# Algebra and its Applications—Spring 2023

Earlier sessions are available here.

## Schedule and Talks

- Jan 12
**Organizational meeting**

- Jan 19 & Jan 26
*Ettore Aldrovandi***Computer based formalization of Group Theory and Categorical Algebra in Homotopy Type Theory.**- I describe some work in progress in collaboration with Keri D’angelo (Cornell — CS) to formalize crossed modules of groups, hence groups, in Homotopy Type Theory using the Agda proof assistant. No previous knowledge of Type Theory or Agda is required, and I will sketch the main concepts: types, univalence and h-level. I will also illustrate how to pass to higher h-levels beyond sets, thus enabling one to do “categorical algebra without categories.”

- Feb 2
- No seminar

- Feb 9
*Mark van Hoeij***Generating Function of the Squares of Legendre Polynomials.**- There are a number of classical formulas for the generating functions of sequences defined with Legendre polynomials, but one case had eluded a closed form formula, namely for the sequence \(\binom{2n}{n} P_n(y)^2\) where \(P_n\) is the \(n\)’th Legendre polynomial. It turns out that the differential equation for its generating function has some highly unusual properties that I will discuss in the talk.

- Feb 16
*Reese Madsen***Quotients of products of upper half planes by certain groups.**- We will show why the quotient of the product of \(n\) upper half planes by a torsion-free discrete subgroup of \(\mathrm{SL}(2,\mathbb{R})^n\) is a manifold and why the space of forms on the quotient manifold is isomorphic to the space of forms on the product of upper half planes that are invariant under the action of the discrete subgroup.

- Feb 23
*Brandon Story***Lorentzian Properties of the Cremona Transformation Segre Zeta Functions**- A discussion on the relationship between standard and determinantal Cremona transformations and their Segre Zeta function. We will show that the homogenization of the numerator of the Segre Zeta function associated to determinantal Cremona transformations is Lorentzian and that the numerator is log concave for all Cremona transformations of \(\mathbb{P}^3\).

- Mar 2
- No seminar

- Mar 9
*Matthew Winters***Reducibility and rational torsion in elliptic curves**- Let \(A\) be a semistable elliptic curve over the rational numbers which is optimal, and \(r\) be a prime greater than three such that \(A\) has good reduction at \(r\). We show that if \(A[r]\) is reducible, then \(A\) has a rational \(r\)-torsion point.

- Mar 16
**No meeting**(Spring Break) - Mar 23
*Matthew Winters***Reducibility and rational torsion in elliptic curves (Part 2)**- Let \(A\) be a semistable elliptic curve over the rational numbers which is optimal, and \(r\) be a prime greater than three such that \(A\) has good reduction at \(r\). We show that if \(A[r]\) is reducible, then \(A\)

- Mar 30
- No seminar

- Apr 6
*Milind Gunjal***Homotopy theory of Model categories**- The talk is based on a paper by W. G. Dwyer and J. Spalinski. In this talk, I define Model categories, see some examples, then I define homotopy category of Model categories in a couple of ways and see some applications of this homotopy theory.

- Apr 13
*Cindy Lester***Generalizing the determinant**- I will discuss a categorification and generalization of the determinant. Specifically, we will start with the determinant of a matrix from undergraduate linear algebra and discuss how it became a functor from the isomorphism classes of an exact category to a Picard groupoid, which gives information about the exact category’s \(K\)-theory. Then, time permitting, I will discuss the version of the determinant functor for triangulated categories.

- Apr 20
*Cindy Lester***Multi-Determinant Functors for Triangulated Categories**- I will discuss current work that extends the notion of determinant functor from triangulated categories to tensor triangulated categories. Specifically, I will discuss some background, motivation and, to account for the multiexact structure of the tensor, a multi-categorical version of the determinant functor. Lastly, time permitting, I will introduce a secondary definition of determinant functor based on cubical shapes.

- Apr 27
**Topological Hochschild Homology, Mac Lane Homology, stable \(K\)-Theory, and the homology of rings**This is an expository talk (mostly). It has been known for a long time that the \(K\)-Theory of a ring has trace maps towards that ring’s Hochschild homology. In the last decade or so of the last century these maps were refined by Bökstedt (with some help from Waldhausen and others) to a much more refined gadged dubbed Topological Hochschild Homology, or THH, for short. It was relatively soon proved that THH is isomorphic to a certain stabilization of \(K\)-Theory—an amazing result in itself. Even more surprising was the proof by Pirashvili, Waldhausen and others that the two were isomorphic to Mac Lane Homology, a homology theory for rings devised by Mac Lane in the late fifties and soon almost completely forgotten, except from some computations by Breen and Bökstedt who both used different techniques from Mac Lane’s.

Mac Lane’s theory controls extensions of rings, for example, and more recently it has been shown to encode the Postnikov invariants for categorical rings. This theory is based on a mysterious and intriguing construction (the \(Q\)-complex), which one hopes to generalize to rings up to homotopy.