AWM at JMM 2011

Minisymposia, Special Session and Workshop Abstracts

Contents

Thursday, January 6, 2011, 2:45 p.m. − 6:15 p.m., Le Galerie 1, 2nd floor, Marriott

AWM Schafer Minisymposium

2:45 p.m. Life in the Trenches with Alice–The Early Years

Mary W. Gray*, American University, Washington DC (1067-01-567)

Let’s fight City Hall—in our case, the mathematical establishment. For many years Alice Schafer knew and I knew, that women could do mathematics, and that given the chance, we could prove it. In the early years, it took some courage, and lots of persistence, to see that as many women as possible got that chance. Alice’s short stature, Southern accent and Southern grace, led many a mathematician, male and even sometimes female, to underestimate her persistence and her commitment. Madeline Albright is credited with saying that there is a special place in hell for women who do not help other women. That is a place ever unknown to Alice. The successes were many, the struggles eternal.

3:15 p.m. Sparse Regular Random Graphs: Spectra and Eigenvectors

Ioana Dumitriu*, University of Washington;
Soumik Pal, University of Washington (1067-60-2169)

Regular graphs are widely studied in connection to Markov chains and expanders, as well as networks. It is interesting, therefore, to understand what a ”typical” object from this class looks like, and what kind of properties it exhibits. This can be accomplished, in many cases, by studying the random regular graph.
Of the quantities that characterize the random regular graph, we will focus on eigenvalues and eigenvectors, particularly in the regime when the size of the graph, as well as the degree, grow to infinity (the latter, much slower than the former). We will describe the limiting shape of the empirical eigenvalue distribution, list some eigenvector properties, and mention what is known about other models.
This is joint work with Soumik Pal.

3:45 p.m. Bose-Einstein condensation, the NLS, and a phase transition

Kay L Kirkpatrick*, Courant Institute/Paris IX Dauphine (1067-82-496)

Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called Bose-Einstein condensation, that behaves like a giant quantum particle. Recently we’ve been able to make the rigorous probabilistic connection between the physics of the microscopic dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrodinger equation (NLS).
I’ll mention joint work with Benjamin Schlein and Gigliola Staffilani on the two-dimensional cases for Bose-Einstein condensation–and the periodic case is especially interesting, because it uses techniques from analytic number theory and has applications to quantum computing. I’ll also describe new work with Sourav Chatterjee about a phase transition for the invariant measures of the NLS, work which sheds light on typicality of blow up as well as a controversial conjecture of Lebowitz, Rose, and Speer.

4:15 p.m. Do the primes behave independently?

Melanie Matchett Wood*, American Institute of Mathematics and Stanford University (1067-11-529)

In number fields, finite extensions of the rational numbers, rational primes such as 2,3,5 may no longer be prime–sometimes they factor in the extension. We ask the question of whether such factoring is independent for different primes. The question can be answered completely for extensions that have abelian Galois group, but the answer is more subtle than one originally expects.

4:45 p.m. Traces and topological fixed point theory

Kate Ponto*, University of Kentucky (1067-55-1226)

Fixed point theorems are very common in many areas of math. In algebraic topology, these results are often comparisons of invariants defined using different techniques. For example, the Lefschetz fixed point theorem is the identification of algebraic and geometric invariants. I’ll describe an approach to these comparisons that is very different from the standard proofs. This perspective uses traces and has many advantages. One of the most important is that it generalizes easily.

5:15 p.m. Panel Discussion: Getting Started as a Research Mathematician

Saturday January 8, 2011, 8:00 a.m. – 10:50 a.m., Nottoway Room, 4th Floor, Sheraton

AMS-AWM Special Session on Hopf Algebras and Their Representations, I

8:00 a.m. Right coideal subalgebras of Nichols algebras and the Duflo order of the Weyl groupoid

Hans-Juergen Schneider*, Ludwig-Maximilians-Universitaet Muenchen;
Istvan Heckenberger, Philipps-Universitaet Marburg (1067-20-1153)

