# Algebra and its Applications—Spring 2024

## Schedule and Talks

- Jan 18
**Organizational meeting** - Jan 25
*Brandon Doherty***Cofibration category of directed graphs for path homology**- Cofibration categories are categories equipped with designated classes of morphisms, called cofibrations and weak equivalences, satisfying certain properties which allow for the convenient construction of a homotopy category. Similar in concept to model categories, though weaker, cofibration categories present homotopy theories having small homotopy colimits. In this talk, I will describe a cofibration category structure on the category of directed graphs, whose weak equivalences are maps inducing isomorphisms on the path homology groups defined by Grigor’yan, Lin, Muranov, and Yau. This talk is based on joint work with Carranza, Kapulkin, Opie, Sarazola, and Wong, arXiv:2212.12568.

- Feb 1
*Franquiz Caraballo Alba***Matroids: from Linear Independence to Hyperplane Arrangements**- Matroids are a combinatorial object generalizing the concept of linear independence in sets of vectors. In this talk we will develop the definition of a matroid, using the study of linearly independent sets in a vector space as a starting point. We will then study how operations on matroids correspond to operations on a vector space, with a view towards how to assign a matroid to a hyperplane arrangement in projective space. Finally, we will discuss other combinatorial structures arising from matroids; the lattice of flats and building sets of this lattice. This talk will serve as background for a future talk on the Chern-Schwartz-MacPherson class of the complement of a hyperplane arrangement.

- Feb 8
*Franquiz Caraballo Alba***Chern-Schwartz-MacPherson classes**- The Chern-Schwartz-MacPherson (csm) class of a variety \(X\) is a generalization of the Chern class of the tangent bundle of \(X\) when \(X\) is possibly singular. In this talk, we will develop the intuition behind csm classes and discuss some computation techniques, which specialize to the case of wonderful models of hyperplane arrangements.

- Feb 15

- Feb 22
*Marcus Lawson***Global \(p\)-Curvatures of Linear Recurrence Operators**- Linear Recurrence Operators appear as objects of interest in the study differential equations, number theory, QFT and a variety of other areas. One property that we may look at is the \(p\)-Curvature. If for all but finitely many primes, the characteristic polynomial of the \(p\)-Curvature of an operator is the image of some fixed polynomial over \(\mathbb Q\), then we say that our operator has a global \(p\)-Curvature. It would appear that many linear recurrences that appear in the OEIS have a global \(p\)-Curvature. The purpose of our study is to gain necessary and sufficient criteria for an operator to have a global \(p\)-Curvature. We do this by first developing techniques in Maple to compute these objects, and then using our experimental data as a starting point towards developing an understanding of the criteria we are seeking.

- Feb 29
*Maxime Ramzi*(Copenhagen)**From Hochschild homology to traces and back**- Traces in symmetric monoidal categories are a generalization of the trace of a matrix, and they enjoy a number of pleasant properties reminiscent of the usual trace, such as cyclic invariance. In this talk, I will explain how these properties are encoded in (topological) Hochschild homology and how in turn, structural properties of Hochschild homology can be used to infer (calculational) properties of traces and related objects in classical algebraic topology. Time permitting, I will explain how to go back, and extract calculational properties of Hochschild homology. This talk is in part based on joint work with Carmeli, Cnossen and Yanovski, and partly based on joint work with Klein and Malkiewich.

- Mar 7
*Brandon Story***Multidegrees of Monomial Cremona Transformations**- Multidegrees are an important sequence of natural numbers associated to a rational map of projective schemes that are closely related to Segre classes. In this talk, we will discuss how one may compute the multidegrees of a rational monomial map and how questions concerning monomial Cremona transformations can be thought of in terms of volumes of certain simplices.

- Mar 14
**Spring Break** - Mar 21
*Chris Kapulkin*(UWO)**Calculus of fractions for higher categories**- A central objective of (abstract) homotopy theory is to understand
the localization of a category at a class of weak equivalences. While
the localization is always known to exist, it is typically very
difficult to compute. One case in which a workable model for the
localization can be described is when the class of weak equivalences
satisfies “calculus of fractions,” introduced by P. Gabriel and M.
Zisman in their 1967 book.

I will report on joint work with D. Carranza and Z. Lindsey (arXiv:2306.02218) that generalizes calculus of fractions to higher category theory. We show that for higher categories satisfying our condition the localization can be computed via a marked version of Kan’s Ex functor. These results have since been applied in several areas, including combinatorics (joint with D. Carranza and J. Kim) and string topology (A. Blumberg and M. Mandell), but we continue to look for new applications.

- Mar 28
*Paolo Aluffi***An explicit generating function for the betti numbers of \(\overline{\mathcal{M}_{0,n}}\)**- The variety \(\overline{\mathcal{M}_{0,n}}\) parametrizes stable rational curves with \(n\) marked points. This is a central object in algebraic geometry, as the most studied and best understood moduli space of curves. Explicit constructions of this variety have been known for several decades, and recursion formulas for its betti numbers were obtained more than 30 years ago, but (to our knowledge) a more explicit expression for the betti numbers was not available. We obtain just such an expression, in the form of an explicit generating function for the class of \(\overline{\mathcal{M}_{0,n}}\) in the Grothendieck group of varieties. As an application, we prove an asymptotic form of log concavity for the Poincaré polynomial of \(\overline{\mathcal{M}_{0,n}}\).

- Apr 4
*Milind Gunjal***Introduction to Stable Homotopy Theory**- In this talk, we will observe the phenomenon of stable homotopy groups of spheres, and we will try to generalize it for a bigger setting by defining spectra. We will also discuss some interesting properties of spectra that make them so useful.

- Apr 11
*Piotr Pstrągowski*(Harvard)**The even filtration**- The even filtration, introduced by Hahn-Raksit-Wilson, is a canonical filtration attached to a commutative ring spectrum which measures its failure to be even. Despite its simple definition, the even filtration recovers many arithmetically important constructions, such as the Adams-Novikov filtration of the sphere or the Bhatt-Morrow-Scholze filtration on topological Hochschild homology. I will describe a linear variant of the even filtration which is naturally defined on associative rings.

- Apr 18
*Heba Badri Bou KaedBey***Solving Third Order Linear Difference Equations in Terms of Second Order Equations**- Classifying order 3 linear difference operators over \(\mathbb{C}(x)\) that are solvable in terms of lower order difference operators. In this talk, I will focus on one of the cases of this classification and give the algorithm we developed. I will give an example from OEIS (The On-Line Encyclopedia of Integer Sequences) where our algorithm produces an output that proves a conjecture from Z.-W. Sun.

- Apr 25
*Matthew Winters***Rational torsion and reducibility for abelian varieties associated to newforms**- Let \(f\) be a newform and \(A\) its associated abelian variety. We have shown before that for certain primes \(r\), if \(A\) is an optimal semistable elliptic curve with reducible torsion subgroup \(A[r]\), then \(A\) has rational \(r\)-torsion. In this talk we define necessary terms and how this result generalizes to other abelian varieties. We will sketch the proof and discuss the differences that arise when \(A\) is not necessarily an elliptic curve.