ISU Algebra Seminar (organized by L. Ionescu)

Fall 2007    Spring 2008    Fall 2008   Spring 2009  Spring 2014

Spring 2014 - Tuesdays @2:00 p.m. STV 136A
1/28: L. Ionescu - The Prime Number Theorem (base camp for RH)
2/11: G. Seelinger: Introduction to vector space partions over finite fields
2/18: L.I. PNT and Chebyshev's estimates
2/25: (no talk @2) - see presentation @3pm on p-groups and cohomology
3/4: LI: PNT and Riemann Zeta Function
3/11 spring break
3/18 LI: Riemann-Mangoldt Exact Formula
4/1 LI: Rationals, Farey sequence & modular group
4/8 Alberto Delgado On Baumslag-Solitar groups,  A Baumslag-Solitar group has the presentation (i) <x,t ; t^{-1}xt = x^{-1}>. This also presents (ii) the fundamental group of the Klein bottle and (iii) a graph of groups with one integral vertex-group and one inverting edge-group.
4/15 A. Delgado - On BS-groups / Part II
4/22 LI or SC on Mersenne primes
...

Fall 2007 - Thursdays @10:00 a.m. WIH 22A
9/6: L. Ionescu - On Rota-Baxter algebras I (after hep-th/0407082)
9/13: skipped;
9/20: L. Ionescu: - On RBA II: Spitzer's identity
9/27: no seminar
10/11 F. Akman: Beyond Hopf algebras I (after Loday's talk, Paris, spring 2007)
10/18 F. Akman: Beyond Hopf algebras II (dendriform algebras and Dynkin's theorem)
10/25 L. Ionescu: On RBAIII: Splitting a RB-algebra (A possibly new result)
11/1 R. Dijkgraff, The Quantum Geometry of Strings (MSRI Video)
11/8 L. Ionescu: On RBA IV: Splitting an RB-algebra (after hep-th/0407082 - the \chi map)
11/15 cancelled
11/22 Thanksgiving break - no seminar
11/29 Tim D. Comar REGULAR CONFORMATIONS AND REGULAR STICK NUMBERS OF KNOTS AND LINKS
12/6 W. Zhao - A Vanishing Conjecture on Differential Operators with Constant Coefficients
12/13 Final's week - no seminar.

Winter break

Spring 2008 - Thursday @ 1:00 p.m. Room: WIH 111
1/24 G. Yamskulna - Classification of irreducible modules of a vertex algebra $V_L^+$
when $L$ is a non-degenerate even lattice of an arbitrary rank.
1/31 L. Ionescu - On Rota-Baxter algebras and Lie group decompositions.
2/7 External Speaker - MSRI Video:
Yair Minsky, Introductory topics in Kleinian groups and hyperbolic 3-manifolds Start Video
2/14 Invited speaker (ISU Kepler Speakers Series),. Ray Kurzwel, The acceleration of technology in the 21st century.
2/21 Fusun Akman, What the Hecke is a Differential Operator?
Differential operators of higher orders that act on commutative associative algebras (such as the algebras of polynomials or smooth functions) have been defined by Grothendieck. Even if we didn’t know the definition, we would be inclined to say “I know one when I see one.” Generalizations to noncommutative and even nonassociative algebras, as well as modules, have been made, but not in a universal way. We will talk about the three older definitions (Grothendieck, Koszul, Akman) that coincide for a commutative associative algebra (proof by Akman and Ionescu), and compare them with Ginzburg and Schedler’s definition for associative algebras (“Differential Operators and BV Structures in Noncommutative Geometry”, math.QA/0710.3392). For starters, twisted commutative algebras (e.g. free tensor algebras) and their differential operators will be defined. Then (oh, joy!) differential operators on commutative “wheelgebras” will be considered. Thus, the differential operators of noncommutative geometry will act not on a general associative algebra A, but on a “Fock space” F(A) which is defined functorially (and happens to be a commutative wheelgebra).

