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Research Seminars: Graduate Student Colloquium

Spring 2021

Time & Location: All talks are on Tuesdays at 5:00 PM unless otherwise noted.
Organizer: Houser, Hayden


April 13

Title: A Gentle Introduction to Moduli Spaces
Corey Wolfe | Tulane University

Abstract: Many spaces of classical interest in algebraic geometry may naturally be regarded as moduli space: flag varieties, Hilbert schemes, moduli spaces of abelian varieties, and so many more! In this talk, we set aside these sometimes intimidating spaces and explore moduli spaces of more familiar objects: lines and triangles. Using concrete examples, we will get acquainted with these moduli spaces and their “natural” geometric structure.

Time: 5:00pm
Zoom ID: 954 0971 2190


March 30

Title: The Representation Theory of Angular Momentum in Quantum Mechanics
Zachary Bradshaw | Tulane University

Abstract: Often, the theory of Lie groups, Lie algebras, and their representations make an appearance in physics. The underlying symmetries responsible for this appearance are exploited by physicists to reduce the complexity of a given problem. For example, the rotational symmetry in the Hydrogen atom is used to split the 3-dimensional problem of finding the eigenvalues of the associated Hamiltonian into two parts: finding a solution for the angular part (a 2-dimensional problem) and finding a solution for the radial part (1-dimensional). These solutions are then pieced back together. In this talk, I will discuss the theory of angular momentum in quantum mechanics from the perspective of the representation theory of Lie groups and Lie algebras.

Time: 5:00pm
Zoom ID: 954 0971 2190


March 23

Title: Convex sets associated to algebraic objects
Thai Nguyen | Tulane University

Abstract: I will talk about a way to associate convex sets to polynomials and ideals in polynomial rings. This idea can be traced back to an idea of Issac Newton (~1676) that was used to prove Newton-Puiseux theorem. In 1996, Okounkov introduced a way to associate convex sets to algebraic varieties in order to show the log-concavity of the degrees of those algebraic varieties. Later on, this construction was studied systematically by the works of Kaveh-Khovanskii and Lazarsfeld-Mustata (~2009). It has become a very active research area recently and is now known as the theory of Newton-Okounkov bodies. Among numerous applications of this theory, I will present a beautiful one, which is to count the number of solutions of a system of (general) polynomials (so-called Kouchnirenko-Bernstein theorem that was proved many years ago). I will also discuss how this relates to my current project, involving Newton polyhedra and symbolic polyhedra.

Time: 5:00pm
Zoom ID: 954 0971 2190


March 16

Title: Ulam´s problem
John Lopez | Tulane University

Abstract: In this talk we will take a look to Ulam´s problem, which is related to the length (l_n) of the longest increasing subsequence in a random permutation of {1, 2, 3,...,n}. This question has been studied by a variety of increasingly technically sophisticated methods over the last 30 years. We will study the asymptotic behavior of the expected value of l_n, (E(l_n)), by analyzing the limit of E(l_n)/sqrt(N). Specifically, we will derive a lower bound for this limit using some combinatorial arguments and will also introduce some probabilistic tools which are used to determine its existence

Time: 5:00pm
Zoom ID: 954 0971 2190


March 9

Title: Spinors and Representation Theory of SO(n)
Hayden Houser | Tulane University

Abstract: In this introductory talk, we will discuss the relationship between irreducible representations of semisimple Lie algebras and their associated Lie groups. In particular, we explore how the construction of irreducible representations of an even orthogonal Lie algebra requires consideration of the representations of Spin(n), the universal covering group of SO(n).

Time: 5:00pm
Zoom ID: 954 0971 2190


March 2

Title: An Intro to Knot Invariants
Will Tran | Tulane University

Abstract: The word "knot" and phrase "knot theory" have been intimidating students since the early 1900s. We may see the word "knot" and immediately shut down, thinking "I don't know enough topology to study knots." It turns out that understanding knot invariants -- a primary focus in knot theory -- requires little to no topology at all! In fact, with some undergraduate linear algebra and an open mind, you too can get started on understanding knot theory. In this talk, we'll compute elementary knot invariants such as p-colorability, the knot determinant, and the Alexander Polynomial. If time permits, we'll see how other knot invariants like the Gauss Linking Integral, Linking Number, and Moebius Energy can connect to number theory, algebra, analysis, biology, and physics.

Time: 5:00pm
Zoom ID: 954 0971 2190


February 2

Title: Living your best life with symbolic math software
Dana Ferranti | Tulane University

Abstract: I've only recently started using symbolic math software in my research and I wish I started earlier. This talk will focus on a few ways that one can use symbolic math software to make their workflow more efficient, as well as its limitations. For my examples I will be using the open source SymPy, however, the concepts will be applicable to other software.

Time: 5:00pm
Zoom ID: 954 0971 2190


February 9

Title: Linearity Testing
Victor Bankston | Tulane University

Abstract: We will discuss techniques for deciding if a Boolean function is linear.

Time: 5:00pm
Zoom ID: 954 0971 2190


February 23

Title: A tutorial on Peirce's Graphical Logic
Alex Nisbet | Tulane University

Abstract: Charles S. Peirce was in important figure in the development of the ``algebra of logic'' in the second half of the 19th and early 20th century. In particular, he is known for his contributions in developing Boole's system, the logic of relatives, and quantification. Indeed, the algebraic notation he used is more-or-less what we use today, after some symbolic changes by Peano. Less well-known is Peirce's system of graphical logic, which he preferred to the algebraic notation. Here I will give a brief tutorial on the alpha and beta parts of this system. The alpha part corresponds to propositional logic and the beta to first order logic with equality.

Time: 5:00pm
Zoom ID: 954 0971 2190