09/05/2025 – Tan Özalp: Friedman’s Borel diagonalization theorem
Abstract: Do logicians just use Cantor’s diagonal argument over and over again? Well, sometimes they prove they can’t.
09/12/2025 – Fuxiang Yang: The Fundamental Theorem of Symmetric Functions
Abstract: Let the symmetric group $\mathfrak{S}n$ act on the polynomial ring $\mathbb{Z}[x_1,\dots,x_n]$ by permuting the variables. The ring of symmetic functions $\Lambda_n$ defined by $\Lambda_n = \mathbb{Z}[x_1,\dots,x_n]^{\mathfrak{S}_n}$ is a well-studied object in Combinatorics. Define the $r$-th elementary symmetric function $e_r$ to be [e_r = \sum{i_1 < \cdots < i_r} x_{i_1}\cdots x_{i_r}.] We will show that $\Lambda_n = \mathbb{Z}[e_1,\dots,e_n]$.
09/19/2025 – Gavin Dooley: The structure of the Turing degrees
Abstract: Some mathematical problems can be solved by an algorithm, but others cannot. Among those that cannot, some of them are “more” noncomputable than others, an idea that is made formal by the notion of “relative” computability. Relative computability induces a degree structure on the set of mathematical problems. What does this structure look like?
09/26/2025 – Jaziel Torres: When Mathematicians Collide: The Calculus Wars and the Battle for the Foundations
Abstract: The history of mathematics is not only a story of theorems and discoveries, but also of rivalries that shaped its trajectory. This talk recounts two of the most famous feuds in mathematical history: the priority dispute between Isaac Newton and G.W. Leibniz over the invention of calculus, and the foundational quarrel between David Hilbert and L.E.J. Brouwer over formalism versus intuitionism.
We will see how questions of intellectual credit, personal rivalry, and philosophical conviction ignited controversies that spilled beyond mathematics into institutions, reputations, and national pride. From Newton’s behind-the-scenes maneuvers in the Royal Society against Leibniz, to Hilbert’s decisive expulsion of Brouwer from the editorial board of Mathematische Annalen, these conflicts reveal how power and personality can shape the development of mathematics.
10/03/2025 – Jason Mitrovich: A Measure Theoretic Characterization of Commutativity
Abstract: What is the probability that two group elements commute? If $G$ is Abelian then every element commutes with every other element so it’s 100%. What happens when $G$ is not Abelian? In 1968 Erdős and Turán showed that for finite groups the probability is bounded by 62.5%. Moreover, they conclude that this percentage is sharp i.e. it cannot be improved. Shortly after in 1973, Gustafson extended this result to compact Hausdorff topological groups. In this talk, we will first explore some examples before stating Gustafson’s generalization. Since Gustafson’s statement relies on the Haar measure, we will take a moment to review the key ideas around the Haar measure before giving a sketch of the proof. After seeing an application or two, we will state (without proof) a result from Guralnick and Wilson from 2000 further linking these probability type theorems to other group theoretic properties.
10/10/2025 – Panel: Nonacademic jobs
Abstract: This seminar will be a panel discussion featuring some former Notre Dame students in mathematics who are currently working in a job outside of academia, or in academia but not as a math professor.
The panelists will include:
Jeremy Mann (Notre Dame PhD, 2019, Data Scientist)
John Siratt (Notre Dame PhD, 2024, Public Sector Researcher)
Danny Orton (Notre Dame PhD, 2019, Arcfield, Technical Specialist, Physicist)
10/17/2025 – Yuyan He: TBA
Abstract: TBA
10/24/2025 – FALL BREAK: NO TALK
10/31/2025 – Katherine Novey: TBA
Abstract: TBA
11/07/2025 – Sam Heard: TBA
Abstract: TBA
11/14/2025 – Pengkun Huang: TBA
Abstract: TBA
11/21/2025 – Roger Murray: TBA
Abstract: TBA
11/28/2025 – THANKSGIVING BREAK: NO TALK
12/05/2025 – Luis Atzin: TBA
Abstract: TBA
12/12/2025 – Harrison Gimenez: TBA
Abstract: TBA