possible after World War II due to the destruction and reconstruction of bridges.

**Takeaways:** Joshi’s talk emphasizes the intertwining of mathematics with trust dynamics in technology and society, showcasing how historical mathematical challenges laid the groundwork for contemporary applications. The central message is that while mathematics offers tools for understanding complex networks, human interpretation remains essential for navigating trust and biases in these systems.

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