# Math Calendar

 Categories: Choose a category... View all categories _______________________ Biomath Seminar Colloquium Computational Analysis Seminar Departmental Student Awards Faculty Meeting Graph Theory and Combinatorics Seminar Informal von Neumann Algebras Seminar Noncommutative Geometry and Operator Algebras Seminar Partial Differential Equations Seminar Subfactor Seminar Symplectic and Differential Geometry Seminar Topology and Group Theory Seminar Undergraduate Seminar Universal Algebra and Logic Seminar Vandy Math Club RSS

###### » Seminar Pages

September 2, 2013 3:10 pm (Monday)

## Structure and Hamiltonicity of 3-Connected Graphs with Excluded Minors

Mark Ellingham, Vanderbilt University
Location: Stevenson 1432

Tutte showed in 1956 that all 4-connected planar graphs are hamiltonian. But examples of 3-connected planar graphs (triangulations, even) that are not hamiltonian have been known since at least the 1930s. So we may ask what additional conditions can be imposed on 3-connected planar graphs to make them hamiltonian. We show that K_{2,5}-minor-free 3-connected planar graphs are hamiltonian. Using similar techniques we investigated the hamiltonicity, and in fact the general structure of 3-connected K_{2,4}-minor-free graphs, leading to a complete characterization of all K_{2,4}-minor-free graphs. Recently we found a way to shorten our proof using some results of Guoli Ding and Cheng Liu. We discuss these results as well as some new directions. For example, computations by Gordon Royle suggest that although there are infinitely many nonhamiltonian 3-connected K_{2,6}_minor-free planar graphs, it may be possible to characterize them. The results of Ding and Liu may lead to characterizations of H-minor-free graphs for other small 2-connected graphs H. This is joint work with Emily Marshall, Kenta Ozeki and Shoichi Tsuchiya.

September 4, 2013 4:10 pm (Wednesday)

## On Pairs of Finitely Generated Subgroups in Free Groups

Alexander Olshanskii, Vanderbilt University
Location: Stevenson 1310

We prove that for arbitrary two finitely generated subgroups A and B having infinite index in a free group F, there is a subgroup H of finite index in B such that the subgroup < A,H > generated by A and H has infinite index in F. The main corollary of this theorem says that a free group of free rank r > 1 admits a faithful highly transitive action, whereas the restriction of this action to any finitely generated subgroup of infinite index in F has no infinite orbits.

September 5, 2013 4:10 pm (Thursday)

## Departmental Welcome Event

Location: Stevenson 5211

Please join us for the annual welcome meeting, which will be an opportunity for us to be introduced to new faculty and students. The event will be followed by a reception in the graduate student area. All faculty, students, and staff are invited and encouraged to attend.

September 6, 2013 4:10 pm (Friday)

## Subfactors with Prescribed Fundamental Groups

Arnaud Brothier, Vanderbilt University
Location: Stevenson 1432

In a joint work with Stefaan Vaes we study fundamental groups for subfactors. We consider a large class of subgroups $\mathcal S$ of $\mathbb R_+^\times$ that contains all countable subgroups and some uncountable groups with any Hausdorff dimension between 0 and 1. Using Popa's deformation/rigidity theory, we construct a hyperfinite subfactor of index 6 with its fundamental group equal to $G$, for any group $G$ in $\mathcal S$. Those subfactors are indexed by ergodic probability measure preserving transformation. This proves in particular that there are unclassifiably many subfactors at index 6 with the same standard invariant. Furthermore, using those technics we provide an explicit uncountable family of non outer conjugate actions on the hyperfinite II$_1$ factors for any non amenable group.

September 10, 2013 4:10 pm (Tuesday)

## Absorbing Subalgebras: Where to Find Them, and How to Use Them

Alexandr Kazda, Vanderbilt University
Location: Stevenson 1310

While studying the complexity of the Constraint Satisfaction Problem, Libor Barto and Marcin Kozik discovered the idea of absorption. If B is an absorbing subalegebra of the algebra A, then many kinds of connectivity properties of A are also true for B. This is very useful for proofs by induction, and absorption has since played a role in several other universal algebraic situations. After giving a taste of how absorption works, we would like to talk about our current project (which is a joint work with Libor Barto): How to decide, given an algebra A with finitely many basic operations and some B subalgebra of A, if B absorbs A.
September 11, 2013 4:10 pm (Wednesday)

Nathaniel Pappas, University of Virginia
Location: Stevenson 1310

The rank gradient and p-gradient are group invariants which assign some real number greater than or equal to -1 to a finitely generated group. Mark Lackenby first defined rank gradient and p-gradient as means to study 3-manifold groups. Rank gradient has connections with other group invariants from other fields of mathematics such as cost and L2 Betti numbers. The question of what values can be obtained as the rank gradient of some finitely generated group has remained open. I will discuss the related problem of determining what values can be achieved by the p-gradient as well as how to compute rank gradient and p-gradient of free products with amalgamation over an amenable subgroup and HNN extensions with amenable associated subgroup.

