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 (submodules, 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, Ktheories, 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 modularinvariance 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 Mincut Maxflow on graphs is one specific goal  a visual exemplification of the theory. Beyond this "old stuff", the theory of Noncommutative 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(ab) 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 nondegenerate. Frobenius Lie algebras are important in the study of coadjoint orbits, the classical YangBaxter equation, and symplectic Lie groups. An interesting open question question is to characterize when L(ab) is Frobenius. It turns out that L(ab) 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(ab) 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 pblocks, discovered a conjugacy class of psubgroups naturally asssociated with each block, and made several conjectures relating the characters in a block to the associated psubgroups. Most of these conjectures remain open. I will describe one approach to the socalled height zero conjecture which is due to Reinhard Knorr and requires an attack on the integral representations of the pgroup. 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 xyplane 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. 