Algebra & Combinatorics Seminar:   2024–25

Abstract. In this talk, we will discuss about duadic codes (duadic group algebra codes) over some special class of finite rings. These codes are a family of abelian codes, which are themselves generalization of cyclic codes.

We will discuss about duadic codes of odd length over $\mathbb{Z}_4+u\mathbb{Z}_4, u^2=0$ and over $\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2, u^2=v^2=0, uv=vu$. We study these codes by considering them as a class of abelian codes and using the Fourier transform approach. In general, we will consider the algebraic structure of abelian codes over these rings. Some properties of the torsion and residue codes of abelian codes are studied. We will discuss about some results related to self-duality and self-orthogonality of duadic codes. Some conditions on the existence of self-dual augmented and extended codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ as well as over $\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2$ will be determined.

We will also discuss about a new Gray map over $\mathbb{Z}_4+u\mathbb{Z}_4$ under which an abelian code over $\mathbb{Z}_4+u\mathbb{Z}_4$ is an abelian code over $\mathbb{Z}_4$. We have obtained five new linear codes of length $18$ over $\mathbb{Z}_4$ from duadic codes of length $9$ over $\mathbb{Z}_4+u\mathbb{Z}_4$ as images of Gray map and a new map defined from $\mathbb{Z}_4+u\mathbb{Z}_4$ to $\mathbb{Z}_4^2$. The parameters of these codes are $[18, 4^42^{10}, 4], [18, 4^52^8, 4], [18, 4^42^5, 8], [18, 4^02^9, 8]$ and $[18, 4^22^5, 6]$. The code with parameters $[18, 4^02^9, 8]$ is self-orthogonal.

We will then discuss about abelian codes over $\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2$, and their Gray images.

Abstract. Frobenius or the $p$-th power map is crucial in defining singularity classes in characteristic $p > 0$, especially those appearing in the birational classification of algebraic varieties. On the other hand, the obstruction to smoothness is homological, according to a celebrated theorem of Serre. In this talk, we will show that Frobenius witnesses this homological obstruction to smoothness. This will explain the effectiveness of Frobenius in detecting singularities, from a homological point of view. The key will be to produce (explicit) generators of the bounded derived category of a variety in characteristic $p > 0$ from perfect complexes using the Frobenius pushforward functor. Our results recover earlier characterizations of smoothness using Frobenius, such as Kunz's theorem. Part of the talk will report a joint work with Matthew Ballard, Patrick Lank, Srikanth Iyengar and Josh Pollitz.

Abstract. A key objective of extremal combinatorics is to investigate various conditions on combinatorial structures (such as graphs, set systems, and simplicial complexes) that guarantee the existence of specific substructures. In this talk, I will concentrate on two central topics within this theme of extremal combinatorics:

1. Turán problems and
2. Embedding spanning subgraphs.

1. Dirac's theorem, which establishes a sharp minimum degree condition on a graph to ensure the existence of a Hamilton cycle, and
2. Theorems on various orientations of Hamilton cycles in tournaments.

Abstract. I will begin by reviewing some basic facts about vector bundles on the Grassmannian $Gr(r,n)$ and state the Borel–Weil–Bott theorem. The space of maps from a smooth projective curve $C$ to $Gr(r,n)$ is compactified by the Quot scheme. In this talk, we define $K$-theoretic invariants involving Euler characteristics of vector bundles over these Quot schemes. We show that these invariants naturally fit into a topological quantum field theory. Additionally, we demonstrate that the genus-zero invariants recover the quantum $K$-ring of $Gr(r,n)$, and provide a novel approach for deriving explicit formulas.

Abstract. We prove a bijection between the branching models of Kwon and Sundaram, conjectured by Lenart–Lecouvey. To do so, we use a symmetry of the Littlewood–Richardson coefficients in terms of the hive model. Along the way, we introduce a new branching rule with flagged hives. This talk is based on a joint work with Dr. Jacinta Torres.

Abstract. We show that the partial sums of the long Plücker relations for pairs of weakly separated Plücker coordinates oscillate around 0 on the totally nonnegative part of the Grassmannian. Our result generalizes the classical oscillating inequalities by Gantmacher–Krein (1941) and recent results on totally nonnegative matrix inequalities by Fallat–Vishwakarma (2024). In fact we obtain a characterization of weak separability, by showing that no other pair of Plücker coordinates satisfies this property.

Weakly separated sets were initially introduced by Leclerc and Zelevinsky and are closely connected with the cluster algebra of the Grassmannian. Moreover, our work connects several fundamental objects such as weak separability, Temperley–Lieb immanants, and Plücker relations, and provides a very general and natural class of additive determinantal inequalities on the totally nonnegative part of the Grassmannian. This is joint work with Daniel Soskin.

Abstract. We discuss an algebraic version of Schoenberg's celebrated theorem [Duke Math. J., 1942] characterizing entrywise matrix transforms that preserve positive definiteness. Compared to the classical real and complex settings, we consider matrices with entries in a finite field and obtain a complete characterization of such preservers for matrices of a fixed dimension. When the dimension of the matrices is at least 3, we prove that, surprisingly, the positivity preservers are precisely the positive multiples of the field's automorphisms. Our work makes crucial use of the well-known character-sum bound due to Weil, and of a result of Carlitz [Proc. Amer. Math. Soc., 1960] that leads to characterizing the automorphisms of Paley graphs. This is joint with Dominique Guillot and Himanshu Gupta.

Abstract. Algebraic Statistics is a relatively new field of research where tools from Algebraic Geometry, Combinatorics and Commutative Algebra are used to solve statistical problems. A key area of research in this field is the Gaussian graphical models, where the dependence structure between jointly normal random variables is determined by a graph. In this talk, I will explain the algebraic perspectives on Gaussian graphical models and present some of my key results on understanding the defining equations of these models. In the end, I will talk about the problem of structural identifiability and causal discovery and how algebraic techniques can be implemented to tackle them.