2/28 Jinjia Li, “Intersection multiplicity and Serre’s multiplicity conjecture”, in Room 128 Williams Hall.
3/4 Iana Anguelova, Centre de Recherches Mathematique, "Vertex algebras: from super to quantum", Room Williams Hall 128.
3/6 Sunil K. Chebolu, University of Western Ontario, "The unreasonable effectiveness of homotopy theory in algebra and representation theory".

Spring break

3/20 Fusun: What the Hecke is a Differential Operator II.
3/24 Dr. Pisheng Ding, from St. John's University in New York, Shape Distortion (Room: WIH 112, at 12:00 pm)
Abstract: There are numerous examples of analytic or meromorphic functions which map some lines or rays into other lines or rays. On
the other hand, except for linear polynomials, no functions analytic on a triangle can map all three of its sides into three other lines.
This is one of the lemmas that we establish to show that linear polynomials are the only analytic functions which ever map some
polygon onto another polygon. Thus, unless it is a linear polynomial, a function analytic on a closed polygon never preserves its shape,
despite the fact that every analytic function is locally conformal, or shape-preserving, at all noncritical points. Looking at it another
way, there is no analytic mapping between any two dissimilar polygons, despite the fact that the Riemann Mapping Theorem guarantees the
existence of many conformal equivalences between the interiors of any two polygons. The proof rests on some special arguments for the
rectangle case and the triangle case, and proceeds somewhat unexpectedly by induction on the number of sides of the polygon.

3/27 W. Zhao, Noncommutative Symmetric Systems over Associative Algebras I.
4/3 W. Zhao, Noncommutative Symmetric Systems over Associative Algebras II.
4/10 L. Ionescu,
1) On NC-Symmetric systems and Rota-Baxter algebras;
2) From Lie Theory to Deformation Theory, Part I Lie Groups after Graeme Segal, Lectures on Lie Groups

4/19 CVE Conference Talk, College of Business Rm 353, 4:25-4:45 p.m.
L. Ionescu: What is a space-time coordinate system on a graph?

4/24 This week's seminar has been moved to Friday:

4/25 Friday at 3:00 p.m., WIH 308, Visitor presentation:
Speaker: Gene Freudenberg from Western Michigan University,
Title: Locally Nilpotent Derivations in Affine Algebraic Geometry

Abstract. We first review the role of locally nilpotent derivations in some of the central questions of affine algebraic geometry: the Abyhankar-Sathaye Embedding Conjecture, the Affine Cancellation Problem, and the Dolgachev-Weisfeiler Conjecture. We then turn attention to the fascinating examples of Bhatwadekar-Dutta (1992) and Vénéreau (2000), where these questions remain open. In particular, there are polynomials f in C [x,y,z,u]  such that B is a C^2-fibration over the subring A=C [x,f]. This condition implies B/fB is a polynomial ring, meaning that f defines a hyperplane in C^4. However, it is not known if B=A[P,Q] for P,Q in B (Dolgachev-Weisfeiler Conjecture), or even if f is a variable of B (Abhyankar-Sathaye Conjecture). In 2004, the author showed that f is stably an A variable, specifically, B[t]=A[X,Y,Z] for an indeterminate t over B. Therefore, the locally nilpotent A-derivation d/dt on B[t] has slice t and kernel B. This leads to the following beautiful question: If R is a commutative affine domain over C, and theta is a locally nilpotent R-derivation of R[X,Y,Z] with a slice, do there exist P,Q in R[X,Y,Z] such that the kernel of theta equals R[P,Q]? We explore this question, with particular attention to the case R=\C [a,b], a polynomial ring in two variables.

5/1 L. Ionescu, Part I (cont) The building blocks of Lie groups: SU2, SO3, SL2(R)

Summer break.

Fall 2008 Tuesdays @2:00 pm, FSA 312

8/26 L. Ionescu: On a few points of homological algebra
1) Introduction from Algebra V by Yu. Manin & S. Ghelfand 2) Kontsevich & Formal pointed manifolds, 3) Formal manifolds & Path integrals (project)

Handwritten notes: p1 p2 p3 p4

9/2 S. Chebolu: An introduction to the Bloch-Kato conjecture (Room STV 313A)

The Bloch-Kato conjecture is a very deep conjecture about the  continuous cohomology of  the absolute Galois groups. It connects  fascinating topics in arithemetic, geometry, and
algebra. This  conjecture at the prime 2 is called the Milnor conjecture, and was  proved by Voevodsky for which he won he won the Fields medal in 2002.  Very recently (early
2008) the Bloch-Kato conjecture has been settled  by Voevodsky and Rost.
In this talk I will present some of the background material necessary  to understand this conjecture, and explain the conjecture and some  history surrounding it.