September 12, 2013 4:10 pm (Thursday)

## Non-Asymptotic Approach in Random Matrix Theory

Mark Rudelson, University of Michigan
Location: Stevenson 5211

Random matrix theory studies the asymptotics of the spectral distributions of families of random matrices, as the sizes of the matrices tend to infinity. Derivation of such asymptotics frequently requires analyzing the spectral properties of random matrices of a large fixed size, especially of their singular values. We will discuss several recent results in this area concerning matrices with independent entries, as well as random unitary and orthogonal perturbations of a fixed matrix. We will also show an application of the non-asymptotic random matrix theory to estimating the permanent of a deterministic matrix. Tea at 3:30 pm in SC 1425.

September 13, 2013 4:10 pm (Friday)

## Existence, Regularity and Convergence Results for the Compressible Euler Equations

Marcelo Disconzi, Vanderbilt University
Location: Stevenson 1307

We study the problem of inviscid slightly compressible fluids in a bounded domain. We find a unique solution to the initial-boundary value problem and show that it is close to the analogous solution for an incompressible fluid. Furthermore we find that solutions to the compressible motion problem in Lagrangian coordinates depend differentiably on their initial data, an unexpected result for non-linear equations. This is joint work with David G. Ebin.
September 17, 2013 4:10 pm (Tuesday)

## The Definable Principal Subcongruences Problem is Undecidable

Matthew Moore, Vanderbilt University
Location: Stevenson 1310

For each Turing machine T, we construct an algebra A'(T) such that the variety generated by A'(T) has definable principal subcongruences if and only if T halts, thus proving that the property of having definable principal subcongruences is undecidable. Using this, we present another proof that A. Tarski's finite basis problem is undecidable.

September 18, 2013 3:10 pm (Wednesday)

## Compressed Sensing with Equiangular Tight Frames

Matt Fickus, Air Force Institute of Technology
Location: Stevenson 1307

Compressed sensing (CS) is changing the way we think about measuring high-dimensional signals and images. In particular, CS promises to revolutionize hyperspectral imaging. Indeed, emerging camera prototypes are exploiting random masks in order to greatly reduce the exposure times needed to form hyperspectral images. Here, the randomness of the masks is due to the crucial role that random matrices play in CS. In short, in terms of CS's restricted isometry property (RIP), random matrices far outshine all known deterministic matrix constructions. To be clear, for most deterministic constructions, it is unknown whether this performance shortfall (known as the "square-root bottleneck") is simply a consequence of poor proof techniques or, more seriously, a flaw in the matrix design itself. In the remainder of this talk, we focus on this particular question in the special case of matrices formed from equiangular tight frames (ETFs). ETFs are overcomplete collections of unit vectors with minimal coherence, namely optimal packings of a given number of lines in a Euclidean space of a given dimension. We discuss the degree to which the recently-introduced Steiner and Kirkman ETFs satisfy the RIP. We further discuss how a popular family of ETFs, namely harmonic ETFs arising from McFarland difference sets, are particular examples of Kirkman ETFs. Overall, we find that many families of ETFs are shockingly bad when it comes to RIP, being provably incapable of exceeding the square-root bottleneck. Such ETFs are nevertheless useful in variety of other real-world applications, including waveform design for wireless communication and algebraic coding theory.

September 18, 2013 4:10 pm (Wednesday)

## Relatively Hyperbolic Groups with the Falsification by Fellow Traveler Property

Yago Antolin Pichel, Vanderbilt University
Location: Stevenson 1310

The falsification by fellow traveler property is a property of the Cayley graph of a group. It was introduced by Neumann and Shapiro, it has several implications. For example, it implies the regularity of the language of geodesics, the rationality of the growth series or having a quadratic Dehn function. I will explain how to find a generating set with the falsification by fellow traveler property for groups relatively hyperbolic to groups with a generating set with this property. This is a joint work with Laura Ciobanu.

September 20, 2013 3:10 pm (Friday)

## Optimal Regularity in the Thin Obstacle Problem with Lipschitz Variable Coefficients

Mariana Smit Vega Garcia, Purdue University
Location: Stevenson 1307

We will describe the lower-dimensional obstacle problem for a uniformly elliptic, divergence form operator $L = div(A(x)\nabla)$ with Lipschitz continuous coefficients and discuss the optimal regularity of the solution. Our main result states that, similarly to what happens when $L = \Delta$, the variational solution has the optimal interior regularity. We achieve this by proving some new monotonicity formulas for an appropriate generalization of Almgren's frequency functional. This is joint work with Nicola Garofalo.