This is a report on recent joint work with I. Heckenberger. The first main problem of the classification of pointed Hopf algebras is the structure of Nichols algebras over group algebras. We are studying systematically the Nichols algebra of a Yetter-Drinfeld module over any Hopf algebra (with bijective antipode) which is a finite direct sum of finite- dimensional irreducible Yetter-Drinfeld modules. In this general context in recent joint work with I. Heckenberger and N. Andruskiewitsch we define reflection maps and a Weyl groupoid. Under mild assumptions we associate a generalized root system (in the sense of Heckenberger and Yamane) to the Nichols algebra. Using these invariants it is possible to decide when the Nichols algebra is finite-dimensional. We obtain a coproduct formula which seems to be new even for the classical quantum groups. Then we describe the right coideal subalgebras of the Nichols algebra by words in the Weyl groupoid. As a special case we obtain a proof of a recent conjecture of Kharchenko which says that the number of right coideal subalgebras of the plus part of the quantum group of a semisimple Lie algebra is the order of the Weyl group.

8:30 a.m. Conjugacy classes for Hopf algebras

Miriam Cohen*, Ben Gurion University;
Sara Westreich, Bar Ilan University (1067-16-422)

An important instance of structure constants exists for finite groups and the way their class sums multiply this is connected to the associated character table.
We shall discuss the meaning of these concepts from the point of view of Hopf algebras and their duals and thus give a generalization of conjugacy classes and class sums for semisimple Hopf algebras H having a commutative character ring (quasitriangular Hopf algebras have this property) and a formula for their associated structure constants. When H is also factorizable these constants turn out to equal the fusion rules up to rational scalar multiples.
We also show a connection between the conjugacy classes and the commutator subspace of H. This connection boils down to a known connection for finite group algebras

9:00 a.m. Secondary cohomology for Hopf algebras

Mihai D. Staic*, DePaul University (1067-16-618)

For a commutative Hopf algebra A we introduce a new cohomology theory. This is connected with the work of Khalkhali and Rangipour on cyclic modules for Hopf Algebras. If one thinks about the usual cohomology of a Hopf Algebra as the cohomology corresponding to the first level of a “Postnikov tower”, then this new cohomology theory deals with the second level (therefore the need for the Hopf algebra to be commutative).

9:30 a.m. Classification of isomorphism types of a class of abelian extensions, by Y. Kashina and L. Krop

Leonid Krop*, DePaul University;
Yevgenia Kashina, DePaul University (1067-16-429)

We determine the isomorphism types in the class of Hopf algebra extensions of a cyclic group C2 of order 2 by an arbitrary finite, elementary 2- group G. Put AutC2(G) for the group of C2– linear automorphisms of G. Our main result asserts existence of a bijection between the orbits of AutC2(G) in the group of Hopf algebra extensions Opext(kC2, kG) and the isomorphism types of algebras in our class. In the special case of commutative or cocommutative extensions H, assuming
G has rank n, the number of isotypes is n + 1 if H is cocommutative, and 3n+(−1)n+2⁄2 if H is commutative.

10:00 a.m. On classification of certain abelian extensions

Yevgenia Kashina*, DePaul University;
Leonid Krop, DePaul University (1067-16-2356)

This talk is based on joint work with Leonid Krop on classification of isomorphism types of a class of abelian extensions. We consider an abelian extension of a group algebra of a cyclic group of order 2 by a group algebra of an elementary abelian 2-group. In this talk we will describe the concrete Hopf algebra structures of such abelian extensions.

10:30 a.m. On Multigraded combinatorial Hopf algebras

Samuel K Hsiao, Bard College;
Gizem Karaali*, Pomona College (1067-16-793)

We develop a theory of multigraded combinatorial Hopf algebras modeled on the theory of graded combinatorial Hopf algebras developed by Aguiar, Bergeron, and Sottile (2006). In particular we introduce the notion of canonical k-odd and k-even subalgebras associated with any multigraded combinatorial Hopf algebra, extending simultaneously the work of Aguiar et al. and Ehrenborg.

