AWM Anniversary Conference at Brown University 2011

Special Session Abstracts

Contents

Saturday Morning 10:15 – 12:15

Algebraic Geometry and Commutative Algebra I

  • Organizers:
  • Claudia Miller (Syracuse University)
  • Mara Neusel (Texas Tech University)
  • Janet Striuli (Fairfield University)

Resolutions over quadratic complete intersections

Irena Peeva, Cornell University

This talk is on the structure of the linear part of a minimal free resolution over a quadratic complete intersection.

Tensor complexes

Christine Berkesch, Duke University

I will discuss two examples where free resolutions appear in algebraic geometry, in the study of determinantal varieties and the construction of resultants for multilinear systems of equations. I will then
present a new construction for building multilinear free resolutions from tensors that simultaneously generalizes these examples. This is joint work with Daniel Erman, Manoj Kummini, and Steven Sam.

Cohomology over quasi-complete intersections

Ines Henriques, University of California at Riverside Adela Vraciu, University of South Carolina

We introduce and study a class of homomorphisms of commutative noetherian rings, which strictly contains the class of locally
complete intersection homomorphisms, while sharing many of its remarkable properties. This is joint work with L. Avramov and L. Sega.

The weak Lefscheftz property for monomial complete intersection ideals in positive characteristic

Adela Vraciu, University of South Carolina

We find explicit necessary and sufficient conditions for the ring k[x,y,z,w]/(x^d, y^d, z^d, w^d) to have the weak Lefschetz property when k is a field of positive characteristic. This is joint work with Andy Kustin.

Conservation Laws — Analytical and Numerical Approaches I

  • Organizers:
  • Katarina Jegdic (University of Houston – Downtown)
  • Barbara Keyfitz (The Ohio State University)

A kinetic equation for an elastic model, via a Hamiltonian lattice with two dimensional displacement field

Charis Tsikkou, Ohio State University

We work on a harmonic lattice with a nonquadratic on-site potential, and two dimensional displacement field, in the context of non linear elasticity. We use Wigner function, Gaussian decoupling and apply the kinetic theory of L. Boltzmann.

Weak transport theory and the vortex-wave system

Helena Nussenzveig Lopes, UNICAMP

The vortex-wave system is a coupling of the two-dimensional vorticity equation with the point-vortex system. It is a model for the motion of a finite number of concentrated vortices moving in a distributed vorticity background. In this talk we discuss existence of a weak solution to this system with an initial background vorticity in Lp, p > 2, up to the time of first collision of point vortices. We also discuss the existence of particle trajectories for this flow.

On the structure of solutions of nonlinear hyperbolic systems of conservation laws

Monica Torres, Purdue University

In this talk we show that a Liouville theorem for systems of conservation laws yields the existence of strong traces on hyperplanes of bounded solutions for the one dimensional isentropic Euler equations.

Global solutions for transonic self-similar 2-dimensional Riemann problems

Eun Heui Kim, California State University – Long Beach

We discuss the recent development of two-dimensional self-similar transonic Riemann problems. More precisely we discuss analytical results and numerical results on a simplified model system — the nonlinear wave system.

Strong regular reflection for two-dimensional isentropic gas dynamics equations

Katarina Jegdic, University of Houston, Downtown

We consider a Riemann problem for two-dimensional gas dynamics equations that gives rise to strong (or transonic) regular reflection. We write the problem in self-similar coordinates and we obtain a free boundary problem for the reflected shock and a subsonic state. We prove existence of a solution using various fixed point arguments and theory of second order elliptic equations with mixed boundary conditions.

Cryptography I

On Probabilistic Proofs

Shafi Goldwasser, MIT

Abstract not available

Flavors and applications of verifiable random functions

Anna Lysyanskaya, Brown University

A random Boolean function is a function where for every input x, the value f(x) is truly random. A pseudorandom function is one where, even though f(x) can be deterministically computed from a small random “seed” s, no efficient algorithm can distinguish f from a random function upon querying it on inputs x1,…,xn of its choice. A verifiable random function (VRF) is a pseudorandom function that can be verified. That is to say, a VRF consists of four algorithms: Generate, Evaluate, Prove, Verify. Alice chooses uses Generate to pick her function f, Evaluate to evaluate it and compute y=f(x), Prove in order to compute a proof p(x) that y is indeed f(x). Bob can then use Verify in order to ascertain that it is indeed the case that y=f(x). At the same time, whenever Bob is not given a proof p(x) for a particular x, no efficient algorithm allows him to determine whether y=f(x) or is random. In this talk I will give a survey of verifiable random functions and their constructions and applications.

Elliptic Curve Primality Tests for Numbers in Special Forms

Alice Silverberg, University of California, Irvine

In joint work with Alex Abatzoglou and Angela Wong, we use elliptic curves with complex multiplication to give primality proofs for integers of certain forms,
generalizing earlier work of B. Gross and of R. Denomme and G. Savin who dealt with elliptic curves with complex multiplication by $Q(i)$ and $Q(\sqrt{-3})$.

Lattices in Cryptanalysis and List Decoding of Error-correcting Codes

Nadia Heninger, UCSD

Coppersmith’s algorithm is a celebrated technique for finding small solutions to polynomial equations modulo integers, and it has many important applications in cryptography. In this talk, I will show how to use lattices to recover an RSA private key from partial information and to find a large approximate common divisor of several integers. Then we will see how the ideas of this technique can be extended to a more general framework encompassing list decoding of Reed- Solomon, Parvaresh-Vardy, and algebraic-geometric codes. These seemingly different problems are all perfectly analogous when viewed from the perspective of algebraic number theory.

Currents Trends in Probability

  • Organizer:
  • Kavita Ramanan (Brown University)

Abstracts not available

Geometric Group Theory I

Hairy graphs and the cohomology of Out(F_n)

Karen Vogtmann, Cornell University

Kontsevich showed that the homology of Out(F_n) can be identified with the cohomology of a certain Lie algebra A. Morita constructed cocycles from elements of the abelianization of A. We will show that the abelianization of A is strictly larger than Morita conjectured, then use the new information to construct new cocycles out of “hairy graphs”. These include a heretofore mysterious unstable homology class in Aut(F_5) which was found in 2002 by F. Gerlits. (This is joint work with Jim Conant and Martin Kassabov.)

Instability in groups

Moon Duchin, Tufts University

In hyperbolic spaces, geodesics between the same endpoints must stay uniformly close together; this means that in hyperbolic groups, different geodesic spellings of the same word can’t be too far apart in the Cayley graph. In free abelian groups, on the other hand, typical words can have geodesic spellings quite far apart from one another because of the many ways to rearrange letters, even though the model spaces are Euclidean and geodesics there are unique.
We might describe this by staying that hyperbolic groups are geodesically stable while free abelian groups are geodesically unstable. I’ll focus this talk on an intermediate situation arising in the nilpotent case, where the groups break into a stable part and an unstable part.

Growth rate of an endomorphism of a group

Delaram Kahrobaei, City University of New York, Graduate Center and City Tech

Bowen defined the growth rate of an endomorphism of a finitely generated group and related it to the entropy of a map $f:M \mapsto M$ on a compact manifold. In this note we study the purely group theoretic aspects of the growth rate of an endomorphism of a finitely generated group. We show that it is finite and bounded by the maximum length of the image of a generator. An equivalent formulation is given that ties the growth rate of an endomorphism to an increasing chain of subgroups. We then consider the relationship between growth rate of an endomorphism on a whole group and the growth rate restricted to a subgroup or considered on a quotient.We use these results to compute the growth rates on direct and semidirect products. We then calculate the growth rate of endomorphisms on several different classes of groups including abelian and nilpotent. This is a joint work with Kenneth Falconer and Benjamin Fine.

