Welcome to the Department of Mathematics

Department of Mathematics at Illinois State University
ISU Algebra Seminar

The ISU Algebra Seminar will meet for the Fall 2017 semester on Thursdays from 12:00 p.m. to 12:50 p.m. in WIH 021. Please let us know if you would like to give a talk. Everyone is welcome to attend. We make an effort to make the talks accessible for the mathematician on the street. Some of the talks will also be accessible to graduate students.

Announcements of upcoming seminars are sent by e-mail to all members of the ISU math department. If you are not a member of the ISU Math department and if you would like to receive seminar announcements, please send e-mail to gzyamsku@ilstu.edu (with the z removed).

Click on the semester of interest below for a schedule of talks.

Date
Speaker
Title
September 7 Sunil Chebolu Organizational Meeting / Fuch's Problem for finite p-groups, Part I
September 14 Sunil Chebolu Fuch's Problem for finite p-groups, Part II
September 21 Lucian Ionescu Constructible Numbers and Periods (Part I)
September 28 Lucian Ionescu Quaternions, Octonions, and Their Applications
October 5 Neil Christensen (Physics Dept) Algebras in Particle Physics-Part II
October 12 Neil Christensen (Physics Dept) Scattering Amplitudes Without Fields
October 19 Wenhua Zhao Recent Developments on Mathieu Subspaces I
October 26 Wenhua Zhao Recent Developments on Mathieu Subspaces II
November 2 Matthew Speck Mathieu-Zhao Subspaces of Commutative Vertex Algebras
November 9 Gaywalee Yamskulna From Vertex Algebras to Mock Theta Functions
November 16 George Seelinger From Algebraic Geometry to Quantum Groups and Beyond
November 30 Matt Speck and Shelby Bush Counting Mathieu-Zhao Subspaces of (Fp )n
Date
Speaker
Title
February 15 Lucian Ionescu Periods, Feynman integrals and Jacobi sums
February 22 Trevor McGuire (IWU) TBA
March 1 Sunil Chebolu TBA
March 8 Sunil Chebolu TBA
March 22 Alberto Delgado TBA
March 29 George Seelinger TBA
April 6 (Thursday) 12-12:50, STV 436 Neil Christensen (ISU Physics) TBA
April 12 Gaywalee Yamskulna TBA
April 19 Lindsay Henderson TBA
April 26 Matt Speck, Richard Sumitro TBA
Date
Speaker
Title
September 7 Sunil Chebolu Recent Progress on Fuchs' Problem (Part I)
September 14 Sunil Chebolu Recent Progress on Fuchs' Problem (Part II)
September 21 Lucian Ionescu On Discrete De Rham Cohomology
September 28 Lucian Ionescu Gauss and Jacobi sums
October 5 Gaywalee Yamskulna N = 4 Conformal Super Algebra and Mathieu Moonshine Conjecture
October 12 Amita Malik (UIUC) p-adic Properties of Sporadic Apery-like Numbers
October 19 George Seelinger Affine Toric Varieties and Zero Sums - I
October 26 George Seelinger Affine Toric Varieties and Zero Sums - II
November 2 Alberto Delgado Construction of the Mathieu group M24
November 9 Fusun Akman Spectral Graph Theory in Gene Regulatory Networks
November 16 Fusun Akman Spectral Graph Theory in Gene Regulatory Networks: Part II
November 30 Alberto Delgado Using Error-Correcting Codes to construct the Mathieu group M24
Date
Speaker
Title
February 4 Lucian Ionescu The Prime Number Theorem
February 11 George Seelinger Introduction to vector space partions over finite fields
February 18 TBA TBA
February 25 Claudio Quadrelli Galois pro-p groups on a diet of roots of the field
Date
Speaker
Title
August 28 Sunil Chebolu What is special about the divisors of 12?
September 4 Gail Yamskulna Shifted Theory of vertex operator algebras
September 11 Lucian Ionescu What a finite field is, really!
October 23 Timothy Comar (Benedictine University) Biological Models using Impulsive differential equations

We discuss two biological mathematical models using impulsive differential equations. The first model is an SEIRV epidemic model with impulsive vaccination. Conditions are found under which the disease-free periodic solution is stable and under which an endemic solution is stable. We then consider a density-dependent one predator, two-prey model for integrated pest management using impulsive differential equations. We explore conditions under which both prey species would be eradicated, only one species would be eradicated, and both species would remain within controlled population levels.