9/9  Unal Ufuktepe from the Dept. of Math. of Izmir University of Economy.
Unal will talk about “Time Scale Calculus with Mathematica”.  Cookies will follow the talk at 2:00 p.m. in the coffee room, STV 313-D. Here’s the abstract:
Stefan Hilger introduced the calculus on time scales in order to unify continuous and discrete analysis in 1988. The study of dynamic equations is an active area of research
since time scales unifies both discrete and continuous processes, besides many others.
In this talk we give many examples on derivation and integration in time scales calculus with Mathematica.

9/16  S. Chebolu: A new interpretation of the quadratic closure of a field.
Abstract: The quadratic closure of a field F is the smallest subfield of the separable closure of F in which every element is a square.
By refining the Milnor conjecture we obtain a purely cohomological characterisation of the quadratic closure of F.
This is joint work with Jan Minac.

9/23 F. Akman: Applications of the Moebus Inversion Principle (I)

Abstract: The classical Möbius inversion formula of number theory describes how to invert a sum taken over the set of divisors of a positive integer. The seminal paper of Rota (1964) generalizes the idea to sums over any finite partially ordered set (poset) and pioneers the uses of the inversion principle in diverse areas of mathematics. Today MIP is considered a standard tool and is still in full swing. In this expository talk, we will prove the MIP and start with the simplest example that we actually teach in calculus. The remaining applications and analogues will come from set theory (inclusion-exclusion principle), classical number theory (Euler phi function), analytic number theory (Riemann zeta function), discrete mathematics (chromatic polynomial of a graph, number of partitions of a set, number of spanning subsets of a finite vector space), symmetric functions (base change; Waring’s formula), algebraic topology (Euler characteristic), abstract algebra (group theoretical proofs; another proof of Sylow’s third theorem), and Hopf algebras (incidence Hopf algebra of a family of posets). And this does not even begin to cover applications in statistics and physics.

9/30 F. Akman: Part (II)

10/7 No seminar; Instead: MATHEMATICS DEPARTMENT COLLOQUIUM
Speaker:   Mike Martin, Thursday, October 9, 3:00-3:50 p.m., STV 401-A
Topic:  Biomathematics, Title:  Cardiac Dynamics — A Modeling Approach

10/14 No seminar; Instead: MATHEMATICS DEPARTMENT COLLOQUIUM
Speaker:   Hamparsum Bozdogan,  Thursday, October 16, 3:00-3:50 p.m.,  Felmley 231
Title:  Data mining of the cause of heart attack in Reproducing Kernel Hilbert Space (RKHS) using information complexity and the genetic algorithm

10/21 Algebra Seminar and Colloquium: 2-2:50 p.m. FSA 231 (note different building)
Speaker: Roel Willems from Radboud University, Nijmegen, the Netherlands
Topic: Polynomial automorphisms over finite fields

10/28 G. Seelinger: Introduction to algebraic geometry and representations of quivers (Part I)

11/4 G. Seelinger: Part II: Representation varieties.

11/11 L. Ionescu: Overview of deformation quantization I

11/18 G. Seelinger Part III of "Introduction to Algebraic Geometry and Representations of Quivers":
moduli space of quiver representations, projective space, projective curves, and Riemann surfaces.