Group Theory and Its Connections to Representation Theory I

  • Organizers:
  • Terrell Hodge (Western Michigan University)
  • Julianne Rainbolt (St. Louis University)

Abstracts not available

Homotopy Theory and Its Applications I

Algebraic Structures from diagrams

Julie Bergner, University of California, Riverside

We investigate ways to understand various algebraic structures via diagrams satisfying a Segal-type condition. Such diagrams have been used extensively for monoid or category structures in models for (\infty,1)-categories, but here we look at ways to understand other kinds of algebraic objects. This project is joint work with Philip Hackney.

The S^1 equivariant generating hypothesis

Anna Marie Bohmann, Northwestern University

Freyd’s generating hypothesis is a long-standing conjecture in stable homotopy theory, which says that the stable homotopy groups functor is faithful on finite spectra. We give the appropriate generalization to the equivariant context, where we prove that for finite groups of equivariance, the situation is similar to the nonequivariant case. For circle actions, the picture is strikingly different: even the rational S^1 equivariant generating hypothesis fails.

Spectral sequences of operad algebras

Kristine Bauer, University of Calgary

Since the 1950’s, topologists have understood how to use the multiplicative structure of a spectral sequence of algebras to aid in the study of the spectral sequence. In good cases, such a spectral sequence converges as an algebra, so that the $E^\infty$ page again has multiplicative structure. In this talk, we explain what it means for a spectral sequence to be a sequence of operad algebras. We give conditions which ensure that the sequence converges as
operad algebras and look at examples of how this can be useful in detecting certain (co)homological operations. This is joint work with Laura Scull.

Coincidence invariants

Kate Ponto, University of Kentucky

A fixed point of a continuous endomorphism f of a topological space
X is a point x in X so that f(x)=x. A coincidence point of a pair of continuous maps f and g from X to Y is a point x in X so that f(x)=g(x). Coincidence points are a natural generalization of fixed points. I will explain how the formal structure that describes the Lefschetz fixed point theorem also describes a corresponding theorem for (some) coincidence invariants.

Intersection of Mathematics and Mathematics Education: Research and Practice K-20

  • Organizers:
  • Juliana Belding (Harvard University)
  • Ginger Warfield (University of Washington)

Research in Mathematics Education: Looking Back and Looking Forward

Karen Graham, University of New Hampshire

Abstract: This talk will provide an overview of research in mathematics education K-20 over the last 40 years; the major themes and methodologies that have shaped the research and informed practice. We will also look forward to what might be important focus areas in the future.

A Mathematics Education Researcher’s Travels: Perspectives on Mathematics Education Research from Academia, Government, and Professional Societies

Karen King, New York University

Abstract: In this talk, I describe my perspective on research in mathematics education, its past and future, through my travels as a faculty member in departments of teacher education and mathematics across the country, a program officer at the National Science Foundation and now the Director of Research at the National Council of Teachers of Mathematics. Through the lens of my various experiences, I will present one perspective on the future of mathematics education research PK-20 and the role of mathematicians and mathematics educators in that future.

Helping Students with Proving: A Tale of Two Whole Class Teaching Experiments

Annie Selden, New Mexico State University

First, I will discuss a whole class teaching experiment for helping advanced undergraduate and beginning graduate mathematics students construct proofs. This course has been taught seven times since Fall 2007 and each time we are learning something more about students’ proving capabilities. For example, there are certain aspects of proving that mathematicians do automatically, but that students are often unaware of. We define the formal-rhetorical part of a proof to be those aspects of a proof that can be written by examining the logical structure of the statement of a theorem and by unpacking associated definitions.
Examples include writing the first and last lines, “unpacking” the meaning of the last line, and considering what strategy one might invoke to prove that. Writing the formal-rhetorical part of a proof can expose “the real problem(s)” to be solved. We call the remainder of the proof the problem-centered part.
Second, I will discuss a voluntary proving supplement for an undergraduate real analysis class. This has been taught three times to since Fall 2009. Each week, one proof problem was selected or created to “resemble in construction” an assigned homework proof problem that the real analysis teacher intended to grade in detail, and that could be improved subsequently and resubmitted
for additional credit. The supplement proof problem could be solved using actions similar to those useful in proving the corresponding assigned homework proof problem. However, the supplement proof problem was not a template problem, and “on the surface” would often not resemble the assigned proof problem.
The teaching for both the proofs course and the supplement has been informed by our theory of actions in the proving process and by our division of proofs into their formal-rhetorical and problem-centered parts.

TITLE not available

Natasha Speer, University of Maine

ABSTRACT not available

Mathematical Biology — An Overview

  • Organizers:
  • Holly Gaff (Old Dominion University)
  • Maeve L. McCarthy (Murray State University)

Abstracts not available

Recent Advances in Numerical Methods and Scientific Computing I

  • Organizers:
  • Sigal Gottlieb (University of Massachusetts-Dartmouth)
  • Misha Kilmer (Tufts University)

Accelerated Fixed Point Methods for Subsurface Flow Problems

Carol Woodward, LLNL

We investigate effectiveness of an acceleration method applied to the modified Picard iteration for simulations of variably saturated flow. We solve nonlinear systems using both unaccelerated and accelerated modified Picard iteration as well as Newton’s method. Since Picard iterations can be slow to converge, the advantage of acceleration is to provide faster convergence while maintaining advantages of the Picard method over the Newton method. Results indicate that the accelerated method provides a robust solver with significant potential computational advantages.
Authors: P. A. Lott, H. F. Walker, C. S. Woodward, and U. M. Yang

Block Preconditioners for Coupled Fluid Problems

Victoria Howle, Texas Tech

Many important engineering and scientific systems require the solution of extensions to standard incompressible flow models, whether by coupling to other processes or by incorporating additional nonlinear effects. Finite element methods and other numerical techniques provide effective discretizations of these systems, and the generation of the resulting algebraic systems may be automated by high-level software tools such those in the Sundance project, but the efficient solution of these algebraic equations remains an important challenge.
Frequently, the nonlinear equations are linearized by a fixed point or Newton technique, and then the linear systems are solved by a preconditioned Krylov method such as GMRES. In this talk, we discuss important extensions to the methodology and analysis of preconditioning such systems. In particular, we extend existing block-structured preconditioners (such as those of Elman, et al.) to address some of these coupled systems, showing how an effective preconditioner for Navier-Stokes may be combined with one for some other process such as convection-diffusion of temperature to obtain a preconditioner for the Newton linearization of a nonlinearly coupled system such as Bénard convection.

Tensor factorizations in image processing

Misha Kilmer, Tufts University

Consider a collection of two-dimensional images. This set can be stored as a third order tensor (multi-way array). In this talk, we investigate the use of new tensor factorizations as a sum of k, product cyclic tensors, that are suitable for compression of such
multiway data, and which can be tuned to preserve features such as non-negativity.

Order-p Tensors: Factoring and Applications

Carla D. Martin, James Madison University

Tensor decompositions or representations of multiway arranys have been motivated by applications. Given that data is often multidimensional, extending powerful linear algebra concepts to tensors has been crucial for interpretation of the data as well as data compression. In this talk, we describe a factorization that is SVD-like in nature, for general order-p tensors. The idea is best explained using recursive programming, but is implemented directly using the fast Fourier transform. In particular, we present an alternative representation for order-p tensors as a ‘product’ of tensors which is reminiscent of the matrix factorization approach. This leads to a different generalization of the matrix SVD. Furthermore, our framework allowed other matrix factorizations to be extended to tensors. We also present algorithms for computation and potential applications.