October 30 George Seelinger
Date
Speaker
Title
April 3rd Michael Mayers Diagonal Property for rings.
April 10th Amber Anderson Applications of Pick's theorem
April 17th Jill Horne Algebraic coding
Date
Speaker
Title
October 4 Lucian Ionescu On prime numbers and multiplicative duality

1) Dirichlet series is in fact a transform, part of a multiplicative duality between arithmetic functions and multiplicative characters (integers viewed as part of a Hopf ring). 2) The primitives are the prime numbers. A non-trivial partial order on the primes, probably new, will be introduced. 3) Relevance to Riemann hypothesis, distribution of primes and physics will be briefly mentioned; 4) The audience will be questioned about further insight and hints (if any).

October 11 Lucian Ionescu What is an algebraic quantum group?
October 18 Wenhua Zhao Recent Developments of Mathieu Subspaces.

In these two talks we will discuss the following recent developments of Mathieu subspaces: some (new) examples; characterizations of Mathieu subspaces with algebraic radicals; classifications of strongly simple algebras and quasi-stable algebras over fields; etc.

October 25 Wenhua Zhao Recent Developments of Mathieu Subspaces.

In these two talks we will discuss the following recent developments of Mathieu subspaces: some (new) examples; characterizations of Mathieu subspaces with algebraic radicals; classifications of strongly simple algebras and quasi-stable algebras over fields; etc.

November 1 Pisheng Ding (and Lucian Ionescu) The Hopf Algebra of Representative Functions

This is the beginning of x (positive and not necessarily integral) sessions in which expository survey of examples of Hopf algebras and quantum groups will be given. In this session, the junior speaker will explain the construction of the algebra of representative functions on a compact group and how the Hopf algebra structure on it relates to the Tannaka-Krein duality.

November 8 Jon Carlson (Univ. of Georgia) Classifying Thick Subcategories of the Stable Category

This is joint work with Srikanth Iyengar. I will spend most of the lecture talking about what the words mean and why we are interested.

November 15 Lucian Ionescu The group algebra of the rationals is an algebraic quantum group (link to slides )
November 22 Thanksgiving break No talk
November 29 Sunil Chebolu Finite Generation of Tate cohomology: Recent developments I

This is all joint work with Jon Carlson and Jan Minac. In this talk (and possibly the next one if there is enough interest) I will give a report on the recent developments on one of the most fundamental problems in Tate cohomology: When is Tate cohomology finitely generated? I will also explain what Jon Carlson and I proved during his recent visit on a different but related problem.

December 6 -- No talk
December 13 William Cragoe A survery of modern cryptosystems

We will discuss some modern methods of cryptography. The focus of the discussion will be Elliptic curve cryptography. We will also look at some problems that can be solved using cryptography -- Electronic coin tossing and Yao's millionaire problem.

Date
Speaker
Title
Tuesday, February 15 Sunil Chebolu (Illinois State) Undergraduate Research: Problems in Lattice Point Geometry

Pick's theorem is a rare and beautiful gem in classical mathematics. It can be stated as follows. Let P be a simple lattice polygon (i.e. a connected polygon whose vertices have integer coordinates and whose sides do not interest each other). Pick's theorem states that the area A enclosed by P is given A = I + B/2 -1 where I is the number of lattice points in the interior of P, and B is the number of lattice points on the boundary of P. The beauty of this theorem lies in its simplicity and depth. It can be explained easily to any 4rd grader but on the other hand mathematicians are still investigating some of its deep consequences. In the project we shall investigate the possible values of B for a given value of I. We shall do this systematically beginning with triangles and parallelograms. We shall also study how Pick's theorem can be used to derive some surprising and interesting formulas for computing the gcd of two integers. Some related problems in lattice point geometry which we shall explore are questions which ask for the existence of lattice regular n-gons in 3-dimensional space. Students working on this project are: Amanda Buscher, David Driscoll, Eric Larson (Primary presenter), and Laura Poulos

