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Mathematics & Statistics

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Integrable discretizations and soliton solutions of KdV and mKdV equations

Yi Zhang

Zhejiang Normal University

P.R. China

4:00pm-5:00pm

PHY 120

Wen-Xiu Ma

**Abstract**

A method of discreting soliton equations is presented. The method is based on the bilinear formalism. From the bilinear forms of the KdV equation and the mKdV equation, we can obtain a few kinds of new bilinear forms through properly substituting the hyperbolic operator into the Hirota operator. Meanwhile we can get soliton solutions of these new equations by Hirota's method.

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The Zalcman conjecture. Proof for initial coefficients

Samuel L. Krushkal

Bar-Ilan University

Israel

3:00pm-4:00pm

PHY 120

Arcadii Grinshpan

**Abstract**

The famous Zalcman conjecture, which implies the Bieberbach conjecture, states that the coeficients of univalent functions \(f(z)=z+\sum\limits_2^\infty a_nz^n\) on the unit disk satisfy $$\left|a_n^2-a_{2n-1}\right|\le (n-1)^2\text{ for all }n>2,$$ with equality only for the Koebe function. This conjecture was raised more than 40 years ago, but it remained open for \(n>3\).

In the talk, we provide its proof for the initial coefficients with \(n=3,4,5,6\). Our approach is geometric and sheds light on the general situation. It relies on holomorphic homotopy of univalent functions and on comparison of generated singular conformal metrics of negative generalized curvature in the disk. The extremality of Koebe’s function follows from hyperbolic features.

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Recovery Algorithms in Compressive Sensing

Simon Foucart

Drexel University

3:00pm-4:00pm

PHY 120

Lesław Skrzypek

**Abstract**

This talk provides an overview of the field of Compressive Sensing, which aims at recovering sparse vectors from a seemingly incomplete set of linear measurements. We shall focus on the recovery process rather than the measurement process. First, we will describe the popular ‘\(1\)-minimization’ algorithm. In particular, the equivalence of the real and complex settings will be discussed. We will then review the Iterative Hard Thresholding and Compressive Sampling Matching Pursuit algorithms. Finally, we will introduce a new iterative algorithm called Hard Thresholding Pursuit, and we will highlight its advantages, namely simplicity, speed, and theoretical performances.

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The spectrum of Laplacian on Riemann manifolds and its relation to quantum theory

Zoltán I. Szabó

City University of New York

3:00pm-4:00pm

PHY 120

Vilmos Totik

**Abstract**

“*To what extent is the geometry of Riemann manifolds encoded in the spectrum of the Laplacian?*” has been one of the most intensely investigated problems in geometric analysis. This is just a reiteration of Kac's famous original question: “*Can one hear the shape of a drum?*” on a most general level. As it is well known today, the answer to this question is negative. Only very little information can be recovered from the Laplace-spectrum about the geometry of Riemann manifolds.

This talk reviews those constructions of the lecturer which provide isospectral manifolds having different local geometries. Among them the most interesting are those examples where the local isometries act transitively on one of the isospectral manifolds while they do not do so on the other one. Later it has turned out that these examples have a very strong relation to quantum theory, namely, the Laplace operator on the manifolds used in the constructions are identical with the Zeeman operators of elementary particle systems consisting both particles and their anti-particles. The isospectrality can be interpreted as the spectrum being indifferent for particle/anti-particle exchanges (which is the manifestation of the so called C-symmetry principle), while the local geometry drastically changes. For instance, if only similar particles (i.e., particles or anti-particles) are present, the local isometries act transitively on the attached Riemannian manifold, whereas they are never transitive for mixed systems.