Symplectic and Contact Geometry I

  • Organizers:
  • Eleny Ionel (Stanford University)
  • Katrin Wehrheim (MIT)

10:15 Symplectic topology – an introduction

Dusa McDuff, Columbia University

11:15 Conley conjecture for negative monotone symplectic manifolds

Basak Gurel, Vanderbilt

The Conley conjecture, formulated by Conley in 1984, asserts the existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms of tori and was established by Hingston in 2004. Of course, one can expect the conjecture to hold for a much broader class of closed manifolds and this is indeed the case. For instance, by now, it has been proved for all closed, symplectically aspherical manifolds and Calabi-Yau manifolds using symplectic topological methods. Most recently, jointly with Ginzburg, we establish the conjecture for negative monotone, closed symplectic manifolds.
In this talk, we will briefly examine the question of existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms and outline a proof of the Conley conjecture in the negative monotone case.

11:45 Quilts and Floer modules

Sikimeti Ma’u, UC Berkeley

I’ll describe some general A-infinity module structure that appears in Quilted Floer theory. This structure lies behind spectral sequences on hypercuboid complexes encoding relationships between different Quilted Floer homology groups. I’ll illustrate with examples based on compositions of Dehn twists.

Saturday Afternoon 3:15 – 5:15

Advances in Nonlinear Dynamics I

  • Organizers:
  • Alethea Barbaro (UCLA)
  • Andrea Bertozzi (UCLA)

Spatio-temporal feedback control of unstable wave patterns

Mary Silber, Northwestern University

We extend the methods of Pyragas time-delayed feedback control of unstable periodic orbits to the situation where the unstable periodic orbits arise in a symmetry breaking Hopf bifurcation. We consider traveling wave patterns with spatio-temporal symmetries, as well as oscillator patterns
for equivariant Hopf bifurcation problems.

TITLE not available

Nancy Rodriguez, Stanford University

ABSTRACT not available

An evolving network model for gang rivalries in Los Angeles

Alethea Barbaro, UCLA

Gang rivalries is a leading cause of violent crime in many cities. However, there is still much to be understood about how and why these rivalries form. We introduce an agent-based model coupled to an evolving network in order to explore how such rivalries might arise.

TITLE not available

Brittany Erickson, Stanford University

ABSTRACT not available

Algebraic Geometry and Commutative Algebra II

  • Organizers:
  • Claudia Miller (Syracuse University)
  • Mara Neusel (Texas Tech University)
  • Janet Striuli (Fairfield University)

Hypergeometric functions and toric varieties

Laura Matusevich, Texas A&M University

I will discuss structural issues relating to the classical
hypergeometric equations of Horn type. The key idea (due to Gelfand,
Graev, Kapranov and Zelevinsky) is to construct torus equivariant
versions of these differential equations and study them using D-module theoretic techniques. Transferring results back to the original setting has proved challenging. I will report on recent progress in this direction, joint with Christine Berkesch.

Spaces of rational curves on hypersurfaces

Roya Beheshti Zavareh, Washington University in St. Louis

I will discuss some aspects of the geometry of spaces of rational
curves on general Fano hypersurfaces including dimension,
irreducibility, and birational geometry. A part of this talk is based on joint work with Mohan Kumar.

Conformal blocks and the Mori dream space conjecture

Angela Gibney, University of Georgia at Athens

Given a simple Lie algebra, a positive integer, and an appropriately chosen n-tuple of dominant integral weights, one can define a vector bundle on the moduli space of curves whose fibers are the so-called vector spaces of conformal blocks. On moduli spaces of pointed rational curves, first Chern classes of these vector bundles turn out to be semi-ample divisors, and so define morphisms. In this talk I will discuss the simplest examples of these divisors, and discuss how they relate to a big-picture conjecture of Hu and Keel about the birational geometry of the moduli space.

Arrangements of Lines: Syzygies and secants

Jessica Sidman, Mount Holyoke College

We discuss arrangements of lines that are curves of genus g embedded in projective space. We produce embeddings whose ideals are generated by products of linear forms. We also discuss the syzygies of these line arrangements as well as the syzygies of their secant varieties.

Combinatorics and Graph Theory I

  • Organizers:
  • Ruth Haas (Smith College)
  • Ann Trenk (Wellesley College)

Some problems on list-coloring planar graphs/h4>

Joan P. Hutchinson, Macalester College

A graph G is said to be L-list-colorable when each vertex v is assigned a list L(v) of colors and G can be properly colored so that each v receives a color from L(v). Typically the lists L may vary from vertex to vertex. A graph is said to be k-list- colorable when it can be L-list-colored whenever every list L(v) contains at least k colors. A celebrated theorem of C. Thomassen proves that every planar graph can be 5-list-colored. An unresolved question of M.O. Albertson asks whether there is a distance d > 0 such that whenever a set P of vertices of a planar graph G are precolored and are mutually at distance at least d from one another, the precoloring extends to a 5-list-coloring of G. In this talk we give some partial affirmative answers to Albertson’s question and investigate the extent to which Thomassen’s theorem and Albertson’s question are best possible. This talk includes joint work with co-authors Maria Axenovich, Alice M. Dean, and Michelle A. Lastrina.

Ribbon graphs and twisted duality

Jo Ellis-Monaghan, St. Michael’s College

Abstract: We consider two operations on the edge of an embedded (or ribbon) graph: giving a half-twist to the edge and taking the partial dual with respect to the edge. These two operations give rise to an action of ${S_3}^{e(G)}$, the ribbon group of $G$, on $G$. We show that this ribbon group action gives a complete characterization of duality in that if $G$ is any cellularly embedded graph with medial graph $G_m$, then the orbit of $G$ under the group action is precisely the set of all graphs with medial graphs isomorphic (as abstract graphs) to $G_m$. We then show how this group action leads to a deeper understanding of the properties of, and relationships among, various graph polynomials, Petrie duals, and some knot and link invariants. This is joint work with Iain Moffatt.

On generalized configurations

Brigitte Servatius, WPI

Abstract: We consider generalized configurations as defined by Konrad Zindler (1890) and study their degree of irregularity and their automorphism group. We give a construction of a family of such configurations of arbitrarily large degree of irregularity. This is joint work with Tomaz Pisanski.

Symmetry Breaking in Matroids

Speaker: Jenny McNulty, The University of Montana

Abstract: What can we do to an object to break its symmetry?” That is, how can we restrict the object in some way so that the only automorphism is trivial? We examine two approaches. The first involves distinguishing the elements of the object while the second involves fixing some of the elements. Distinguishing and fixing numbers were originally defined for graphs. We are interested in the extension of these ideas to matroids. We give a sampling of fixing and distinguishing number results for matroids.

Geometric Group Theory II

Length functions of right-angled Artin groups

Ruth Charney, Brandeis University

Morgan and Culler proved that a minimal action of a free group on a tree is determined by its length function. This theorem has played an important role in the study of automorphism groups of free groups. Right-angled Artin groups may be viewed as higher dimensional analogues of free groups. Motivated by an interest in their automorphism groups, we prove a 2-dimensional analogue of Culler and Morgan’’s theorem for right-angled Artin groups acting on CAT(0) rectangle complexes. (Joint work with Max Margolis)

The geometry of right-angled Artin subgroups of mapping class groups

Johanna Mangahas, Brown University

I’ll describe joint work with Matt Clay and Chris Leininger. We give sufficient conditions for a finite set of mapping classes to generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmueller space is a quasi-isometric embedding for both of the standard metrics. As a consequence, we produce infinitely many genus h surfaces (h at least 2) in the moduli space of genus g surfaces (g at least 3) for which the universal covers are quasi-isometrically embedded in the Teichmueller space.