Saturday January 8, 2011, 8:00 a.m. – 11:00 a.m., La Galerie 1, 2nd Floor, Marriott

AWM Hay Minisymposium

8:00 a.m. Mathematics, Educational Research, and STEM Education Policy: Challenges and Opportunities in the Intersection

Joan Ferrini-Mundy*, National Science Foundation and Michigan State University (1067-97-2203)

Drawing on examples from my own experience, I will reflect on how making mathematics more central in educational research and in STEM education policy could improve endeavors in both arenas. I will describe a large-scale, multi- district K-12 mathematics and science education research and development project – Promoting Rigorous Outcomes in Mathematics and Science Education (PROM/SE) – where the project team confronted issues of curriculum coherence in mathematics. Using current national context and initiatives as examples, I will propose ways in which mathematics education research might better inform STEM education policy development and implementation. Specifically, the relationship of educational research to improvement in K-12 STEM education, undergraduate education, standards, assessments, and STEM education policy will be explored.

8:30 a.m. Addressing Challenges in the Common Core: Mathematics Specialists in Elementary and Middle Schools

Patricia F Campbell*, University of Maryland (1067-97-1622)

The Common Core State Standards for Mathematics (CCSS) both reposition and refocus mathematics objectives in the K-8 curriculum, raising demands on teachers already challenged by calls to increase student achievement. Successful implementation of the CCSS will depend not only on assessment and curriculum development, but also on teachers’ knowledge and instructional practice. Recently, mathematics specialists/coaches are being positioned in elementary and middle schools to serve as an on-site resource, addressing teachers’ knowledge of mathematics content and pedagogy while catalyzing and sustaining teachers’ efforts to define and implement meaningful instructional change across a school. This session will highlight some of the challenges raised by the content of the CCSS and report the results of a
collaborative project that utilized a 3-year randomized control-treatment design to investigate the impact of knowledgeable mathematics specialists who served as coaches in elementary schools. Concluding remarks will consider (1) the feasibility of positioning specialists to address some of the demands raised by the CCSS and (2) the implications of mathematicians and mathematics educators working together to develop more rigorous and appropriate content courses for K-8 teachers.

9:00 a.m. The Role of Logic in the K-12 Mathematics Curriculum

Susanna S Epp*, DePaul University (1067-97-2152)

How can we teach children important mathematical facts in ways that are both age-appropriate and intellectually honest? Some informal explanations are both helpful and suggestive of the mathematics that underlies the facts. Other expla- nations help students get right answers on tests but do not provide a sound basis for future understanding. This talk will examine examples of both kinds of explanations in the context of courses for prospective and in-service mathematics teachers.

9:30 a.m. The Power of Interdisciplinary Bridges: Throwing the Net Widely

Deborah Hughes Hallett*, University of Arizona/Harvard Kennedy School (1067-97-1237)

Some students are fascinated by the elegance of mathematics. Others are captured by the connection between mathematics and other fields—science, art, business, or medicine. These interdisciplinary links are important for our own teaching; they are equally important in the K-12 classroom. Perhaps surprisingly, for many students, seeing mathematical ideas in context deepens their mathematical intuition—as well as their appreciation of the power of mathematics. These links provide a vehicle for teachers teaching to the Common Core State Standards, in which fluent understanding is key. For us, interdisciplinary connections provide a bridge for collaboration with K-12 teachers in workshops that explore mathematics in other fields. In this talk, we will look at ways to use interdisciplinary bridges and talk about how develop them, with examples from climate change, oil production, the spread of disease, racial profiling, and drug testing.

10:00 a.m. Panel Discussion: The Mathematical Education of Teachers and the Common Core

Saturday January 8, 2011, 1:00 p.m. – 5:50 p.m., Nottoway Room, 4th Floor, Sheraton

AMS-AWM Special Session on Hopf Algebras and Their Representations, II

1:00 p.m. A Freeness Result Revisited

David E. Radford*, U. of Illinois at Chicago (1067-16-1849)

The relationship between a Hopf algebra over a field and a Hopf subalgebra has been of interest over the years. Finite- dimensional Hopf algebras are free over their Hopf subalgebras by the Nichols-Zoeller Theorem. The first example of a Hopf algebra which is not free over one of its Hopf subalgebras was described by Oberst and Schneider; another example having the same characteristics was given by Takeuchi. These examples depend heavily on the field. An example defined over any field was described by the author. Generalizing the techniques used in the construction of the latter strongly suggests how to construct a class of commutative examples. Commutative Hopf algbebra are projective modules over their Hopf subalgebras by a result of Takeuchi.

