ISU Algebra Seminar 2007-2008

Participants
     Fusun Akman, Lucian Ionescu, George Seelinger, Gaywalee Yamskulna, Wenhua Zhao
 
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.

For previous semesters see my ISU web page