September 20, 2013 4:10 pm (Friday)

## $W^*$-Superrigidity for Arbitrary Actions of Central Quotients of Braid Groups

Ionut Chifan, University of Iowa
Location: Stevenson 1432

For any $n > 4$ let $\tilde B_n = B_n/Z(B_n)$ be the quotient of the braid group $B_n$ through its center. We prove that any free ergodic probability measure preserving (pmp) action $\tilde B_n \ca (X, \mu)$ is virtually $W^*$-superrigid: whenever $L^\infty (X,\mu)\rtimes \tilde B_n =L^\infty (Y,\nu)\rtimes \Lambda$, for an arbitrary free ergodic pmp action $\Lambda \ca (Y, \nu)$ it follows that the actions $\tilde B_n \ca X,\Lambda \ca Y$ are virtually conjugate. Moreover, we prove that the same holds if $\tilde B_n$ is replaced with any finite index subgroup of the direct product $\tilde B_{n_1} \times \cdots \times \tilde B_{n_k}$, for some $n_1, \ldots , n_k > 4$. The proof uses a dichotomy theorem of Popa-Vaes for normalizers inside crossed products by free groups in combination with a $OE$-superrigidity theorem of Kida for actions of mapping class groups. This is based on joint work with A. Ioana and Y. Kida.

September 23, 2013 3:00 pm (Monday)

## Applications of Wenzl's Formula

Zhengwei Liu, Vanderbilt University
Location: Stevenson 1404

We will talk about Wenzl's formula and its general form. We will see its applications to other subfactor concepts, like Jellyfish, spoke subfactors, the embedding theorem, normalizers.

September 25, 2013 3:10 pm (Wednesday)

## Progress on Hard Thresholding Pursuit

Jean-Luc Bouchot, Drexel University
Location: Stevenson 1307

The Hard Thresholding Pursuit algorithm for sparse recovery is revisited using a new theoretical analysis. The main result states that all sparse vectors can be exactly recovered from incomplete linear measurements in a number of iterations at most proportional to the sparsity level as soon as the measurement matrix obeys a restricted isometry condition. The recovery is also robust to measurement error The same conclusions are derived for a variation of Hard Thresholding Pursuit, called Graded Hard Thresholding Pursuit, which is a natural companion to Orthogonal Matching Pursuit and runs without a prior estimation of the sparsity level. In two extreme cases of the vector shape, it is also shown that, with high probability on the draw of random measurements, a fixed sparse vector is robustly recovered in a number of iterations precisely equal to the sparsity level. These theoretical findings are experimentally validated, too.

September 26, 2013 4:10 pm (Thursday)

## A Survey of Algebraic Geometry and Model Theory for Free and Hyperbolic Groups

Olga Kharlampovich, Hunter College, CUNY
Location: Stevenson 5211

I will survey results of Kharlampovich--Miasnikov and Sela on first-order theories of free and hyperbolic groups. I will show that in the presence of negative curvature'' in groups, there exists a robust algebraic geometry and the principal Tarski-type problems are decidable. In particular, there is an algorithm for the elimination of quantifiers (to boolean combinations of AE-formulas). I will also give a description of definable sets in free and hyperbolic groups (joint result with Miasnikov). This solves Malcev's problem from 1965. Tea at 3:30 pm in SC 1425.

September 27, 2013 4:10 pm (Friday)

## Character Rigidity for Lattices and Commensurators

Darren Creutz, Vanderbilt University
Location: Stevenson 1432

Characters on groups (positive definite conjugation-invariant functions) arise naturally both from probability-preserving actions (the measure of the set of fixed points) and unitary representations on finite factors (the trace). I will present joint work with J. Peterson showing the nonexistence of nontrivial characters for irreducible lattices in semisimple groups and for their commensurators. Consequently, any finite factor representation of such a group generates either the left regular representation or a finite-dimensional representation, generalizing our earlier result that every nonatomic probability-preserving action of such groups is essentially free. The key new idea is to use the contractive nature of the Poisson boundary to bring it in operator algebraic setting and along with it the rigidity behavior of lattices in their ambient groups.

September 27, 2013 4:10 pm (Friday)

## Global Analysis of an Age-Structured Multi-Strain Virus Model

Cameron Browne, Vanderbilt University
Location: Stevenson 1307

We consider a general model of a within-host viral infection with multiple virus strains and explicit age-since-infection structure for infected cells.  Existence and uniqueness of solutions, along with asymptotic smoothness of the nonlinear semigroup generated by the family of solutions, are established.  For each viral strain, a quantity called the reproduction number is defined.  The main result is that the single-strain equilibrium corresponding to the virus strain with maximal reproduction number is a global attractor (provided that this maximal reproduction number is greater than unity and all reproduction numbers are distinct).  In other words, the virus strain with maximal reproduction number competitively excludes all other strains.  As an application of the model, HIV evolution is considered and simulations are conducted.

September 30, 2013 3:00 pm (Monday)

## Relationships Between Different Definitions of Conformal Field Theory

James Tener,
Location: Stevenson 1404

We will introduce three mathematical structures intended to capture the notion of a conformal field theory: conformal nets, vertex operator algebras, and Segal CFT. We will discuss the (philosophical) relationship between these different structures, as well as ongoing work to produce examples and theorems that implement these relationships.