1:30 p.m. Hopf algebras of small dimension

Margaret Beattie*, Mount Allison University (1067-16-1322)

Over an algebraically closed field, the problem of classifying all Hopf algebras of some given small dimension, such as 16 or 32, or for a class of dimensions, such as p, pq , pq2, etc, for p, q prime, is a difficult one. General techniques are lacking and progress is slow. In recent years D. Fukuda has introduced some arguments involving dimensions and the coradical filtration that allowed him to complete the classification for dimensions 18 and 30. More recently, Cheng and Ng have worked on the problem of classifying Hopf algebras of dimension 4p and completed the classification for dimensions 20, 28 and 44. This talk will present some extensions of the techniques of Fukuda and some applications of these to the classification problem. This is joint work with G.A.Garcia.

2:00 p.m. The Central Charge of Factorizable Hopf Algebras coming from Bilinear Forms

Yorck Sommerhaeuser*, University of South Alabama (1067-16-2184)

Recent work of Yongchang Zhu and the speaker established the following fact: For a semisimple factorizable Hopf algebra, the value of an integral on the Drinfel’d element and the value of this integral on the inverse Drinfel’d element differ only by a fourth root of unity. If the dimension is odd, they only differ by a sign, and this sign is a plus sign if the dimension is one modulo four, but a minus sign if the dimension is three modulo four.
The authors of this work conjecture that these two integral values always differ only by a sign, even if the dimension is not odd. In the talk, we provide some evidence for this conjecture by proving it for a class of factorizable Hopf algebras coming from bilinear forms. We also show that the conjecture is false for quasi-Hopf algebras, which can be constructed in an analogous way if one replaces bilinear forms with Eilenberg-MacLane cocycles.

2:30 p.m. A q-identity related to a comodule

Andrea Jedwab*, University of Southern California;
Susan Montgomery, University of Southern California (1067-16-1064)

We determine a set of identities that are equivalent to a certain algebra being a comodule over the Taft algebra. We then show that the algebra is in fact a comodule algebra by giving a direct combinatorial proof of the identities.
These identities involve the q-binomial coefficients, where q is a primitive nth root of unity and n2 is the dimension of Taft algebra.

3:00 p.m. The Lie product in the continuous Lie dual of the Witt algebra

Earl J. Taft*, Rutgers University;
Zhifeng Hao, South China University of Technology (1067-17-212)

Let k be a field of characteristic zero. The simple Lie algebra W1=Der k[x], the one-sided Witt algebra, has a basis ei=x(i+1)d/dx for i at least -1). For each i, the wedge of e0 and eisatisfies the classical Yang-Baxter equation, giving W1 the structure of a coboundary triangular Lie bialgebra (W1)(i). The continuous Lie dual of (W1)(i) is also a Lie bialgebra, and has been identified with the space of k-linearly recursive sequences by W. Nichols [J. Pure Appl. Alg. 68(1990), 359-364]. Let f=(fn) and g=(gn) be linearly recursive sequences in the continuous linear dual of (W1)(i), [f,g] their Lie product. For each n, the n-th coordinate of [f,g] has been described in terms of the coordinates of f and of g [E. J. Taft, J. Pure Appl. Alg. 87(1993), 301-312], but it was an open problem to give a recursive relation satisfied by [f,g] in terms of recursive relations satisfied by f and by g. We give such a relation here. Analogous results hold for the two-sided Witt algebra W=Der k[x,x(−1)].

3:30 p.m. Indicators for the Drinfel’d doubles of certain groups

Marc Keilberg*, University of California San Diego (1067-16-1876)

Frobenius-Schur indicators and their generalizations have proven to be a useful invariant of Hopf algebras, as well as other algebraic objects. For the group algebra of a finite group, these indicators are well-known to be integers. The same result is conjectured to hold for the Drinfel’d double of a finite group. However, little is known of connections between the structure theory of the group and the values of the indicators for its double. We discuss several families of groups for which these indicators have been computed and the patterns that have emerged.

