09/04/2023 – Matt Scalamandre: A Solomon-Tits Theorem for Rings
The classical Solomon–Tits theorem states that a spherical Tits building over a field is homotopy equivalent to a wedge of spheres of the appropriate dimension. In this talk, we’ll define a Tits complex that makes sense for an arbitrary ring, and prove a Solomon-Tits theorem when R either satisfies a stable range condition, or is the ring of S-integers of a number field. We will discuss applications to the cohomology of principal congruence subgroups of SL_n(Z), and some results about the top homology of this complex (an analogue of the classical Steinberg representation).
09/11/2023 – Matt Weaver: On the equations of Rees algebras of ideals in hypersurface rings
The search for the defining equations of Rees algebras is a classical problem within both commutative algebra and algebraic geometry. Geometrically, the Rees ring can be realized as the homogeneous coordinate ring of the blowup of a variety along a subvariety. Algebraically, the Rees algebra records data regarding all powers of an ideal and their various syzygies. In this talk, we will discuss how classical techniques can be updated and how methods from computational algebra can be employed to study the Rees algebra of an ideal in a hypersurface ring. Moreover, we will discuss current projects expanding on these ideas, along with possible directions for future work.
09/18/2023 – Connor Malin: Koszul duality from a homotopical perspective
We introduce the basics of abstract homotopy theory and use it to set up the theory of Koszul duality. We revisit some of the classical results in the theory of differential graded algebras and more modern results in topological settings.
10/02/2023 – Lorenzo: 9 Out Of 10 Topologists Baffled! Find Out How To Get Rid Of Bothersome Pathological Spaces With This One Weird Trick
The definition of a topological space formulated in the 20s has been a source of fruitful developments in many areas of mathematics, including but not limited to topology (duh), analysis, set theory, probability theory, and model theory. Algebraic topologists later realized, however, that this definition did not lend itself well to the study and computation of algebraic invariants of spaces, and chose instead to work with a restricted class of spaces that could alternatively be described in a simple combinatorial way. In this talk I will introduce one of the modern combinatorial descriptions of spaces — simplicial sets — and demonstrate some of its advantages and disadvantages. If time permits we will explicitly construct BG, the classifying space for principal G-bundles, using simplicial sets.
10/09/2023 – Khoi Nguyen: A Panoramic View of (Basic) Riemannian Geometry
In this talk, I will give a gentle introduction to the field of Riemannian geometry. Starting with the geometry of surfaces in R^3, I will discuss an intuitive picture for many concepts that are central to Riemannian geometry. I will discuss vector fields, metric, connection, and curvature for general Riemannian manifolds, all without the formalism of indices and tensor calculus. If time permits, I will give an idea of what I am currently doing for my oral exam (and research).
10/23/2023 – Cory Gillette: TBD
I like Hopf algebras, and you should too. In fact, you probably already do, and maybe just don’t know it yet. We will define these, show how many familiar structures fit the definition, and give some explanation of the nice features that come for free. As time permits, we will discuss some special types of Hopf algebras, illustrated by examples. This talk should appeal to topologists and algebraists, but if analysts show up, I will mutter something under my breath about locally compact Hausdorff groups and quickly move on.
10/30/2023 – Karim Boustany: To Be Or Not To Be (Symplectic)
Symplectic topology exhibits strange phenomena. Indeed, the question of what exactly is symplectic behavior can have some very subtle answers, fitting into the framework of a “rigid/flexible” dichotomy. This dichotomy leads to valuable insights as well as the definition of interesting invariants. In this talk, I will discuss some instances of this dichotomy, and in the process showcase some cool results from the field. No prior knowledge of symplectic topology is assumed.
11/06/2023 – Jason Mitrovich: the resolution of the yamabe problem
The Yamabe problem is concerned with searching for conformal metrics from a prescribed scalar curvature. In this talk, I will first discuss some background regarding conformal metrics and scalar curvature. Then I will discuss Yamabe’s original motivation and how this problem is connected to Poincaré’s conjecture. With this, I will sketch out a proof for the Yamabe problem when we consider a restricted class of Riemannian manifolds. With the remaining time, I will discuss some progress made in other cases of interest.
11/27/2023 – Jake Zoromski: A History of Mathematics for Instructors
The goal of this talk is to explain how the concepts of calculus, as taught in a standard Calc 1 course, came to be. We will trace their historical development from ancient methods of finding area, to the birth of calculus as a field with Newton and Leibniz, to modern notions of limit and number, and beyond.
12/04/2023 – Panel: Non-academic jobs
This seminar will be a panel discussion featuring four former and current Notre Dame graduate students in mathematics who are currently working in a job outside of academia, or in academia but not as a math professor.
The panelists are:
Benjamin Jones ( Notre Dame PhD, 2007, Senior Applied Scientist at Amazon Web Services, Portland)
Ben Lewis (Notre Dame PhD, 2017, Data Scientist at Shopify)
John Siratt (Current Notre Dame PhD student, researcher in the Formal Methods Program at NASA)
Taylor Ball (Notre Dame PhD, 2021, Department of Defense, Applied Mathematics Researcher)