Tuesday, February 22 Sunil Chebolu (Illinois State) Undergraduate Research: n-Numbers Game

The 4-numbers game which can be described as follows: start with a unit square with four integers at the four corners. In the first step, form a new square connecting the midpoints of the four sides of the given square and assign to the four corners of this new square the absolute differences of the corresponding corners of the original square. For example, if the first square has numbers (2, 6, 1, 9), then the next square has numbers (4, 5, 8, 7). In the following interations of the game we get (1, 3, 1, 3), (2, 2, 2, 2), and (0, 0, 0 , 0). Thus starting with any square, we play the game until we reach a square with numbers (0, 0, 0, 0). Several questions arise very naturally here. When is a game finite? What integers can be lengths of games? How does the length of a game change under permutation of the numbers of the original square? etc. In the same way one can also play an n-numbers game starting with a regular n-gon. It can be shown that every n-numbers games has finite length if and only if n is a power of 2, say 2^k. Therefore the 2^k-numbers deserve special attention. The goal of the project is to investigate the theory of 2^k-numbers games after reading the 4=2^2-numbers game theory given in Chapter 1. So far, all this discussion is over the integers. In another direction we shall investigate, starting with the 2^2 case, these games with elements in some finite fields. Students working on this project are:Ahreum Han, Kimberly Knapik, Dan Sobodas, Ellen Sparks (Primary presenter), Akshata Vaidya.

Tuesday, March 1 Fusun Akman (Illinois State) Comparing Integer and Gaussian Partitions

A finite vector space is a finite-dimensional vector space over a finite field. There is a deep relationship between finite vector spaces and finite sets, which has not been explored properly. Sets are sometimes called vector spaces over the field with one element for the following superficial reason: a finite vector space V=V(n,q) of dimension n over the finite field with q elements has certain numerical properties, which are reduced to the numerical properties of a set S(n) with n elements when q is replaced by 1. In a previous paper, we examined the relationship of the set partitions of S(n) and the subspace partitions of V(n,q). In our current work, we compare the properties of the integer partitions of n and of Gaussian partitions associated with V(n,q). To illuminate these concepts at the set level, we note that {{1,2},{3},{4}} is a set partition of {1,2,3,4} and (2|1|1) is the corresponding integer partition of 4. The number of integer partitions of n, denoted by p(n), is known as the partition function. On the other hand, a subspace partition of V is a collection of subspaces of V such that each 1-D subspace of V appears in exactly one subspace in this collection. If we record the dimensions of the subspaces in a subspace partition of V, then we obtain a Gaussian partition of the total number of 1-D subspaces of V. The Gaussian partition function gp(n,q) is by definition the number of all Gaussian partitions associated with V(n,q), and is arguably the most natural q-analogue of p(n). We compute gp(n,q) for n less than or equal to 5 as a polynomial in q, and find a lower bound for gp(6,q). For each n, we observe that the polynomial yields p(n) when q is replaced by 1. Then we make the obvious conjecture, and investigate a second q-analogue of p(n). This is ongoing (and fun) joint work with Papa Sissokho, and the talk is accessible to undergraduates.

Tuesday, March 8 Spring Break no talk
Tuesday, March 15 Gaywalee Yamskulna (ISU) Development in the theory of shifted vertex operator algebras (talk cancelled)

In their Shifted Vertex Operator Algebra paper, Dong and Mason exhibited large number of Z-graded regular vertex operator algebras by shifting conformal vectors of vertex operator algebras associated to positive definite even lattices. In this talk, we will discuss the following question: Suppose V is an N-graded regular vertex operator algebra such that dim V_0 1, is it possible to construct a vertex operator algebra of CFT type by shifting the conformal vector of V?

Tuesday, March 22 Sunil Chebolu, ISU A Strong Generating Hypothesis

Let G be a finite p-group and let k be a field of characteristic p. A kG-linear map between kG-modules is called a strong ghost map if it induces the zero map in Tate cohomology when restricted to each subgroup of G. We formulate the strong generating hypothesis as the statement that every strong ghost between finitely generated kG-modules factors through a projective module, i.e., it is trivial in the stable module category. In joint work with Jon Carlson and Jan Minac, we have identified the class of p-groups for which this strong generating hypothesis holds. I will present an overview of this work and also our motivation for studying this problem.