4:00 p.m. Drinfeld centers of graded fusion categories

Shlomo Gelaki, Technion-Israel Institute of Technology;
Deepak Naidu*, Texas A&M University;
Dmitri Nikshych, University of New Hampshire (1067-00-465)

The Drinfeld Center is an important construction in the theory of tensor categories. It is a way of producing a braided tensor category from a (not necessarily braided) tensor category. In this talk, I will describe how the structure of the Drinfeld center of a fusion category can be understood in terms of a smaller and more transparent category. As an application, I will give a criterion for a fusion category to be group-theoretical and apply it to the Tambara-Yamagami categories to produce non group-theoretical semisimple Hopf algebras, extending a construction of Nikshych.

4:30 p.m. Fusion categories of dimension pq2

David A Jordan*, Massachusetts Institute of Technology;
Eric Larson, Harvard University (1067-18-1455)

In this talk, we completely classify integral fusion categories – and consequently, semisimple Hopf algebras – of dimension pq2, where p and q are distinct primes. This is the simplest class of integer dimensions where non-group- theoretical categories arise; on the other hand, we prove that all semi-simple Hopf algebras of dimension pq2 are group- theoretical.

5:00 p.m. Cocycle deformations, calculus, and extensions

Mitja Mastnak*, Saint Mary’s University (1067-16-1258)

If H is a Hopf algebra over a ground field k, then a multiplicative cocycle on H is a unital linear map σ : H ⊗ H → k satisfying the identity

(ε ⊗ σ) ∗ (σ(id ⊗ m)) = (σ ⊗ ε) ∗ (σ(m ⊗ id))

in the convolution algebra Homk (H ⊗ H ⊗ H, k). If σ is such a map, then one can construct the cocycle twist Hσ of H by conjugating the multiplication m in H by σ, that is, mσ = σ ∗ m ∗ σ−1
I will describe various methods for computing multiplicative cocyles. These include exponential and q-exponential maps as well as cleft Hopf algebra extensions. Applications to the Andruskiewitsch-Schneider classification of pointed Hopf algebras will be considered. The talk is based on joint work with Luzius Grunenfelder.

5:30 p.m. Hopf algebraic approach to Picard-Vessiot theory

Akira Masuoka*, University of Tsukuba (1067-16-891)

Galois theory for differential or difference equations is called Picard-Vessiot theory. M. Takeuchi [J. Algebra 122(1989), 489–509] proposed a sophisticated, Hopf algebraic approach to the Picard-Vessiot theory for differential equations. K. Amano and I [J. Algebra 285(2005), 743–767] extended Takeuchi’s theory so that it can apply to difference or mixed equations as well; see also the recent expository paper by the three of us which is contained in Handbook of Algebra Vol.6, 2009, Elsevier/North Holland. I will explain a Hopf-Galois theoretic idea of our approach as well as relevant new results by myself and by others.

Saturday January 8, 2011, 1:00 p.m. – 5:30 p.m., La Galerie 1, 2nd Floor, Marriott

AWM Michler-Mentoring Minisymposium

1:00 p.m. A Meeting of Algebra and Geometry in Decorated Graphs

Rebecca F Goldin*, George Mason University, (1067-51-592)

How many different lines intersect four fixed lines generically placed in R? Such questions in enumerative geometry have been translated into equivalent questions about the ring structure of algebraic invariants associated to some special symplectic manifolds. These algebraic questions have in turn been translated into combinatorial questions about an algebra associated to decorated graphs. We will show how this dictionary of works for a set of nice algebraic varieties and discuss how it can be generalized to a larger set of manifolds.

1:30 p.m. An optimal metrization theorem for topological groupoids

Irina Mitrea*, University of Minnesota (1067-00-1870)

Metrization theorems (i.e. the question whether a certain topology is induced by a metric) play a basic role in many areas of mathematics including topology, functional analysis, analysis on spaces of homogeneous type, partial differential equations, etc.
In this talk I will discuss a sharp general metrization theorem in the setting of abstract groupoids (groupoids have been introduced by Brand in the 1920’s as ageneralization of groups which also include arbitrary sets). This theorem contains as particular cases several basic metrization results such as Alexandroff-Urysohn metrization theorem in Topology, the Aoki-Rolewicz metrization theorem in Functional Analysis and the Macias-Segovia metrization theorem in Harmonic Analysis.

