Graduate Colloquium

The graduate colloquium hosts a wide range of speakers, from students to professors, with talks at a variety of levels, from expository to research-level. For students, this is a great opportunity to cement material that you may be currently learning/doing research in, and simultaneously gain valuable speaking experience. No matter what your intended career trajectory, the ability to give a comprehensible talk is essential. This semester, speakers can decide whether they would like to receive anonymous feedback from the attendees after their talk is over in order to see what they are doing right and not-so-right.

Speakers:

Talk Title: Determinantal Ideals

Abstract: In this talk, we will take a look at so-called determinantal ideals; that is, ideals generated by all minors of a given size of some matrix M. The study of these ideals borrows techniques from algebra, combinatorics, and representation theory, and has a very long history. We will see how these ideals are unusually well behaved, yet simultaneously intractably complicated. Finally, we will see how determinantal ideals can be generalized by thinking of the indexing sets of the associated minors as being parametrized by a simplicial complex, or even more generally, a hypergraph.

Talk Title: Algebras with Straightening Law

Abstract: Algebras with Straightening Law (or ASLs) were introduced by Eisenbud, DeConcini, and Procesi as a means of studying homological properties of determinantal varieties. After giving the definition of an ASL and presenting some examples, I will demonstrate how a commutative algebraist might use the fact that an algebra is an ASL to prove that it is Cohen-Macaulay (a very desirable property). This talk will assume only undergraduate algebra.

Talk Title: Quasipolynomials

Abstract: A quasipolynomial is a function from the integers to the integers whose coefficients are periodic functions. Many very basic constructions turn out to be quasipolynomials, such as floors and ceilings of rational polynomials. Quasipolynomials emerge naturally in solutions of enumerative problems in algebra, geometry and combinatorics. I will overview how they allow you to count the number of lattice points in a convex polytope and encode the Hilbert function of a variety. We’ll find a Smith Normal Form over the ring Z[t] even though it's not a PID!

Talk Title: Proper Diameter for Graphs

Abstract: Given an edge coloring of a graph G, a properly colored path in G is a path in which no two consecutive edges have the same color. The coloring is called properly connected if there exists a properly colored path between every pair of vertices. The proper distance between a pair of vertices is the length of the shortest properly colored path connecting them. The proper diameter of a graph is the longest proper distance between any pair of vertices. Since the proper diameter is a function of the given coloring, we can construct a set of proper diameters for G ranging across all properly connected colorings. We consider the maximum of this set. If G has n vertices, a trivial upper bound on the proper diameter is n-1, but this upper bound is not attainable for all graphs. In this talk, we characterize all graphs that attain the upper bound n-1.

Talk Title: Relationships Among Quadratic Fields, Binary Quadratic Forms, and Modular Forms

Abstract: A binary quadratic form is a homogeneous polynomial of degree two in two variables with integer coefficients. Consider the binary quadratic forms Q0(x,y)=x2+xy+6y2, Q1(x,y)=2x2+xy+3y2, and Q2(x,y)=2x2−xy+3y2. We say that a quadratic form represents an integer n if the equation Q(x,y)=n has an integer solution. Let r(Q,n) denote the number of integer solutions to this equation. In this talk, we will use the theory of Quadratic Fields and Modular Forms to determine explicit formulas for r(Q,n) for each of the three binary quadratic forms Q0, Q1, and Q2.

Q&A with the Graduate Director

Talk Title: Burnside's Theorem

Abstract: The theorem was discovered by William Burnside in 1904 which states that any finite group of order p^a*q^b is soluble. Although this theorem can be proven without Representation Theory, it was not done until the early 1970s. In this talk, we will go over what a representation of a finite group G is, some properties of the entries in its character tables, and how we can use this information to prove Burnside's Theorem.

  • 3/30/21: No Talk (wellness day)

  • 4/6/21: Drew Meier (local)

Talk Title: Anti-Ramsey Numbers and Matroids

Abstract: In this talk, we will visit various variations on the theme of Ramsey Numbers with the goal of discussing the recently popular Anti-Ramsey Number of $t$ edge-disjoint rainbow spanning trees. In an effort to gain an intuition behind one of the most powerful theorems for this Anti-Ramsey Number (a Nash-Williams type result), we include a brief discussion of the incredible world of matroids. We then turn to an extension lemma which allows one to extend edge-disjoint rainbow spanning forests to edge-disjoint rainbow spanning trees. We conclude with further generalizations of this number along with directions for future research.

Talk Title: Approximating Entanglement Measures

Abstract: Entanglement is a fundamental concept of quantum mechanics. Entanglement measures are a common tool used to quantify the amount of entanglement present in a system. Many entanglement measures are defined using the convex roof construction, but such measures are often difficult to compute directly. I will discuss the existence of optimal pure state ensembles for a certain class of entanglement measures defined by the convex roof construction. I will also present a quantum algorithm that can approximate optimal pure state ensembles for such measures. Moreover, in order to make the talk accessible to all, I will first cover some basics of quantum mechanics.

  • 4/20/21:

Panel Discussion on Qualifying/Comprehensive Exams