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: From small doubling to small “n-pling”
Abstract: For a given finite subset A of an abelian group, the Plünnecke–Ruzsa inequality bounds the size of nA-mA in terms of |A+A|/|A|. I will state several theorems where the inequality played an important role, and give Petridis’ simplified version of the original proofs which keeps the use of the Plünnecke graph, but eliminates the tensor product trick.
10/24/2025 – FALL BREAK: NO TALK
10/31/2025 – Katherine Novey: Failure of Ordinary TQFTs to Distinguish Homotopy Type
Abstract: Topological Quantum Field Theories (TQFTs) provide manifold invariants. There has been significant work done over the past couple of decades investigating the sensitivity of these invariants and which manifolds they distinguish, but this question remains largely open. Work by David Reutter and Chris Schommer-Pries has shown that ordinary TQFTs can distinguish stable diffeomorphism classes of closed, connected, even-dimensional manifolds subject to certain finiteness conditions. In particular, simply connected, closed, smooth $6$-manifolds with finite $\pi_2$ are diffeomorphic if and only if they cannot be distinguished by ordinary TQFTs. The question of whether or not this result holds for all simply connected $6$-manifolds was open. I will show that the answer to this question is no by presenting a pair of simply connected $6$-manifolds with infinite $\pi_2$ that are indistinguishable by ordinary TQFTs. This counter-example can be extended to show that ordinary TQFTs cannot distinguish the homotopy type of simply-connected closed manifolds in all dimensions $\geq 6$ and non simply-connected closed $5$-manifolds. Along the way, I will discuss bordisms with various structures and present a new bordism category of “formally smooth” manifolds required to build the counter example referenced above.
11/07/2025 – Sam Heard: What is a Cluster Algebra?
Abstract: In this talk, we will illustrate an example of a cluster algebra: the coordinate ring of the Grassmannian $\mathbf{Gr}_k(m)$ (the set of k-dimensional subspaces of $\mathbb{C}^m$).We will use this example to motivate the definition of a cluster algebra, as well as general properties of cluster algebras and connections to total positivity.
11/14/2025 – Pengkun Huang: Finite Extensions of Q_p
Abstract: Let p be a prime number. We can filter the ring of integers by powers of p, and equip the ring of integer by the p-adic topology. Fortunately, Z is not complete with respect to this topology, so we can do a completion and get the p-adic integer. In this talk, we will build up p-adic integers by the Witt vectors. They are harder than the usual constructions, and we’re doing in this way not only because mathematicians just like to make things harder, we will see this turns out to be helpful in understanding a class of finite extensions of Q_p.
11/21/2025 – Khoi Nguyen: (Not WIRED) 4 levels of Caclculus of Variations in Geometry
Abstract: One of the most elegant ideas in math is the idea of optimization of a function. The tools of carrying out this procedure are manifold (pun intended), one of which is the calculus of variation. Mimicking the famous WIRED 5 level series, I will attempt to explain this idea in 4 levels that (hopefully) is easily understandable.
11/28/2025 – THANKSGIVING BREAK: NO TALK
12/05/2025 – NO TALK
12/12/2025 – Harrison Gimenez: Coxeter n-cubes
Abstract: Given a Coxeter system (W,S) and element w of W, the left inversion set of w is the set of positive roots that are mapped to negative roots by the inverse of w. Coxeter squares are commutative square diagrams whose edges describe how left inversion sets are mapped to other left inversion sets by elements of W. We investigate properties of higher dimensional analogues of Coxeter squares called Coxeter n-cubes, which are hyper-cubical diagrams the two-dimensional faces of which are Coxeter squares. In the symmetric groups, we describe relationships between Coxeter n-cubes, full binary trees with n+1 leaves, and bigrassmannian permutations.