2:00 p.m. Heat kernel analysis on infinite-dimensional curved spaces

Masha Gordina*, University of Connecticut (1067-58-660)

The heat kernel analysis has long been an essential tool in diverse areas of mathematics such as analysis, geometry, and probability, as well as in physics. We will review recent developments of the subject in the case of infinite-dimensional curved spaces.

2:30 p.m. Interplay of Combinatorics and Topology through Posets

Patricia L. Hersh*, North Carolina State University (1067-05-1124)

This talk will focus on how partially ordered sets help record topological structure, including mentioning some limitations in how much they can capture. I will briefly discuss my work on discrete Morse theory for order complexes of partially ordered sets and how this has been used e.g. to count by inclusion-exclusion. Then I’ll turn things around and discuss more recent work on how topological structure of a stratified space can sometimes be gleaned from combinatorics of its closure poset combined with codimension one topology. This is used to show that certain stratified spaces arising from combinatorial representation theory are regular CW complexes homeomorphic to balls. Familiarity with this area will not be assumed.

3:00 p.m. The Evolution of Spatio-Temporal Models of Tumor Angiogenesis

Trachette L. Jackson*, University of Michigan (1067-92-525)

Motility – random, directed and collective – is a fundamental property of cells. Coordinated motility of endothelial cells that reside on the inner surface of blood vessels leads to a critical bifurcation point in cancer progression: tumor angiogenesis. Successful angiogenesis is a consequence of integration across multiple levels of biological organization, and several temporal and spatial scales. A major challenge facing the cancer research community is to integrate known information in a way that improves our understanding of the mechanisms driving tumor angiogenesis and that will advance efforts aimed at the development of new therapies for treating cancer. In this talk, the evolution of spatio-temporal mathematical models of tumor angiogenesis will be explored and recent advances will be highlighted.

3:30 p.m. Weak and numerical solutions for coupled Navier-Stokes, Darcy and transport equations

Beatrice Riviere*, Rice University (1067-65-2025)

The coupling of porous media flow with free flow arises in many applications including the industrial filtration problems and the environmental problems of contaminated aquifers through rivers. In this multiphysics couplings, the free flow is characterized by the Navier-Stokes equations whereas the porous media flow is described by the Darcy equations. Interface conditions such as the Beavers-Joseph-Saffman’s law are prescribed at the interface between the two different physical flows. A transport equation satisfied by the contaminant concentration is coupled to the flow problem via the fluid viscosity and the velocity field.
In this work, we first study the well-posedness of weak solutions to the coupled problems. By varying the interface condition for the balance of forces, we construct two weak solutions using a Galerkin approach. Second we define and analyze several numerical schemes based on classical finite element methods and discontinuous Galerkin methods. Convergence of the schemes is also verified numerically. Numerical solutions for non homogeneous porous media are presented.

4:00 p.m. Panel Discussion: Mentors Count!

Sunday January 9, 2011, 8:00 a.m. – 10:20 a.m., Mardi Gras E, 3rd Floor, Marriott

AWM Workshop: Research Presentations by Recent Ph.D.s, I

8:00 a.m. Tensor category of integrable modules over ??∞, ??∞, and ??∞

Elizabeth Dan-Cohen*, Jacobs University Bremen;
Ivan Penkov, Jacobs University Bremen;
Vera Serganova, U.C. Berkeley (1067-17-244)

We find an interesting category of representations of the three simple finitary Lie algebras, ??∞, ??∞, and ??∞. The modules in this category are not only integrable weight modules, but also together with their dual modules have finite Loewy length. We are able to describe the injective modules in this category, and show that the category corresponds to a particular Koszul ring.

