Department of Mathematics

We prove that every 3-manifold possesses a $C^1$, volume-preserving flow with no fixed points and no closed trajectories. The main construction is a volume-preserving version of the Schweitzer plug. We also prove that every 3-manifold possesses a volume-preserving, $C^\infty$ flow with discrete closed trajectories and no fixed points (as well as a PL flow with the same geometry), which is needed for the first result. The proof uses a Dehn-twisted Wilson-type plug which also preserves volume.

We prove limit theorems for small-scale pair dispersion in velocity fields with power-law spatial spectra and wave-number dependent correlation times. This result establishes rigorously a family of generalized Richardson's laws with a limiting case corresponding to Richardson's $t^3$ and 4/3-laws.

The reviewers contacted by the editors to evaluate this work have been unable to confirm that the main results are correct. Flaws that were identified by the reviewers in earlier versions of the paper have been addressed by the author. Although it is possible that future research will uncover a significant mistake in this paper or show that the conclusions are in error, I believe that publishing it may benefit the readership of the Journal and stimulate further work in mathematical physics on an important topic.

Compressed sensing has a wide range of applications that include error correction, imaging, radar and many more. Given a sparse signal in a high dimensional space, one wishes to reconstruct that signal accurately and efficiently from a number of linear measurements much less than its actual dimension. Although in theory it is clear that this is possible, the difficulty lies in the construction of algorithms that perform the recovery efficiently, as well as determining which kind of linear measurements allow for the reconstruction. There have been two distinct major approaches to sparse recovery that each present different benefits and shortcomings. The first, L1-minimization methods such as Basis Pursuit, use a linear optimization problem to recover the signal. This method provides strong guarantees and stability, but relies on Linear Programming, whose methods do not yet have strong polynomially bounded runtimes. The second approach uses greedy methods that compute the support of the signal iteratively. These methods are usually much faster than Basis Pursuit, but until recently had not been able to provide the same guarantees. This gap between the two approaches was bridged when we developed and analyzed the greedy algorithm Regularized Orthogonal Matching Pursuit (ROMP). ROMP provides similar guarantees to Basis Pursuit as well as the speed of a greedy algorithm. Our more recent algorithm Compressive Sampling Matching Pursuit (CoSaMP) improves upon these guarantees, and is optimal in every important aspect.

We derive new perturbation bounds for eigenvalues of Hermitian matrices with block structures. The structures we consider range from a standard 2-by-2 block form to block tridiagonal and tridigaonal forms. The main idea is the observation that an eigenvalue is insensitive to componentwise perturbations if the corresponding eigenvector components are small. We show that the same idea can be used to explain two well-known phenomena, one concerning extremal eigenvalues of Wilkinson's matrices and another concerning the efficiency of aggressive early deflation applied to the symmetric tridiagonal QR algorithm.

An effective family of spectral curves appearing in Hitchin fibrations is determined. Using this family the moduli spaces of stable Higgs bundles on an algebraic curve are embedded into the Sato Grassmannian. We show that the Hitchin integrable system, the natural algebraically completely integrable Hamiltonian system defined on the Higgs moduli space, coincides with the KP equations. It is shown that the Serre duality on these moduli spaces corresponds to the formal adjoint of pseudo-differential operators acting on the Grassmannian. From this fact we then identify the Hitchin integrable system on the moduli space of Sp(2m)-Higgs bundles in terms of a reduction of the KP equations. We also show that the dual Abelian fibration (the SYZ mirror dual) to the Sp(2m)-Higgs moduli space is constructed by taking the symplectic quotient of a Lie algebra action on the moduli space of GL-Higgs bundles.

This paper focuses on determining the volumes of permutation polytopes associated to cyclic groups, dihedral groups, groups of automorphisms of tree graphs, and Frobenius groups. We do this through the use of triangulations and the calculation of Ehrhart polynomials. We also present results on the theta body hierarchy of various permutation polytopes.

We encode the binomials belonging to the toric ideal $I_A$ associated with an integral $d \times n$ matrix $A$ using a short sum of rational functions as introduced by Barvinok \cite{bar,newbar}. Under the assumption that $d,n$ are fixed, this representation allows us to compute the Graver basis and the reduced Gr\"obner basis of the ideal $I_A$, with respect to any term order, in time polynomial in the size of the input. We also derive a polynomial time algorithm for normal form computation which replaces in this new encoding the usual reductions typical of the division algorithm. We describe other applications, such as the computation of Hilbert series of normal semigroup rings, and we indicate further connections to integer programming and statistics.