Tuesday, March 29 no talk
Tuesday, April 5 no talk
Tuesday, April 19 Rachel Morrison The Burnside Ring of a Finite Group
Tuesday, April 19 Allisha Langdon Divisions of Finite Groups
Thursday, April 21 Sunil Chebolu, Illinois State, (in STV 120) What is special about the divisors of 24?
Friday, May 6 Maria Paduret (STV 120) Quadratic Reciprocity in the Ring of Eisentein Integers
Date
Speaker
Title
Tuesday, August 17 Keir Lockridge (Wake Forest University) Homological dimensions of ring spectra

Homological dimensions are important invariants in classical ring theory. The idea is to compute the maximum number of steps required to build any module out of a fixed class of basic building blocks (e.g., projective or flat modules). This process requires a certain amount of rigid structure in the category of modules (sub-modules, quotients, kernels, etc.), which is an abelian category. In contrast, the ring analogues that arise in algebraic topology, known as structured ring spectra, do not have abelian module categories. Their module categories are examples of triangulated derived categories, where familiar algebraic constructions are `weak.' Fortunately, there is a way to associate a triangulated derived category to a classical ring, and in this talk I will discuss how one can use derived categorical formulations of classical homological dimensions to study dimensions for ring spectra. Important examples of ring spectra include cobordism theories, K-theories, and the spectrum tmf of topological forms.

Tuesday, August 24 Geoff Mason (Univ. of California at Santa Cruz) In pursuit of modularity

There is a deep conjecture concerning the modular-invariance of certain classes of vertex operator algebras. I will discuss recent efforts to understand this using techniques from various areas of mathematics: algebra, number theory and differential equations. (No technical expertise about VOAs is expected or required.)

Tuesday, August 31 Sunil Chebolu (Illinois State University) The Witt Cancellation Theorem

I will introduce quadratic forms and motivate the fundamental problem of classifying equivalence classes of forms by tying it up with classical number theory. In joint work with Dan McQuillen and Jan Minac we are investigating some fundamental theorems on quadratic forms developed by Witt in 1965. I will also present our new approach to the Witt Cancellation Theorem which is the key ingredient in constructing the Witt ring of Quadratic forms.

Tuesday, September 7 Sunil Chebolu (Illinois State University) The Witt Cancellation Theorem II

I will introduce quadratic forms and motivate the fundamental problem of classifying equivalence classes of forms by tying it up with classical number theory. In joint work with Dan McQuillen and Jan Minac we are investigating some fundamental theorems on quadratic forms developed by Witt in 1965. I will also present our new approach to the Witt Cancellation Theorem which is the key ingredient in constructing the Witt ring of Quadratic forms.

Tuesday, September 14 Lucian Ionescu (Illinois State University) Hodge theory on projective manifolds

I will extract topics from the book The Hodge Theory on Projective Manifolds by Mark Andrea de Cataldo. Hodge theory and duality contains for example the correspondence between homology and harmonic forms, and applications to electromagnetism and deformation theory. The example of electric circuits, and the connection with Min-cut Max-flow on graphs is one specific goal - a visual exemplification of the theory. Beyond this "old stuff", the theory of Non-commutative Hodge structures have been recently studied by Kontsevich a.a. PS: Wanted: volunteers to present related topics (or from the book).

Tuesday, September 21 Jim Parr (Illinois State University) Subgroups of Q^2 I

We develop a description of the subgroups of the rational plane that gives insight into their structure, their endomorphism rings and some of their peculiarities.

Tuesday, September 28 Jim Parr (Illinois State University) Subgroups of Q^2 II

We develop a description of the subgroups of the rational plane that gives insight into their structure, their endomorphism rings and some of their peculiarities.

Tuesday, October 5 Sunil Chebolu (Illinois State University) On a small quotient of the big absolute Galois group

Let G be the absolute Galois group of a field that contains a primitive pth root of unity. This is a profinite group which is a central object of study in Galois theory. In joint work with Ido Efrat and Jan Minac we have show that a remarkably small quotient of this big group determines the entire Galois cohomology of G. As application of this result, we give new examples of profinite groups that are not realisable as absolute Galois groups of fields. I will present an overview of this work.

