### 2023–2024 Academic Year

**Organized by**: Anirban Bhaduri (abhaduri@email.sc.edu) and Pankaj Singh (pksingh@email.sc.edu).

If you wish to participate as a presenter in this seminar, kindly get in touch with
either of the organizers

As active researchers, we come across many new ideas and theories which help us in understanding our field in a better way. The seminar presents a great platform to discuss these ideas with our peers and help each other to grow into a better mathematician/Researcher. It also offers an opportunity to bring new problems among the fellow graduate students to create an active environment of collaboration.

This page will be updated as new seminars are scheduled. Make sure to check back each week for information on upcoming seminars.

**Speaker: **Scotty Groth

**Abstract: ***Legendre polynomials* were first discovered/formalized by Adrien-Marie Legendre in 1782. They arise naturally
as coefficients when modeling systems involving gravitational potential, electric
potential, and several other settings of interest in physics. Applications aside,
Legendre polynomials as an object of pure math provide a rich bounty of questions
worth investigating. One question in particular -- are all Legendre polynomials (beyond
trivial factors) always irreducible?

Suppose one wants to determine if a polynomial \(f(x)\in Q [x]\) is irreducible.
Of course, there are several methods for such a task. One method is through Newton
polygons.

In this talk, we give a brief overview of Newton Polygons, and how they relate to
the factorization of polynomials over \(Q\). Further, we discuss how Newton Polygons
can be used in addressing the irreducibility of Legendre Polynomials.

**Speaker:** Pankaj Singh

**Abstract: **Tropical geometry is a combinatorial analogue of algebraic geometry. In this talk
we will use weighted metric graphs with legs to study the topological properties of
a particular subset of the moduli space of tropical curves called the double ramification
(DR) cycles A substantial amount of work has been done on the algebraic double ramification
cycle since it was introduced

**Speaker: **Bailey Heath

**Abstract:** The problem of bounding the size or complexity of certain algebraic objects is a
natural one, and in this talk we discuss the maximal orders of finite groups of integral
matrices. This problem was completed in 2006 thanks, in part, to the classification
of finite simple groups. After briefly discussing these results, we will explore
Friedland's 1997 (mostly) elementary proof of the answer for sufficiently large dimensions.

**Speaker:** Alexandros Kalogirou

**Abstract:** We narrow down the possibilities for exact and almost exact coverings of the integers
by arithmetic progressions.

**Speaker: **Aditya Iyer

**Abstract: **Demi-primes are integers that, when doubled, are one more than a prime. An interesting
conjecture relating demi-primes to graphs is implied by one of the most famous open
problems in number theory: Shinzel’s Hypothesis-H. In this talk we will go over a
proof of this implication

**Speaker:** Uttaran Dutta

**Abstract: **In my next talk, we’ll take a step back. I’ll explain what we mean by a moduli space
and show you how we actually build one using examples. Starting from the very basics,
I will try to make the definitions as intuitive as possible. Prerequisites will be
some basic knowledge of Algebraic Geometry, except for the last example.

**Speaker:** Uttaran Dutta

**Abstract:** Stability is a fundamental concept in Algebraic Geometry, particularly in the construction
of moduli spaces. I will start with the notion of stability for vector bundles on
a smooth projective curve, which can naturally be extended to coherent sheaves. This
will motivate our definition of stability functions on an abelian category. Our goal
will be to extend this to an arbitrary triangulated category. Although this may seem
very abstract, we will see that the ideas are very much motivated from classical theory.
At the end we will see that doing this we get a lot of freedom to deformour stability
functions, and that the space of stability conditions form a complex manifold.

**Speaker:** Matthew Booth

**Abstract:** An Artinian graded algebra A is said to have the Weak Lefschetz Property (WLP) if
multiplication by a generic linear form has maximal rank in every degree. This definition
was motivated by a 1980 result of Stanley, and it has since found connections to problems
in algebraic geometry, commutative algebra, and combinatorics. We shall survey some
properties and known results of WLP, with a particular emphasis on a recent paper
of Dao & Nair characterizing WLP from degree 1 to degree 2 of an Artinian monomial
algebra in terms of an associated simplicial complex. In another direction, a pair
of nonzero elements (x, y) in a ring R is called an exact pair of zero divisors if
\(Ann_R(x) = (y)\) and \(Ann_R(y)=(x)\) . (Such pairs of elements are interesting
if for no other reason than affording a construction of nonfree totally reflexive
modules.) If time permits, we will examine some necessary and sufficient conditions
for the existence of exact zero divisors in quadratic algebras with a fixed Hilbert
function. Of the work done in this area, the contributions of Conca, Yoshino, Kustin
& Vraciu, and Christensen et al are especially noteworthy.

**Speaker:** Pat Lank

**Abstract:** This talk will be an overview of strong generation and level in the derived category
of bounded complexes with coherent cohomology for a Noetherian scheme. We will discuss
down to earth examples for those who have an introductory background in commutative
algebra or algebraic geometry. Hopefully, this will convince the audience that the
machinery discussed can be viewed as a noncommutative analog of regularity in the
flavor of a classical result by Auslander-Buchsbaum-Serre.

**Speaker:** Jonathan Smith

**Abstract:** Within classical algebraic geometry, the Riemann-Roch theorem for curves relates
an arbitrary divisor to the curve’s canonical divisor and genus. In 2007, M. Baker
and S. Norine derived an analogous theorem for linear systems of divisors on finite
graphs. The Riemann-Roch theorem for graphs has combinatorial applications, but it
also acts as a starting point for a more robust dictionary between algebraic geometry
and graph theory. In this talk, we will introduce linear systems of divisors, state
the Riemann-Roch theorem for graphs, investigate an application or two, and mention
some other notions from algebraic geometry that can be reformulated for graphs. This
talk should be accessible for graduate students that do not have a background in algebraic
geometry.