A graphical expansion formula for non-commutative matrix integrals with values in a finite-dimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their non-orientable counterpart called Moebius graphs. The contribution of each graph is an invariant of the topological type of the surface on which the graph is drawn. As an example, we calculate the integral on the group algebra of a finite group. We show that the integral is a generating function of the number of homomorphisms from the fundamental group of an arbitrary closed surface into the finite group. The graphical expansion formula yields a new proof of the classical theorems of Frobenius, Schur and Mednykh on these numbers.

The strong Macdonald theorems state that, for $L$ reductive and $s$ an odd variable, the cohomology algebras $H^*(L[z]/z^N)$ and $H^*(L[z,s])$ are freely generated, and describe the cohomological, $s$-, and $z$-degrees of the generators. The resulting identity for the $z$-weighted Euler characteristic is equivalent to Macdonald's constant term identity for a finite root system. We calculate $H^*(\mathfrak{p} / z^N \mathfrak{p})$ and $H^*(\mathfrak{p}[s])$ for $\mathfrak{p}$ a standard parahoric in a twisted loop algebra, giving strong Macdonald theorems that take into account both a parabolic component and a possible diagram automorphism twist. In particular we show that $H^*(\mathfrak{p} / z^N \mathfrak{p})$ contains a parabolic subalgebra of the coinvariant algebra of the fixed-point subgroup of the Weyl group of $L$, and thus is no longer free. We also prove a strong Macdonald theorem for $H^*(\mathfrak{b}; S^* \mathfrak{n}^*)$ and $H^*(\mathfrak{b} / z^N \mathfrak{n})$ when $\mathfrak{b}$ and $\mathfrak{n}$ are Iwahori and nilpotent subalgebras respectively of a twisted loop algebra. For each strong Macdonald theorem proved, taking $z$-weighted Euler characteristics gives an identity equivalent to Macdonald's constant term identity for the corresponding affine root system. As part of the proof, we study the regular adjoint orbits for the adjoint action of the twisted arc group associated to $L$, proving an analogue of the Kostant slice theorem.

We study the set of infinite volume ground states of Kitaev's quantum double model on $\mathbb{Z}^2$ for an arbitrary finite abelian group $G$. It is known that these models have a unique frustration-free ground state. Here we drop the requirement of frustration freeness, and classify the full set of ground states. We show that the ground state space decomposes into $|G|^2$ different charged sectors, corresponding to the different types of abelian anyons (also known as superselection sectors). In particular, all pure ground states are equivalent to ground states that can be interpreted as describing a single excitation. Our proof proceeds by showing that each ground state can be obtained as the weak$^*$ limit of finite volume ground states of the quantum double model with suitable boundary terms. The boundary terms allow for states which represent a pair of excitations, with one excitation in the bulk and one pinned to the boundary, to be included in the ground state space.

We consider a problem of optimizing convex functionals over matroid bases. It is richly expressive and captures certain quadratic assignment and clustering problems. While generally NP-hard, we show it is polynomial time solvable when a suitable parameter is restricted.

The reviewers contacted by the editors to evaluate this work have been unable to confirm that the main results are correct. Flaws that were identified by the reviewers in earlier versions of the paper have been addressed by the author. Although it is possible that future research will uncover a significant mistake in this paper or show that the conclusions are in error, I believe that publishing it may benefit the readership of the Journal and stimulate further work in mathematical physics on an important topic.

Understanding principles of neurolocomotion requires the synthesis of neural activity, sensory feedback, and biomechanics. The nematode C. elegans is an ideal model organism for studying locomotion in an integrated neuromechanical setting because its neural circuit has a well-characterized modular structure and its undulatory forward swimming gait adapts to the surrounding fluid with a shorter wavelength in higher viscosity environments. This adaptive behavior emerges from the neural modules interacting through a combination of mechanical forces, neuronal coupling, and sensory feedback mechanisms. However, the relative contributions of these coupling modes to gait adaptation are not understood. Here, an integrated neuromechanical model of C. elegans forward locomotion is developed and analyzed. The model consists of repeated neuromechanical modules that are coupled through the mechanics of the body, short-range proprioception, and gap-junctions. The model captures the experimentally observed gait adaptation over a wide range of mechanical parameters, provided that the muscle response to input from the nervous system is faster than the body response to changes in internal and external forces. The modularity of the model allows the use of the theory of weakly coupled oscillators to identify the relative roles of body mechanics, gap-junctional coupling, and proprioceptive coupling in coordinating the undulatory gait. The analysis shows that the wavelength of body undulations is set by the relative strengths of these three coupling forms. In a low-viscosity fluid environment, the competition between gap-junctions and proprioception produces a long wavelength undulation, which is only achieved in the model with sufficiently strong gap-junctional coupling. The experimentally observed decrease in wavelength in response to increasing fluid viscosity is the result of an increase in the relative strength of mechanical coupling, which promotes a short wavelength.