8:30 a.m. Prime Ideals in Birational Extensions of Two-Dimensional Power Series Rings

Christina L. Eubanks-Turner*, University of Louisiana at Lafayette;
Serpil Saydam, University of Louisiana at Monroe;
Melissa Luckas, University of Nebraska-Lincoln (1067-13-230)

In this talk we describe the prime spectrum, the set of prime ideals, for certain two-dimensional polynomial and power series rings. Our main result is the characterization of those partially ordered sets that arise as prime spectra of simple birational extensions of a power series ring in one indeterminate with coefficients in a countable infinite Dedekind domain that has infinitely many maximal ideals.

9:00 a.m. Freeness of Arrangement Bundles

Amanda C Hager*, USMA West Point (1067-55-273)

A hyperplane arrangement is a finite collection of hyperplanes in any vector space. I investigate the relationship between the topology of the complement of an arrangement and certain algebraic objects related to an arrangement such as the module of derivations. In particular, Falk and Proudfoot conjectured that this module is well-behaved with respect to fibrations of the complement. I will give results toward this conjecture.

9:30 a.m. Divisibility properties and recursions for the Hilbert series of Macdonald polynomials

Elizabeth M Niese*, Virginia Tech;
Nicholas Loehr, Virginia Tech (1067-05-227)

In this presentation we look at F˜μ(q, t), the Hilbert series of Macdonald polynomials. We use the combinatorial definition ??of F˜μ to prove that F˜μ is divisible by certain factors. To prove this bijectively we introduce a recursion for two-column ?shapes along with several combinatorial operations on the fillings which generate F˜μ. This recursion also leads to a ?fermionic formula which expresses F˜(2n)(q,t) as a sum indexed by perfect matchings.

10:00 a.m. Fourier-Jacobi coefficients of Eisenstein series on unitary group and the application in Iwasawa main conjecture

Bei Zhang*, Northwestern University (1067-11-260)

In this talk, I will explain my work about the calculation of the Fourier-Jacobi expansion of Eisenstein series on U(3,1), or more generally on any non quasi-split unitary group. I relate the Fourier-Jacobi coefficient of the Eisenstein series with special values of L-functions. It can help verify the existence of certain p-integral Eisenstein series on U(3,1) which does not vanish modulo p. This is a crucial step towards the main conjecture for GL2 × K× using Eisenstein congruence method.

Sunday January 9, 2011, 2:30 p.m. – 3:50 p.m., Mardi Gras E, 3rd Floor, Marriott

AWM Workshop: Research Presentations by Recent Ph.D.s, II

2:30 p.m. Predicting Tumor Response to Vascular-Targeting Therapies using a Mathematical Model

Jana Gevertz*, The College of New Jersey (1067-92-188)

A relatively novel use of mathematics in the field of cancer research is in drug discovery process. In this talk, I will briefly describe a validated hybrid cellular automaton model of tumor growth and I will illustrate the models ability to predict the anti-tumor activity of several vascular-targeting compounds of known efficacy. Following model validation, I will demonstrate how the model can be used to make predictions about clinically-untested treatment protocols.

3:00 p.m. Steklov-Neumann Eigenproblems and Nonlinear Elliptic Equations with Nonlinear Boundary Conditions

Nsoki Mavinga*, University of Rochester, NY
M. N. Nkashama, University of Alabama at Birmingham (1067-35-261)

We will present existence results for nonlinear elliptic equations with nonlinear boundary conditions. We introduce the notion of ‘eigenvalue-lines’ in the plane; these eigenvalue-lines join each Steklov eigenvalue to the first eigenvalue of the Neumann problem with homogeneous boundary condition. The nonlinearities involved asymptotically stay, in some sense, below the first eigenvalue-lines or in a quadrilateral region enclosed by two consecutive eigenvalue-lines. The proofs are based on variational methods.

3:30 p.m. Compatibility of Slender Body Theory and Surface Traction

Eva M Strawbridge*, University of Chicago (1067-92-201)

I will present a careful argument for the compatibility of slender body theory, specifically Kirchhoff rod theory, and surface traction due to viscous drag from a Stokes flow. This is the first careful analysis of this theory and will show the precise case when the two theories are compatible and when they are not. This work has direct applications to mathematical biology, in particular DNA mechanics and dynamics and flagellar motion of sperm and microorganisms.