Tuesday, October 12 Lucian Ionescu (Illinois State University) Overview of Hodge Theory

The main ideas and some applications of Hodge theory will be explained: harmonic forms and Hodge decomposition, Kahler identities and correspondence with de Rham and Betti homology.

Monday, October 18 (3:00 PM) Tony Giaquinto (Loyola university Chicago) Meanders and principal elements of Frobenius Seaweed Lie algebras

To each pair a=(a1,...,ar) and b=(b1,...bs) of tuples of positive integers with a1+...+ar = b_1+...+bs = n there is an associated Seaweed Lie algebra L(a|b) which is contained in the special linear Lie algebra sl(n). A Lie algebra L is Frobenius if there exists a linear functional F on L such that the induced skew bilinear form B(-,-) sending (x,y) to F([x,y]) is non-degenerate. Frobenius Lie algebras are important in the study of coadjoint orbits, the classical Yang-Baxter equation, and symplectic Lie groups. An interesting open question question is to characterize when L(a|b) is Frobenius. It turns out that L(a|b) is Frobenius whenever its "meander" graph is a single chain, but it is not known what conditions on a and b are sufficient for this to occur. This talk will be a report on this problem, as well as the notion of the "principal element" of L(a|b) and how it relates the Lie algebra structure.

Wednesday, October 27 (7:00 PM) Roger B. Eggleton (Illinois State University) Colourful trees, Sudoku, and Mathematics

It's Fall, so colorful trees are topical. What do they have in common with Sudoku? What do both have to do with Mathematics? This talk will reveal some of the connections, but the audience will be sworn to secrecy, so you have to come if you want to find out.

Tuesday, November 2 Lucian Ionescu (Illinois State University) The mathematics of a 2D "Electromagnetism"

I will discuss the binding between application interface (gauge theory) and mathematics implementation: conformal geometry, residues and connections with homology and Hodge decomposition. The talk will be accessible to Calc III and (especially) Advanced Calculus students, but also instructive regarding how to design a background independent string theory.

Tuesday, November 9 Michael Geline (Northern Illinois University) Sources of irreducible lattices and Brauer's height zero conjecture

Given a prime p and a finite group G, Richard Brauer partitioned the irreducible characters of G into subsets called p-blocks, discovered a conjugacy class of p-subgroups naturally asssociated with each block, and made several conjectures relating the characters in a block to the associated p-subgroups. Most of these conjectures remain open. I will describe one approach to the so-called height zero conjecture which is due to Reinhard Knorr and requires an attack on the integral representations of the p-group. The attack can be reformulated in terms of almost split sequences.

Tuesday, November 16 No talk No talk
Tuesday, November 23 Thanksgiving break No talk
Tuesday, November 30 Robert Bernales (Illinois State University) Density of visible Lattice points [Master's Project]

I will discuss how the density of lattice points in the xy-plane that are visible from the origin is 6/pi^2. Taking this theme further, I will show how a sector of lattice points in the plane remarkably has the same density and also explore the density of lattice points visible from the origin in higher dimensions. Conditions for which squares and cubes constructed of lattice points remaining invisible from the origin will be also examined. Lastly, we will explore how Pi can be calculated from the stars in the heavens from the results. Several branches of mathematics, such as number theory, calculus, analysis, discrete mathematics and statistics will be touched upon in the presentation.

Tuesday, December 7 Maria Paduret (Illinois State University) Quadratic Reciprocity in the Ring of Eisenstein Integers [Master's Project]

I will talk about the Ring of Eisenstein Integers and present the most important properties of its elements and the ring itself. I will then make a parallel between quadratic reciprocity in Z and the Ring of Eisenstein Integers. The interesting part of the presentation will be the comparison between the Law of Quadratic Reciprocity in the 2 rings: The Ring of Integers and the Ring of Eisenstein Integers.

Date
Speaker
Title
Tuesday, January 19 Sunil K. Chebolu A new geodesic proof of a formula of Gauss PREPRINT
Tuesday, Januray 26 Karl Rubin, Ken Ribet, John Conway -- MSRI video Fermat's Last Theorem - The Theorem and Its Proof: An eploration of Issues and Ideas

Short talks by Rob Osserman, Karl Rubin, Ken Ribet, and John Conway on Fermat's last theorem and Andrew Weil's proof of it.

