Graduate Colloquium at University of South Carolina
Schedule
 9/20/22

 Speaker: Gabrielle Tauscheck “An Introduction to Distance Critical Graphs”

 Abstract: Graham and Pollak discovered that the determinant of the distance matrix of any tree is given by \((n1)(2)^{n2}\) independent of the structure of the tree. The simplicity of this equation’s proof relies on the fact that every tree has at least two pendant vertices and deleting these vertices does not change the remaining entries of the distance matrix. We vary this property to instead consider graphs that possess the characteristic such that deleting a vertex does change at least one of the remaining entries of its distance matrix. We call these graphs “distance critical” and present our results so far.
 9/27/22

 Speaker: Mohamed Wafik “Construction of Polynomials with Prescribed Divisibility Conditions on the Critical Orbit”

 Abstract: Let \(f_{d,c}(x)=x^d+c\in\mathbb{Q}[x]\), \(d\ge 2\). We write \(f_{d,c}^n\) for \(\underset{\text{n times}}{\underbrace{f_{d,c}\circ f_{d,c}\circ\dots\circ f_{d,c}}}\). The critical orbit of \(f_{d,c}(x)\) is the set \(O_{f_{d,c}}(0):=\{f_{d,c}^n(0):n\geq 0\}\). For a sequence \(\{a_n:n\geq 0\}\), a primitive prime divisor for \(a_k\) is a prime dividing \(a_k\) but not \(a_n\) for any \(1\leq n < k\). A result of H. Krieger asserts that if the critical orbit \(O_{f_{d,c}}(0)\) is infinite, then each element in \(O_{f_{d,c}}(0)\) has at least one primitive prime divisor except possibly for \(23\) elements. In addition, under certain conditions, R. Jones proved that the density of primitive prime divisors appearing in any orbit of \(f_{d,c}(x)\)$ is always \(0\). Inspired by the previous results, we display an upper bound on the count of primitive prime divisors of a fixed iteration \(f_{d,c}^n(0)\). Fixing \(n\geq 2\), we also, under certain assumptions, calculate the density of primes that can appear as primitive prime divisors of \(f_{2,c}^n(0)\) for some \(c\in\mathbb{Q}\). Further, we show that there is no uniform upper bound on the count of primitive prime divisors of \(f_{d,c}^n(0)\) that does not depend on \(c\). In particular, given \(N>0\), there is \(c\in\mathbb{Q}\) such that \(f_{d,c}^n(0)\) has at least \(N\) primitive prime divisors.
 10/4/22

 Speaker: Aditya Iyer “Newton Polygons and Symmetric Galois Groups”

 Abstract: A well known result in Galois theory, attributed to Hilbert, is that for any positive integer \(n\), there exist irreducible polynomials over \(\mathbb{Q}\), having degree \(n\), with Galois group \(S_n\). Hilbert’s proof, however, was existential, not constructive. Schur gave a constructive proof of the same. In this talk, I will try to shed some light on a simple proof of Schur’s result.
 10/11/22

 Speaker: George Brooks “Introduction to Flag Algebras”

 Abstract: The theory of flag algebras, developed in 2007 by Alexander Razborov, has arisen as a powerful tool with applications in many areas of mathematics. Using the theory, many difficult questions can be reduced to a semidefinite programming problem. With the assistance of computers, this method gives an algorithmic approach to solving open problems in asymptotic extremal combinatorics. In this talk, I will give some motivation by proving a version of Mantel’s Theorem, followed by a discussion of the theory of flag algebras and its applications.
 10/18/22

 Speaker: Jonathan Smith “Automorphisms of del Pezzo Surfaces”

 Abstract: The plane Cremona group over a field \(k\), denoted \(Cr_2(k)\), is the group of birational automorphisms of \(\mathbb{P}^2_k\). This group has garnered interest among algebraic geometers for well over a century. The finite subgroups of \(Cr_2(k)\) are of particular interest, and in 2009, I. Dolgachev and V. Iskovskikh provided an essentially complete classification of the finite subgroups of \(Cr_2(\mathbb{C})\) up to conjugacy. The classification uses the fact that every finite subgroup of \(Cr_2(k)\) is realized as a group of regular automorphisms of a del Pezzo surface or a conic bundle. For an arbitrary field \(k\) of characteristic zero, to classify the finite subgroups of \(Cr_2(k)\), one would like to determine which groups act on some del Pezzo surface over \(k\). In this talk, we examine the structure of the plane Cremona group and del Pezzo surfaces and offer some results that generalize the classification of automorphisms of del Pezzo surfaces to arbitrary fields of characteristic zero.
 10/25/22

 Speaker: Tiju John “Introduction to Quantum Probability”

 Abstract: Development of quantum mechanics in the 1920s and 1930s demanded a new kind of probability theory, which is now known as quantum probability or noncommutative probability. This theory is a generalization of classical probability theory but is drastically different when the random variables are noncommutative. In this talk, we will have an introduction to this subject by drawing comparisons and differences with classical theory. We will avoid technicalities and generalizations as much as possible. This talk will be accessible to all graduate students.
 11/1/22

 Speaker: Shreya Sharma “Classification of smooth Fano 3folds”

 Abstract: A complex projective variety with ample anticanonical divisor \(K_X\) is called a Fano variety. These varieties were first studied by Gino Fano in the first half of the 20th century. More recently, Fano varieties have found importance in the birational classification of algebraic varieties through Mori’s Minimal Model Program. We call a Fano variety of dimension \(3\), a Fano \(3\)fold. The classification for smooth Fano \(3\)folds was first completed by Iskovskikh for Picard rank \(\rho=1\) and by Mori and Mukai in the case of higher Picard rank \(\rho\). In this talk, I will give a few examples of smooth Fano varieties, in particular of dimension \(3\) and discuss some of their properties relevant to the classification. Time permitting, I will present an outline for the classification of smooth Fano \(3\)folds as given by Iskovskikh, Mori, and Mukai.
 11/8/22


 11/15/22

 Speaker: Ruth Luo “Some problems in Extremal Combinatorics”

 Abstract: Extremal combinatorics studies the maximum size of a combinatorial object that avoids some property or structure. One famous area of study is the Turan problem in graph theory which asks how dense a graph can be while avoiding some forbidden subgraph. This talk is a brief survey of Turan problem and some of its applications to other combinatorial problems. In particular, we discuss the development of the powerful Szemeredi’s Regularity Lemma.
 11/22/22

 11/29/22

 Speaker: Matthew Booth “An invitation to infinite Galois theory”

 Abstract: One of the crown jewels in abstract algebra is the Fundamental Theorem of Galois Theory, which gives a onetoone correspondence between intermediate fields of a Galois extension and subgroups of an associated group of automorphisms (the Galois group). This result is familiar to most students after a firstyear graduate course in algebra. However, a key assumption in this correspondence is that the field extension in question be of finite degree. Infinite algebraic extensions do not obey the correspondence as “nicely” as their finite counterparts. The goal of this introductory talk, after a review of the fundamentals and finite extensions, will be to develop the necessary theory to extend (no pun intended) the fundamental theorem to infinite extensions. In particular, we will examine the Krull topology of a Galois group and some of its properties.