11/25 Thanksgiving break -  no seminar

12/2 Sunil Chebolu, Cohomology using stones and sticks,
Abstract: An important and magnificent theory that evolved in the 20th
century is group cohomology. While scads of brilliant theorems have been
proved about group cohomology since its inception, the fundamental problem
of computing group cohomology remained a mystery. Therefore it is natural to
investigate different methods for computing group cohomology. In this talk I
will present a diagrammatic approach (using stones and sticks) for computing
cohomology of finite groups.
In the cases where it is applicable, this method has the advantage that it
reveals some incredibly beautiful connections between representation theory
and cohomology of finite groups. I have some interesting problems (based on
these diagrammatic methods) which are well accessible for graduate students.
So graduate students are most welcome for the talk. For the most part, the
techniques involve only basic facts about groups, rings, and vector spaces.
----------------------------------------------------------------------------

Spring 2009

1/29 G. Yamskulna, Rationality of the vertex algebra $V_L^+$ when $L$ is a non-degenerate even lattice of arbitrary rank.

Abstract:
Vertex algebras are rich algebraic structures which play an important role in
many active and relatively new areas of mathematics and physics. They were
introduced to mathematics in the work of the Fields medalist Richard Borcherds
and developed by others such as Frenkel, Lepowsky and Meurman in 1980's. In
physics, they are part of string theory where they appear as the chiral
algebras of two-dimensional conformal field theory.

In this talk, I will first discuss about one of the main conjectures in the
theory of vertex algebras. Next, I will review the construction of vertex
algebras $V_L^+$ which are one of the most important classes of vertex
algebras. They were originally introduced in the Frenkel-Lepowsky-Meurman
construction of the moonshine module vertex algebra. Finally, I will discuss
about the representation theory of the vertex algebra $V_L^+$ when $L$ is a
non-degenerate even lattice of arbitrary rank.

2/5 In this week's Algebra Seminar Sunil Chebolu will talk about:
How I got a job at ISU with the help of Auslander-Reiten sequences!
Abstract:
Auslander-Reiten sequences are some special short exact sequences of
kG-modules which are very close to being split exact sequences. They were
introduced by Auslander and Reiten in the early 1970s and they proved to be
among the most useful tools in modular representation theory. I will explain
how these innocent looking sequences played an extremely important role in
my post-doctoral research which paved my way to ISU.

2/12 Sunil Chabolu, (cont) Part II.

New Talks Series: "Algebraic Graph Theory" by Norman Biggs.
The realm of algebraic graph theory is mainly the use of linear algebra and group theory to deduce properties of graphs
and the other way around (quiver representations, voltage graphs and Rieman surfaces, 1+1 TGFTs etc.)
The Algebra group has collectively studied many graph-infested topics, such as Feynman diagrams, quivers, Hopf algebras
defined on trees, networks, Hasse diagrams, knots and knot polynomials, Dynkin diagrams, etc.,
but we do not use the same language nor have the same goals as discrete mathematicians.
We hope that this series of talks will act as a Stimulus Package that will please both sides of the aisle.
No background other than Linear Algebra is necessary.

2/19 Fusun Akman, The Spectrum of a Graph, Thursday, February 19, 1:00-1:50 p.m., WIH 313
Abstract: The adjacency matrix A of a graph is a real symmetric matrix that completely determines the graph.
The eigenvalues of A, together with their multiplicities, form the spectrum of the graph. We will study the properties
of the characteristic polynomial of A and the adjacency algebra (the ring of polynomials in A)
while investigating several examples. (Next topic: Regular Graphs and Line Graphs)

2/26 Phichet Jitjankarn, Thursday, February 26, 1:00-1:50 p.m., WIH 313
Talk title: The $C_2$-cofiniteness of the vertex algebra $V_L^+$ when $L$ is a non-degenerate even lattice.

Abstract: The $C_2$ condition was first introduced by Zhu in 1996. He used
this condition, as well as other assumptions, to show modular invariance of
certain trace functions. Since its introduction, the $C_2$-condition has
proven to be an power tool in the study of theory of vertex algebras. In
particular, it has played an important role in the study of structure of
modules of vertex algebras which satisfy it.

In this talk, I will discuss about the $C_2$-condition of the vertex algebra
$V_L^+$ and its irreducible weak modules. Here, $L$ is a non-degenerate even
lattice.

3/5 Who: Fusun Akman When: Thursday, March 5, 1:00-1:50 p.m. Where: WIH 313

Title: Spectrum, Regular Graphs, and Line Graphs

Abstract: We will continue with the properties of the characteristic
polynomial of a graph, study the adjacency algebra, and discuss graphs that
possess some combinatorial regularity; the latter have spectra with
distinctive features. Finally, we will study the spectral connections
between regular graphs and their line graphs.