Spaces of CAT(0) structures on 2- and 3- manifolds

Geneive Walsh, Tufts University

Consider a right-angled Coxeter group whose defining graph is the 1-skeleton of triangulation of $S^1$ or S^2$. Associated to such a group is a reflection orbifold, which is finitely covered by a manifold. When the triangulation is acute, the reflection orbifold has a moduli space of singular “cubed” structures, which are made out of “hyperbolic cubes”. These arise directly from the triangulation of $S^1$ or $S^2$. When the right-angled Coxeter group is a group of reflections in the faces of a right-angled hyperbolic 3-dimensional polyhedron, this moduli space of the reflection orbifold contains the (unique) hyperbolic structure. We also give applications to triangulations of $S^2$. This is joint work with Sam Kim.

Generalized Riley Slices of Parameter Spaces of Kleinian Groups

Linda Keen, Graduate Center and Lehman College CUNY

In this talk we will discuss how to extend the theory of pleating rays for those Kleinian groups in the Riley Slice to groups representing orbifolds. We will show how the dynamics of the boundary of this space can be described by continued fractions. We will also talk about discrete subgroups outside the Riley slice that represent finite volume groups and show how they also correspond to boundary points.

Group Theory and Its Connections to Representation Theory II

Modular Representations: Something old, something new

Bhama Srinivasan, UIC

One of the main problems in the p-modular representation theory of finite groups is to determine the decomposition matrix which relates the complex representations with the p-modular representations of the groups. We will describe how, in the case of groups such as symmetric groups and general linear groups, this problem has been found in recent years to have connections with Lie theory, including quantum groups.

BGG Reciprocity for Current Algebras

Yyjayanthi Chari, UC Riverside

We discuss the category Igr of graded representation with finite-dimensional graded pieces fro the current algebra g ⊗ C[t] where g is a simple Lie algebra. This category has many similarities with the category O of modules for g and we formulate the analogue of the famous BGG duality.

The power of symmetric functions in noncommutative variables

Anne Schilling, University of California, Davis

We show in terms of the k-Murnaghan-Nakayama rule and the fusion algebra that it is often preferable to work with symmetric functions in noncommutative variables instead of their commutative counterparts.

Intersections of Mathematics and Math Education: Research and Practice K-20

Is one of these things not like the others? Comparing Math for America with other national teacher preparation and professional development programs

Katherine Socha, Math for America

Math for America is a 7-year-old professional development program that supports public secondary school mathematics teachers. Despite the similarity of names, MfA differs greatly from the Teach For America model. This talk will explore similarities and differences between MfA, TFA, and a few other nationally recognized teacher development programs that emphasize mathematics teaching.

Common Sense and Mathematics: The Role of Quantitative Reasoning in Teacher Education and K – 12 Instruction

Maura B. Mast, University of Massachusetts, Boston

The book Mathematics and Democracy, published in 2001 by the National Council on Education and the Disciplines, begins its argument for the importance of teaching quantitative literacy with the statement that “The world of the twenty- first century is a world is awash in numbers.” But does K – 12 mathematics education in the United States prepare students for the numeracy – and the numerical common sense – they will need as adults? While the new Common Core standards call for students to “reason abstractly and quantitatively,” it is not clear how teachers will be trained to teach their students to use mathematics in context. In this talk, we will discuss possible approaches to teacher education that include quantitative literacy, as well as new initiatives to teach quantitative literacy in the K – 12 setting.

The Common Core State Standards

Juliana Belding, Harvard University and Ginger Warfield, University of Washington

The Common Core State Standards, launched in 2010 and now adopted by forty- four states, include a set of standards for mathematical practice which likely seem second nature to mathematicians. As a result, we are in a unique position to help design and lead professional development for teachers around these standards, i.e. to provide mathematical experiences in which these standards arise naturally. First we will give some examples and talk briefly about some of the challenges and possibilities of translating such experiences back to the middle and high school classroom. Then we will discuss the implications for assessment, and why it is that we all need to stay acutely conscious of developments on that front.

Model Theory and Its Applications I

  • Organizers:
  • Lynn Scow (University of Illinois at Chicago)
  • Carol Wood (Wesleyan University)

Definable sets in o-minimal fields with convex subrings

Jana Marikova, Western Illinois University

The class of structures (R,V), where R is an o-minimal field and V is a convex subring such that the residue field is o-minimal, is first order axiomatizable. We investigate the question whether these structures yield a good generalization of the T-convex case.

Rational points on definable sets and integer-valued functions

Margaret Thomas, Universität Konstanz

Following the influential theorem of Pila and Wilkie concerning the density of rational and algebraic points lying on sets definable in o-minimal expansions of the real field, Wilkie has conjectured an improvement to the result for sets definable in the real exponential field. We shall review some partial results in this direction, including the proven one-dimensional case of the conjecture, and shall illustrate how this can already be applied in considering the growth behaviour of integer-valued definable functions.

TITLE not available

Laurel Miller-Sims, Smith College

ABSTRACT not available

A strictly simple theory with many many fields

Alice Medvedev, UC Berkeley

An action of (Q, +) on a field F is a collection of automorphisms of F indexed by rational numbers, such that composition of automorphisms corresponds precisely to addition of indices. The model companion QACFA of the theory of fields with an action of (Q, +) is a simple theory, neither stable nor supersimple, closely related to ACFA. The fixed fields of the many automorphisms form an infinite lattice of definable fields between two algebraically closed fields: the type- definable intersection of all fixed fields, and the Ind-definable union of them all. This talk is about these fields and that lattice.

Nonlinear Wave Phenomena I

  • Organizers:
  • Andrea Nahmod (University of Massachusetts-Amherst)
  • Gigliola Staffilani (MIT)

On the Gross-Pitaevskii hierarchies

Natasa Pavlovic, The University of Texas at Austin

The Gross-Pitaevskii (GP) hierarchy is an infinite system of coupled linear non-homogeneous PDEs, which appear in the derivation of the nonlinear Schrödinger equation (NLS). Inspired by the PDE techniques that have turned out to be useful on the level of the NLS, we realized that, in some instances we can introduce analogous techniques at the level of the GP. In this talk we will discuss
some of those techniques which we use to study well-posedness
for GP hierarchies. The talk is based on joint works with T. Chen and with T. Chen and N. Tzirakis.

Title TBD

Magda Czubak, Binghamton University

We establish an interaction Morawetz estimate for the magnetic Schrodinger equation under certain smallness conditions on the gauge potentials. We discuss applications to wellposedness and scattering. This is joint work with J. Colliander and J. Lee.

Dynamics of blow up solutions to the focusing nonlinear Schroedinger equation

Svetlana Roudenko, George Washington University

I will consider the focusing NLS equation in one, two and three space dimensions with different powers of nonlinearities (including cubic and quintic powers) and their global solutions with finite energy initial data.

My discussion will focus on blow up solutions and known types of their dynamics. In particular, I will show that the class of so-called `log-log’ blow up solutions can blow up not only on a single point set but on various geometric sets such as circles, spheres, cylinders, while remaining regular (in the energy space) away from the blow-up core.

