The Mathematics Undergraduate Seminar
(V)

The Fifth Undergraduate Seminar Session

UGS 5 closed out the semester with three talks spanning the foundations of mathematics, the theory of distributed computing, and complex analysis. As always, the seminar gives students the chance to present research they've developed with support from the Math Society, with papers reviewed by faculty before the talks are given.

Speakers & Topics

• Alexei Khovrenkov – Gödel's Incompleteness Theorems: When Mathematics Met Its Limits Traces the search for a perfectly complete and consistent foundation for mathematics, from Leibniz and non-Euclidean geometry through Hilbert's program, and explains how Gödel's two incompleteness theorems showed that any sufficiently powerful formal system will always contain true statements it cannot prove — with consequences that reach into computability theory, AI, and philosophy.
  

  Introducing the Peano arithmetic

  

  Walking through Gödel numbering and the diagonal lemma

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• Dr. Ivan Ovsyannikov – Discussion: Parallel Lines in Hyperbolic Geometry Following Alexei's talk, Dr. Ovsyannikov took the whiteboard to expand on the parallel postulate, unpacking how "parallel" is defined once you move from Euclidean to hyperbolic (Lobachevskian) geometry — where non-intersecting lines through a point split into two special boundary lines, and triangles can be built from mutually parallel sides with no vertices.
  

  Dr. Ivan Ovsyannikov at the whiteboard

  

  Explaining the concepts of hyperbolic geometry

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• Taha Hbirri (2nd year CS) – The FLP Theorem: Why Consensus Is Impossible in an Asynchronous System Starting from Lamport's "happened-before" relation and the problem of ordering events across machines that don't share a clock, this talk builds up to the FLP theorem: the result that no deterministic algorithm can guarantee consensus in an asynchronous distributed system if even one process can fail — a foundational impossibility result for distributed systems.
  

  Introducing the consensus problem in distributed systems

  

  Working through the FLP theorem

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• Pablo Santos Guerrero – The Fundamental Theorem of Algebra and the Geometry of Zeros and Poles Looks at the Fundamental Theorem of Algebra through the lens of complex analysis, connecting the existence of roots of polynomials to the geometric behavior of zeros and poles of complex functions.
  

  Pablo Santos Guerrero introducing the Fundamental Theorem of Algebra

  

  Explaining objects with the same global structure

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