Tuesday, February 2 Georgia Benkart (University of Wisconsin-Madison) -- MSRI video Combinatorial representation theory: Old and New

We will watch the video of Georgia Benkart's (Jan. 18, 2008) talk which she gave during the "Connections for Women: Introductionto the Spring, 2008 programs" workshop. The talk is expository and a historical survery of results.

Tuesday, Febrauary 9 Gaywalee Yamskulna Relationships between Commutative associative algebras, finite dimensional Lie algebras and vertex operator algebras I

I will discuss about the relationships between Commutative associative algebras, finite dimensional Lie algebra and homogeneous subspaces of vertex operator algebras. Later on, I will describe some classifications of vertex operator algebras using commutative associative algebras and Lie algebras.

Tuesday, February 16 Gaywalee Yamskulna Relationships between Commutative associative algebras, finite dimensional Lie algebras and vertex operator algebras II

I will discuss about the relationships between Commutative associative algebras, finite dimensional Lie algebra and homogeneous subspaces of vertex operator algebras. Later on, I will describe some classifications of vertex operator algebras using commutative associative algebras and Lie algebras.

Tuesday, February 23 -- No talk
Tuesday, March 2 Fusun Akman The Lattice of Finite Vector Space Partitions

This is joint work with Papa Sissokho. A partition of a finite vector space V=V(n,q) of dimension n over the finite field GF(q) is a set of nonzero subspaces of V such that the mutual intersections are zero and their union makes up the whole space. We show that (1) The set of all vector partitions of V is a lattice, (2) The number of all vector space partitions of V approach the number of all set partitions of the set with n elements as q goes to 1, and (3) The Mobius number of the vector partition lattice approaches the Mobius number of the set partition lattice as q goes to 1. We also compute the Mobius numbers of the vector partition lattice for small n. The convergence results parallel similar ones between the lattice of subspaces of V(n,q) and the lattice of subsets of the set with n elements, which are well-known, and provide another important example of the mysterious vector space-set relationship (as we all know, sets are vector spaces over the field with one element!). Our proof makes use of “canonical bases” of subspaces of finite vector spaces. Definitions will be supplied for free.

Tuesday, March 9 -- Thanks giving break
Tuesday, March 16 -- No talk
Tuesday, March 23 Sunil Chebolu Progress report on the generating hypothesis

Freyd's generating hypothesis is a very fundamental and deep statement about the category of finite spectra. It is a conjecture due to Peter Freyd (1965) which states that the stable homotopy functor on the category of finite spectra is faithful. An unbelievable consequence of this conjecture is that it reduces the study of finite spectra to that of graded modules over the homotopy ring of the sphere spectrum. Therefore this conjecture stands as a central problem in the homotopy theory which is still open. To the best of my knowledge there hasn't been any progress on this conjecture in the recent years. However, there has been lots of developments on analogues and variations of this conjecture on other axiomatic stable homotopy categories including equivariant stable homotopy categories, derived categories, and the stable module categories of finite groups. This talk will be a survey of these results with particular emphasis on the stable module categories. This is joint work with Jon Carlson and Jan Minac.

Tuesday, March 30 George Seelinger Galios actions and the Brauer group I
Tuesday, April 6 George Seelinger Galois actions and the Brauer group II
Tuesday, April 13 -- No talk
Tuesday, April 20 -- No talk
Tuesday, April 27 Mobashera Hallam Rooted lower subtractive partial ordering for cyclic Galois extensions

Let K/F be a G-Galois extension of fields. We can construct weak 2-cocyles as described by D. Haile. Weak 2-cocyles can be used to construct weak crossed product algebras. Associated to each weak 2-cocycle is also a partial ordering on the elements of G that satisfies a lower subtractive property. We call such partial ordering rooted lower subtractive. We will describe some properties of the structure of these rooted lower subtractive partial orderings for cyclic Galois groups and give some algorithms for generating such partial orderings. These properties can then be translated to properties of the weak crossed product algebras constructed from the weak 2-cocycles corresponding to these rooted lower subtractive partial orderings.