From the Geometry of Einstein-Maxwell Spacetimes in General Relativity to Gravitational Radiation

Lydia Bieri, University of Michigan, Ann Arbor

A major goal of mathematical General Relativity (GR) and astrophysics is to precisely describe and finally observe gravitational radiation, one of the predictions of GR. In order to do so, one has to study the null asymptotical limits of the spacetimes for typical sources. Among the latter we find binary neutron stars and binary black hole mergers. In these processes typically mass and momenta are radiated away in form of gravitational waves. D. Christodoulou showed that every gravitational-wave burst has a nonlinear memory. In this talk, we discuss the null asymptotics for spacetimes solving the Einstein-Maxwell (EM) equations, compute the radiated energy and derive limits at null infinity and compare them with the Einstein vacuum (EV) case. The physical insights are based on geometric-analytic investigations of the solution spacetimes.

Number Theory I

  • Organizers:
  • Alina Bucur (University of California San Diego)
  • Michelle Manes (University of Hawaii)

TITLE

Melaine Matchett Wood, AIM & University of Wisconsin

ABSTRACT

Descent on elliptic surfaces and transcendental Brauer elements

Bianca Viray, Brown University

Transcendental elements in the Brauer group are notoriously difficult to compute. Wittenberg and Ieronymou have worked out explicit representatives for 2-torsion elements of elliptic surfaces, in the case that the Jacobian fibration has rational 2-torsion. We use ideas from descent to develop techniques to study the 2- torsion elements of elliptic surfaces without an assumption on the 2-torsion of the Jacobian.

Computations with Coleman integrals

Jennifer Balakrishnan, Harvard University

The Coleman integral is a p-adic line integral that can encapsulate valuable information about the arithmetic and geometry of curves and abelian varieties. For example, certain integrals allow us to find rational points or torsion points; certain others give us p-adic height pairings. I’ll present a brief overview of the theory, describe algorithms to calculate some of these integrals, and illustrate these techniques with numerical examples computed using Sage.

Constructing genus 2 curves for cryptography

Kristin Lauter, Microsoft Research

Jacobians of genus 2 curves can be used in cryptography, but constructing
curves which are appropriate to use requires deep methods from number theory, including the theory of complex multiplication. This talk will explain some of these methods, including Igusa class polynomials and the conjecture of Bruinier and Yang.
Bruinier and Yang conjectured a formula for an intersection number on the arithmetic Hilbert modular surface, CM(K).Tm, where CM(K) is the zero-cycle of points corresponding to abelian surfaces with CM by a primitive quartic CM field K, and Tm is the Hirzebruch-Zagier divisors parameterizing products of elliptic curves with an m-isogeny between them. In this talk, we examine fields not covered by Yang’s proof of the conjecture. We give numerical evidence to support the conjecture and point to some interesting anomalies. We compare the conjecture to both the denominators of Igusa class polynomials and the number of solutions to the embedding problem stated by Goren and Lauter.
This project was initiated at the WIN workshop in Banff in November 2008, and is joint work with: Bianca Viray, Jennifer Johnson-Leung, Adriana Salerno, Erika Frugoni and Helen Grundman.

Recent Advances in Numerical Methods and Scientific Computing II

Conservative high order semi-Lagrangian hybrid finite element-finite difference methods for the Vlasov equation

Jingmei Qiu, Colorado School of Mines

We propose a novel conservative hybrid finite element-finite difference method for the Vlasov equation. The methodology uses semi-Lagrangian discontinuous Galerkin (DG) for spatial advection, and semi-Lagrangian finite difference WENO for velocity acceleration/deceleration. Such hybrid method enjoys the advantage of DG in its compactness and ability in handling complicated geometry and that of the WENO in its robustness in resolving sharp gradients. Simulation results will be demonstrated to show the performance of the proposed method. Authors: Jingmei Qiu and Wei Guo

Positive Preserving discontinuous Galerkin methods for ideal MHD equations

Fengyan Li, Rensselaer Polytechnic Institute

Ideal MHD system arises in astrophysics and energy physics, it
consists of a set of nonlinear hyperbolic conservation laws. In this presentation, we report our recent progress in developing positive preserving discontinuous Galerkin (DG) and central DG methods for this system. Such methods preserve positivity of density and pressure in the simulation.

Hierarchical Algorithms in Heterogeneous Systems for the Exascale Era

Lorena Barba, Boston University

There is a potentially transformative combination in the use of fast hierarchical algorithms with heterogeneous systems. In particular, GPU-based clusters are likely to be a leading contender in the exascale era, and algorithms that adapt well to this hardware will become key players. Among these, the fast multipole method is particularly favorable for heterogeneous many-core hardware. In fact, this algorithm offers exceptional opportunities to enable exascale applications. Its exascale-suitable features include: (i) intrinsic geometric locality, and local access patterns via particle indexing techniques; (ii) temporal locality achieved via an efficient queuing of GPU tasks before execution; and (iii) global data communication and synchronization, often a significant impediment to scalability, is a soft barrier for the FMM. Recent innovations in this area indicate a potential for unprecedented performance with these algorithms

Dynamics of an elastic rod in a viscous fluid: Stokes Formulation

Sarah Olson, WPI (as of Fall 2011)

The generalized immersed boundary (IB) method simplifies the interaction of a slender, elastic rod with a surrounding fluid by representing the rod by its centerline and keeping track of an evolving orthnormal director basis. In this method, the IB applies torque and force to the surrounding fluid. Additionally, the equations of motion of the IB involve the local angular velocity and the local linear velocity of the fluid. We will
present a Stokes formulation of the generalized immersed boundary method. In this formulation, the fluid velocity resulting from a distribution of regularized forces and torques is expressed in terms of regularized Stokeslets and rotlets. The dynamics of an open and closed rod with curvature and twist in a viscous fluid will be studied as a benchmark problem. This will be compared to previous generalized immersed boundary implementations that solved the full Navier-Stokes equations using finite difference methods.
This work has been in collaboration with Sookkyung Lim at University of Cincinatti, Ricardo Cortez at Tulane University, and Lisa Fauci at Tulane University.

Riemannian Geometry I

  • Organizers:
  • Carolyn Gordon (Dartmouth College)
  • Sarah Greenwald (Appalachian State University)
  • Christina Sormani (CUNY GC and Lehman College)

Complete Ricci-flat Kahler metrics on resolutions of singularities

Bianca Santoro, CCNY, CUNY

ABSTRACT

Orbifold homeomorphism finiteness based on geometric constraints

Emily Proctor, Middlebury College

We will describe a proof that the collection of all compact
n-dimensional orbifolds with sectional curvature uniformly bounded below, diameter bounded above, volume bounded below, and having only isolated singular points contains only finitely many orbifold homeomorphism types.
As a corollary, any isospectral collection of orbifolds with sectional
curvature uniformly bounded below and having only isolated singular points contains only finitely many orbifold homeomorphism types. The proof involves finitely partitioning the collection according to singular data,
and then showing that any sequence from a particular subcollection has a convergent sequence whose limit is also an orbifold. From there, Perelman’s stability theorem can be used to obtain the final result.

Homogeneous spaces with nonnegative curvature

Megan Kerr, Wellesley College

Abstract: We study invariant Riemannian submersion metrics on compact homogeneous spaces $G/H$ where
there is a fibration $K/H \to G/H \to G/K$. Given a triple $(H,K,G)$, we determine whether scaling up the fibers maintains the nonnegative curvature of a normal homogeneous metric on $G/H$. In this work, we use a criterion of Schwachh\”{o}fer and Tapp, exploring the algebraic causes behind their condition.