SPRING Break

3/19  Speaker: Wenhua Zhao, Time: Thursday, March 19, 1:00-1:50 p.m., Place: WIH 313

Talk 1: Some Properties and Open Problems of Hessian Nilpotent Polynomials

Abstract: Hessian nilpotent polynomials are the polynomials whose Hessian
matrix are nilpotent. In this talk, we will discuss some background,
criteria and open problems of Hessian nilpotent polynomials. If time
permits, we will also discuss the graphs associated with homogeneous Hessian
polynomials.

Ref: arXiv:0704.1689, arXiv:0409534.

3/26 Speaker: Wenhua Zhao,  Time: Thursday, March 26, 1:00-1:50 p.m., Place: WIH 313

Talk 2: A Deformation of Commutative Polynomial Algebras

Abstract: We will discuss a deformation of commutative polynomial algebra.
We will show that, even though, this deformation is trivial in the sense of
deformation theory of algebras, it is interesting and highly non-trivial in
the study of some other mathematics objects.

4/2 Speaker: Lucian Ionescu

Title: Algebraic Graph Theory - Ch. 4: Cycles and Cuts, and their relation with homological algebra and dynamics.

4/9 Who: F. Akman, When: Thursday, April 9, 1:00-1:50 p.m., Where: WIH 313

Title: Adjacency Algebra, Strongly Regular Graphs, and the Laplacian (walk into a bar)

Abstract:
This week's algebra seminar features the happy ending of the cliffhanger
from Chapter 3 of "Algebraic Graph Theory" by Biggs: the matrix consisting
of all 1's is an element of the adjacency algebra of a graph iff the graph
is regular and connected (algebra equals geometry). I will also talk about
the spectra of strongly regular graphs and how to make strongly regular
graphs out of a Latin square (is there anything one cannot make out of Latin
squares?).  I will then throw in some comments about the spectrum of the
Laplacian of a graph, to round off our recent discussions.

4/16 Sunil Chebolu will give the following talk:
Title: How is a sphere like a field in which -1 is a sum of squares?

Abstract:
This talk is going to be a mix of topology and arithmetic. I will begin by
introducing Milnor K-theory  of a field to give a characterization of fields
in which -1 is a sum of squares. I will then define the stable homotopy
groups of spheres and discuss a celebrated result of Nishida on the
multiplicative structure of the stable homotopy groups of spheres.  Finally,
I will draw an analogy between these results (belonging to different
subjects!) to answer the ostensibly preposterous, yet intriguing, riddle
posed in the title of my talk: how is a sphere like a field in which -1 is
a sum of squares?

4/23 Who: Josh Hallam, When: Thursday, April 23, 1:00-1:50 p.m.Where: WIH 313
Title: Group Algebras and Their Genetic Significance
Abstract:
The study of genetic inheritance first started with the work of Gregor
Mendel and was continued by population geneticists in the early 1900's. In
1939 I.H.M. Etherington studied inheritance from an algebraic point of view.
This work was continued by several other mathematicians throughout the 20th
century, when new types of nonassociative algebras (baric, train, genetic,
and special train) of genetic significance were developed and studied.

In this talk, we explore algebras with genetic significance, and then show
where group algebras fit into the hierarchy of these algebras based on the
properties of the group and the field.

_________________________________________________________

4/30 Who: Sunil Chebol When: Thursday, April 30, 1:00-1:50 p.m.Where: WIH 313

Title: A Theorem of Artin and Schreier

Abstract:
I will prove a celebrated theorem of Artin and Schreier which says that a
field F admits an ordering if and only if -1 cannot be expressed as a sum of
squares. In particular, this gives one arithmetical explanation to the
question:  why is there no ordering on the field of complex numbers, while
there is one for the field of real numbers. I will also discuss finite
fields in the light of the Artin-Schreier Theorem. (This will be an
independent talk. I will not assume anything from the previous talk,
although the two are logically connected.)

<end of semester: Spring 2009>

For previous semesters see my ISU web page