Hypersurfaces with nonnegative scalar curvature

Lan-Hsuan Huang, Columbia University

Since the time of Gauss, geometers have been interested in the interplay between the intrinsic metric structure of hypersurfaces and their extrinsic geometry in the ambient space. For example, a result of Sacksteder tells us that if a complete hypersurface has non-negative sectional curvature, then its second fundamental form in Euclidean space must be positive semi-definite.
In a recent joint work with Damin Wu, we study hypersurfaces under a much weaker intrinsic curvature condition. We prove that closed hypersurfaces with nonnegative scalar curvature must be mean convex. This result is optimal in the sense that the scalar curvature cannot be replaced by other k-th mean curvatures. The result and argument have applications to the mean curvature flow, to the positive mass theorem, and to the rigidity theorems.

Sunday Morning, 8:30 – 10:30

Advances in Nonlinear Dynamics II

A new result in blow-up for long-wave unstable thin film equations

Mary Pugh, University of Toronto

This talk will provide an introduction to long-wave unstable thin film equations of the form $$u_t = – (u^n u_{xxx})_x – B (u^m u_x)_x. $$ The exponents $n$ and $m$ determine whether or not finite-time blow-up of the solution might occur. In this talk, we present new results for the critical case $n=m+2$ on the line. This is joint work with Marina Chugunova and Roman Taranets.
A factorization method for non-symmetric linear operator: enlargement of the functional space while preserving hypo-coercivity.
Maria Pia Gualdani, University of Texas at Austin
We present a factorization method for non-symmetric linear operators: the method allows to enlarge functional spaces while preserving spectral properties for the considered operators. In particular, spectral gap and related convergence towards equilibrium follow easily by hypo-coercivity and resolvent
estimates. Applications of this theory on several kinetic equations will be presented.

TITLE not available

Carola-Bibiane Schoenlieb, University of Cambridge

ABSTRACT not available

Compressible fluids with vacuum

Juhi Jang, UC Riverside

I’ll discuss some vacuum states arising in gas dynamics. The rigorous results include the well-posedness of compressible Euler equations with vacuum free boundary and some open problems will be addressed.

Combinatorics and Graph Theory II

The Combinatorialization of Linear Recurrences

Jennifer Quinn, University of Washington, Tacoma

Abstract: Binet’s formula for the nth Fibonacci number, $F_n = \frac{1}{\sqrt{5}} \left( \frac{1+\sqrt{5}}{2}\right) ^n – \frac{1}{\sqrt{5}} \left( \frac{1-\sqrt{5}}{2}\right)^n,$ is a classic example of a closed form
solution for a homogenous linear recurrence with constant coefficients.
Proofs range from matrix diagonalization to generating functions to index-chasing proofs by strong induction. Could there possibility be a better way? A more visual approach? A combinatorial method?
This talk introduces a combinatorial model using weighted tiles. Coupled with a sign reversing involution, Binet’s formula becomes a direct consequence of counting exceptions. But better still, the weightings generalize to find solutions for any homogeneous linear recurrences with constant coefficients.

Matching Algorithms and their Applications

Pallavi Jayawant, Bates College

The classical one-to-one matching problem is about matching two groups of people taking into consideration the ranked preferences of every member of each group over the members of the other group. In general, the expectation is that the resulting matching should be stable in the sense that no two people should get better ranked partners by simply switching their partners. When it comes to matching people to institutions, the one-to-one problem generalizes to the many- to-one matching problem as in this case more than one person can be matched to the same institution. Again there is a notion of stability associated with these matchings. I will talk about these matching problems and the algorithms that help us obtain stable matchings in each case. I will also discuss applications of these algorithms with particular focus on the hospital/resident problem and the ways in which we can try to address some of the issues that arise due to its difference from the classical problem.

Graphs of Polytopes

Marge Bayer, University of Kansas

A well-known theorem of Steinitz says that a graph G is the graph of a 3- dimensional polytope if and only if G is planar and 3-connected. No such characterization is known for the graphs of convex polytopes of higher dimensions. However, it is known that the graph of every d-dimensional polytope is d-connected and contains a subdivision of the complete graph $K_{d+1}$. The nonexpert in polytopes may be surprised to learn that for every $n \geq5$ and every d, $4 \leq d \leq n- 1$, there is a d-dimensional polytope whose graph is $K_n$. The graph of an n-dimensional crosspolytope (generalized octahedron) is the n-partite graph $K_{2,2,\dots , 2}$. William Espenschied determined the dimensions d for which there exists a d-dimensional polytope with graph $K_{2,2,\dots , 2}$. As time permits, we discuss other results and questions about graphs of polytopes.

Extremal set theory and the weak order on the symmetric group

Susanna Fishel, Arizona State University

Extremal set theory is the combinatorial study of families of finite sets: how large or small a family with a given property can be, and what structure is found in extremal cases. For example, the Erdos-Ko-Rado Theorem gives the maximum size of a family of pairwise intersecting k-subsets of {1,,,,n}. We consider similar questions. However, instead of the boolean poset, we study the weak order on the symmetric group, where we replace “intersecting” with “meet at rank at least one.” This is joint work with Glenn Hurlbert, Karen Meagher, Vikram Kamat.

Conservation Laws — Analytical and Numerical Approaches II

Hybrid meshfree methods for hydrodynamics

Nathalie Lanson, University of Waterloo

Meshfree methods, also referred to as particle methods, are a set of lagrangian methods developed for the approximation of conservation laws in many fields of hydrodynamics and solid dynamics. The main advantage of these methods lays in their ability to handle complex situations involving highly distorted systems, such as crash or impact problems. This is due to the fact that no structural mesh is needed. Despite their potential, meshfree methods present some weaknesses. In particular, the use of numerical viscosity to stabilize the schemes can yield unphysical behaviors. In this talk I will present the hybrid renormalized meshfree scheme. I will show how this scheme is built upon a finite volume treatment of the interactions between particles and present simulations underlining the significant improvement that this new scheme offers.

BOUNDED DOMAIN PROBLEM FOR THE MODIFIED BUCKLEY-LEVERETT EQUATION

Chiu-yen Kao, Ohio State University

The focus of the present study is the modified Buckley-Leverett (MBL) equation describing two-phase flow in porous media. The MBL equation differs from the classical Buckley-Leverett (BL) equation by including a balanced diffusive- dispersive combination. The dispersive term is a third order mixed derivatives term, which models the dynamic effects in the pressure difference between the two phases. The classical BL equation gives a monotone water saturation profile for any Riemann problem; on the contrast, when the dispersive parameter is large enough, the MBL equation delivers non-monotone water saturation profile for certain Riemann problems as suggested by the experimental observations. In this talk, we first show that the solution of the finite interval [0, L] boundary value problem converges to that of the half-line [0,+∞) boundary value problem for the MBL equation as L → +∞. This result provides a justification for the use of the finite interval boundary value problem in numerical studies for the half line problem. Furthermore, we extend the classical central schemes for the hyperbolic conservation laws to solve the MBL equation which is of pseudo- parabolic type. Numerical results confirm the existence of non-monotone water saturation profiles consisting of constant states separated by shocks.

Linear Stability Analysis of the Discontinuous Galerkin Method on Non- Uniform Grids

Lilia Krivodonova, University of Waterloo

Applying a discontinuous Galerkin spatial discretization to a hyperbolic PDE results in a system of ODEs for the unknown solution coefficients. This can be solved with a time integration scheme such as for example a Runge-Kutta method. The largest allowable time step depends on the eigenvalues of the spatial discretization matrix (or the Jacobian in the nonlinear case) and the absolute stability region of the ODE solver. In this talk we present an analysis of the eigenvalues of the DG scheme with the upwind flux applied to the one- dimensional scalar advection equation. We derive a formula for the eigenvalues on an N element uniform grid in terms of the sub-diagonal Pade (p+1,p) approximation of exp(-z), where p is the dimension of the finite element basis. This allows us to draw a number of conclusions about the CFL number and stability of the scheme. For example, we obtain a bound on the largest in magnitude eigenvalue and also its asymptotic growth rate with the order of approximation. Then, we analyze the eigenvalues of the DG on non-uniform grids and demonstrate that a CFL condition less than the one prescribed by the local stability condition can be used.

Noise Filtering in Spatial Gradient Sensing and Response during Yeast Cell Polarization

Ching-Shan Chou, Ohio State University

Cells sense chemical spatial gradients and respond by polarizing internal components. This process is disrupted by gradient noise caused by fluctuations in chemical concentration. In this talk, I will discuss how gradient noise affects

spatial sensing and response. In our study, we discovered that a combination of positive feedback, multiple signaling stages, and time-averaging produced good results. There was an important tradeoff, however, because filtering resulted in slower polarization. Using both modeling and experiments, we showed that yeast cells likely also combine the above three filtering mechanisms to achieve impressive spatial-noise tolerance, but with the consequence of a slow response time.

Cryptography II

On a Magic of Math in a Key Exchange Protocol
Tal Rabin, IBM Research</h5
Abstract not available

Public Key Encryption: From Basic Security to Stronger Notions

Tal Malkin, Columbia University</h5
Public key encryption (PKE) allows parties that had never met in advance to communicate over an unsafe channel. The notion was conceived in the 1970s, followed by the discovery that one could provide formal definitions of security for this and other cryptographic problems, and that such definitions were achievable by assuming the hardness of some computational problem (e.g., factoring large numbers). For PKE, the most basic security definition — semantic security — guarantees passive privacy, namely that it is infeasible to learn anything about the plaintext from its encryption. However, as cryptographic applications grew more sophisticated, this level of security is often not sufficient, since it does not protect against active attacks arising in networked environments. Much recent work has focused on achieving stronger security notions for PKE, such as protections against adaptive corruptions, man-in-the-middle attacks (malleability), chosen ciphertext attacks, leakage and tampering attacks. I will review some of the main themes in this line of work, focusing on the example of using any basic PKE as a “black box” to construct a non-malleable PKE.

Functional Encryption: Current Systems and Proof Techniques

Allison Bishop Lewko, University of Texas, Austin</h5

In this talk, we will describe some examples of functional encryption and the challenges that arise in proving their security. We will then discuss the methodology of dual system encryption (recently introduced by Waters) and a few of its applications.

Cryptography Robust against Side Channel Attacks

Yael Tauman-Kalai, Microsoft Research – New England

Traditionally, cryptographers assume that the secret keys are totally hidden from the adversary. However, in reality there are various real-world physical attacks, including, timing and power attacks, which allow an adversary to (continually) leak information about the secret keys. In addition, there are various attacks, including heat and EM radiation attacks, which allow an adversary to (continually) tamper with the secret keys.
Recently, there has been a large and growing body of work, which tries to secure cryptographic systems against such, so called, side-channel attacks.
In this talk, I will survey some of these results, and focus on two recent results, which show how to construct cryptographic schemes that are secure even against an adversary that continually leaks (bounded) information about the secret key, and continually tampers with the secret key.
These results are based on joint work with Zvika Brakerski, Jonathan Katz and Vinod Vaikuntanathan, and on joint work with Bhavana Kanukurthi and Amit Sahai.

Geometry and Combinatorics in Flag Manifolds and Beyond

  • Organizers:
  • Rebecca Goldin (George Mason University)
  • Julianna Tymoczko (University of Iowa)

Abstracts not available

Homotopy Theory and Its Applications II

Spectra associated to symmetric monoidal bicategories

Angelica Osorno, University of Chicago

In this talk, we show how to construct a spectrum from a symmetric monoidal bicategory, using Segal’s Gamma-spaces. As an
example, we use this machinery to show the group-like symmetric monoidal bigroupoids model stable homotopy 2-types.

Partial Approximation Towers for Functors

Rosona Eldred, University of Illinois, Urbana-Champaign

We call attention to constructions in Goodwillie’s Calculus of homotopy functors that lead to a new partial approximation tower, and its relationship with certain cosimplicial resolutions. For functors of spaces and spectra, this equivalence gives a greatly simplified construction for the Taylor tower of functors under fairly general assumptions.

Algebraic model structures

Emily Riehl, Harvard University

Cofibrantly generated model categories have an algebraic model structure, which is a Quillen model structure in which the functorial factorizations define monands and comonads, inducing a fibrant replacement monad and a cofibrant replacement comonad. In the presense of this structure, the (co)fibrations can be regarded as (co)algebras for the (co)monad. The (co)algebra structures witness the fact that a particular map is a (co)fibration and can be used to construct a canonical solution to any lifting problem. For example, the algebraic structure for Hurewicz fibrations is a path lifting function; for Kan fibrations it is a choice of filler for all horns. We describe a few features of this theory and then define and characterize algebraic Quillen adjunctions, in which the funtors must preserve algebraic (co)fibrations, not simply ordinary ones. We conclude with a brief discussion of new work defining monoidal and enriched algebraic model structures that gives particular emphasis to the role played by “cellularity” of certain cofibrations.

Computations in Algebraic K-Theory

Teena Gerhardt, Michigan State University

Techniques from equivariant stable homotopy theory are often key to algebraic K-theory computations. In this talk I will describe joint work with Vigleik Angeltveit, Mike Hill, and Ayelet Lindenstrauss, yielding new computations of algebraic K-theory groups using equivariant methods. In particular, we consider the K-theory of truncated polynomial algebras in
several variables.

Model Theory and Its Applications II

GENERALIZED POWER SERIES AND REAL CLOSED FIELDS

Karen Lange, Wellesley College

An integer part I for an ordered _eld R is a discrete ordered subring containing 1 such that for all r 2 R there exists a unique i 2 I with i _ r < i+1. Mourgues and Ressayre [1] showed that every real closed field R has an integer part by constructing a special embedding of R into a _eld khhGii of generalized power series. Let k be the residue field of R, and let G be the value group of R. The field khhGii consists of elements of the form _g2Sagg where ag 2 k and the support of the power series S _ G is well ordered. Ressayre [2] showed that every real closed exponential _eld has an integer part I that is closed under 2x for
positive elements of I using the same approach as in [1]. However, he had to choose more carefully the value group G and the embedding of R into khhGii. We demonstrate that these alterations cause Ressayre’s construction in the exponential case to be much more complex than Mourgues and Ressayre’s original construction.This is joint work with Paola D’Aquino, Julia Knight, and Salma Kuhlmann.
References
[1] M. H. Mourgues and J.-P. Ressayre, \Every real closed _eld has an integer part,” J. Symb. Logic, vol. 58 (1993), pp. 641-647.
[2] J.-P. Ressayre, \Integer parts of real closed exponential _elds,” in Arithmetic, Proof Theory, and Computational Complexity, Oxford Logic Guides, vol. 23 (1993), pp. 278-288.

Solution Spaces to Linear Equations in Valued D-Fields

Meghan Anderson, Harvard University

A D-field is a field endowed with a derivative-like operator obeying a (possibly) twisted Leibniz rule. Both difference and differential fields can be seen as D- fields. When a valuation is introduced, interacting with the operator in the proper way, both cases can be considered in the same structure, with good model theoretic properties. We’ll look at the solution spaces to linear equations in such structures.

Independence results in the model theory of infinitary logics

Monica van Dieren, Robert Morris University

Initial results in the development of model theory of infinitary logic were splattered with set theoretic assumptions and sometimes turned to be independent of ZFC. Later on set theoretic assumptions continued to show up in model theoretic results for non-first order logics because they served as a stand- in for compactness. We will provide a brief history of the interplay between set theory and model theory and highlight some recent advancements.

Invariant Measures on Countable Models

Rehana Patel, Harvard University/Wesleyan University

The Erdos-Renyi random graph construction can be seen as inducing a probability measure concentrated on the Rado graph (sometimes known as the countable “random graph”) that is invariant under arbitrary permutations
of the underlying set of vertices. A natural question to ask is: For which
other countable combinatorial structures does such an invariant measure exist? Until recent work of Petrov and Vershik (2010), the answer was not known even for Henson’s countable homogeneous-universal triangle-free graph.
In this talk we will provide a characterization of countable relational structures that admit invariant measures, in terms of the notion of definable closure. This leads to new examples and non-examples, including a complete list of homogeneous graphs satisfying our criterion, as well as certain directed graphs and partial orders. Joint work with Nathanael Ackerman and Cameron Freer.

Nonlinear Wave Phenomena II

Boundary layers for channel and pipe flows

Anna Mazzucato, Penn State University

We present recent results concerning the vanishing viscosity limit and the analysis of the associated
boundary layer for certain classes of incompressible Couette flows in channels and pipes.

Uniqueness of solitary water waves

Vera Hur, University of Illinois, Urbana-Champaign

I will speak on the uniqueness issue of solitary waves on the free surface of
a two-dimensional steady flow of water over a finite bed, acted upon by gravity. I will begin by a brief account of the solitary water wave problem as a nonlinear pseudo-differential equation involving the Dirichlet-Neumann operator. I will mention existence/non-existence of solutions. After briefing on the non- uniqueness and the instability results of waves near the extremal form, I will describe my recent work on the non-degeneracy of the linearized equation and its implication for uniqueness for small-amplitude waves.

Smooth global solutions for the two dimensional Euler-Poisson system

Xiaoyi Zhang, The University of Iowa

The Euler-Poisson system is a fundamental two-fluid model to describe the dynamics of the plasma consisting of compressible electrons and a uniform ion background. By using the dispersive Klein-Gordon effect, Yan Guo first constructed a global smooth irrotational solutions in the three dimensional case. It has been conjectured that same results should hold in the two dimensional case. The main difficulty in 2D comes from the slow dispersion of the linear flow and certain nonlocal
resonant obstructions in the nonlinearity. We develop a new method to overcome these difficulties and construct smooth global solutions for the 2D Euler-Poisson system.

A rigorous justification of the modulation approximation to the 2D full water wave problem

Sijue Wu, University of Michigan, Ann Arbor

We consider the 2D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Given wave packet initial data of the form εB( εα)e^{ikα} for k > 0, we show that the modulation of the solution is a profile traveling at group velocity and governed by a focusing cubic nonlinear Schrodinger equation, with rigorous error estimates in Sobolev spaces. As a consequence, we establish existence of solutions of the water wave problem in Sobolev spaces for times of order O(ε^{−2}) provided the initial data differs from the wave packet by at most O(ε^{3/2}) in Sobolev spaces. These results are obtained by directly applying modulational analysis to the evolution equation with no quadratic nonlinearity constructed in [1] and by the energy method. This is a joint work with Nathan Totz.
[1] S. Wu. Almost global wellposedness of the 2-D full water wave problem. Invent. Math., 177(1):45–135, 2009.

Number Theory II

Classification and Symmetries of a Family of Continued Fractions With Bounded Period Length

Renate Scheidler, University of Calgary

It is well-known that the continued fraction expansion of a quadratic irrational is symmetric about its centre; we refer to this symmetry as horizontal. However, an additional vertical symmetry is exhibited by the continued fraction expansions arising from a certain one-parameter family of positive integers known as Schinzel sleepers. This talk investigates the period lengths as well as both the horizontal and vertical symmetries of this family. We also outline a method for generating all Schinzel sleepers. This is joint work with Kell Cheng, Richard Guy and Hugh Williams.

Ramanujan bigraphs associated with $SU(3)$ over a $p$-adic field

Cristina Ballantine, College of the Holy Cross with Dan Ciubotaru

We use the representation theory of the quasisplit form $G$ of $SU(3)$ over a $p$-adic field to investigate whether certain quotients of the Bruhat–Tits tree associated to this form are Ramanujan bigraphs. We show that a quotient of the tree associated with $G$ (which is a biregular bigraph) is Ramanujan if and only if $G$ satisfies a Ramanujan type conjecture. This result is analogous to the seminal case of $PGL_2(\mathbb{Q}_p)$ considered by Lubotzky, Phillips and Sarnak. As a consequence, Rogawski’s classification of the automorphic spectrum of the unitary group in three variables implies the existence of certain infinite families of Ramanujan bigraphs.

L-functions and periods

Brooke Feigon, University of East Anglia

I will describe recent work relating L-functions to periods.

TITLE

Marie-France Vigneras, Institut de Mathematiques de Jussieu

ABSTRACT

Riemannian Geometry II

Asymptotic geometry of the Heisenberg group

Moon Duchin, Tufts University

ABSTRACT

Self-shrinkers of Mean Curvature Flow

Lu Wang, MIT

ABSTRACT

Scalar flat Kahler metrics on non compact toric surfaces

Rosa Sena-Dias, Instituto Superior Técnico, Lisbon

ABSTRACT

Resonance for loop homology on spheres

Nancy Hingston, College of New Jersey

ABSTRACT

Symplectic and Contact Geometry II

8:30 Contact Invariant in Sutured Floer Homology

Gordana Matic, University of Georgia

We describe an invariant of contact structures in Sutured Floer Homology and show some applications to non-fillability of contact structures. This is joint work with Ko Honda and Will Kazez.

9:00 Transverse invariants in Heegaard Floer homology

Vera Vertesi, MIT

Using the language of Heegaard Floer knot homology recently two invariants were defined for Legendrian knots. One in the standard contact 3-sphere defined by Ozsvath, Szabo and Thurston in the combinatorial settings of knot Floer homology, one by Lisca, Ozsvath, Stipsicz and Szabo in knot Floer homology for a general contact 3–manifold. Both of them naturally generalizes to transverse knots. In this talk I will give a characterization of the transverse invariant, similar to the one given by Ozsvath and Szabo for the contact invariant. Namely for transverse braids both transverse invariants are given as the bottommost elements with respect to the filtration of knot Floer homology given by the axis. The above characterization allows us to prove that the two invariants are the same in the standard contact 3–sphere. This is a joint work with J. Baldwin and D.S. Vela-Vick.

9:30 Legendrian Contact Homology in Seifert Fibered Spaces

Joan Licata, Stanford University

Seifert fibered spaces can be viewed as circle bundles over surface orbifolds. We consider when these manifolds can be equipped with contact forms whose Reeb orbits realize the Seifert fibers, and we construct a combinatorial invariant modeled on Legendrian contact homology for Legendrian knots in these contact manifolds. This is joint work with J. Sabloff.

10:00 Topological Invariants of Orbifolds

Tara Holm, Cornell

We present techniques for computing the various cohomology and K-theory rings associated to an orbifold X that arises as a symplectic quotient M//G. We will pay particular attention to the example of a weighted projective space, where many computations simplify.