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The Evolution of the Nonlinear Evolution Operator

Athanassios Kartsatos

3:00pm-4:00pm

PHY 120

**Abstract**

We all know Lagrange’s variation of constants formula for scalar first-order inhomogeneous linear differential equations. This formula gives rise to the first known evolution operator for linear problems. The same formula goes over to perturbed linear systems of differential equations and linear parabolic time-dependent problems in Banach spaces involving linear time-dependent densely defined operators. However, it is not widely known that there exist nonlinear evolution operators for purely nonlinear timedependent problems involving nonlinear accretive and \(m\)-accretive operators. We will go over such problems for ordinary as well as functional differential equations in Banach spaces. Connections will also be made between parabolic and elliptic problems involving nonlinear operators of monotone type.

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Geometry, algebra, and dynamics of the pentagram map

Sergei Tabachnikov

Penn State

4:00pm-5:00pm

PHY 120

Dmitry Khavinson

**Abstract**

Introduced by R. Schwartz almost 20 years ago, the pentagram map acts on plane \(n\)-gons, considered up to projective equivalence, by drawing the diagonals that connect second-nearest vertices and taking the new \(n\)-gon formed by their intersections. I shall survey recent work on the pentagram map, in particular, I shall demonstrate that the map is completely integrable. I shall also show that the pentagram map is a discretization of the Boussinesq equation, a well known completely integrable partial differential equation. An unexpected relation between the spaces of polygons and combinatorial objects called the \(2\)-frieze patterns (generalizing the frieze patterns of Coxeter) will be revealed. Eight new configuration theorems of projective geometry will be demonstrated. The talk will be illustrated by computer animation.

Based on joint work with S. Morier-Genoud, V. Ovsienko and R. Schwartz.

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Making Sense of Non-Hermitian Hamiltonians

Carl M. Bender

Physics Department

Washington University, St. Louis

3:00pm-4:00pm

PHY 120

Razvan Teodorescu

**Abstract**

The average quantum physicist on the street believes that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under combined matrix transposition and complex conjugation) in order to guarantee that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian \(H=p^{2}+ix^{3}\), which is obviously not Dirac Hermitian, has a real positive discrete spectrum and generates unitary time evolution, and thus it defines a fully consistent and physical quantum theory.

Evidently, the axiom of Dirac Hermiticity is too restrictive. While \(H=p^{2}+ix^{3}\) is not Dirac Hermitian, it is \(PT\)-symmetric; that is, invariant under combined space reflection \(P\) and time reversal \(T\). The quantum mechanics defined by a \(PT\)-symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics is extended into the complex domain, new kinds of theories having strange and remarkable properties emerge. Some of these properties have recently been verified in laboratory experiments. If one generalizes classical mechanics into the complex domain, the resulting theories have equally remarkable properties.

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Best Approximation and Smoothness

Zeev Ditzian

University of Alberta

CANADA

4:00pm-5:00pm

PHY 120

Vilmos Totik

**Abstract**

Recent and new results concerning relations between the rate of best approximation and the smoothness of the function approximated will be discussed. Relations between different levels of smoothness will also be treated. Different systems of orthogonal functions will be considered. Geometric properties of the underlying spaces and their in influence on the relations will be emphasized.

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Fox Colorings and Cocycle Invariants of Roll-Spun Knots

Shin Satoh

Kobe University

JAPAN

3:00pm-4:00pm

PHY 120

Masahiko Saito

**Abstract**

The Fox coloring number is one of elementary invariants for knotted surfaces in \(4\)-space as well as circles in \(3\)-space. We determine the coloring numbers for the family of knotted \(2\)-spheres called “roll-spun knots”. Furthermore, we prove that the cocycle invariant of any roll-spun knot associated with a Fox coloring is trivial.

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Braids and knots in mathematics, and a \(4\)-dimensional generalization

Masahide Iwakiri

Osaka City University

JAPAN

3:00pm-4:00pm

PHY 120

Masahiko Saito

**Abstract**

A knot is an embbeded circle in the \(3\)-dimensional space. Since it is complicated to observe all knots, and braids are represented in a systematic form, we often use braids to study knots. We can study all knots by Alexander's theorem: “any knot can be represented as a closure of a braid”.

We also study \(4\)-dimensional generalizations of knots and links, which are called surface-knots and surface braids. A surface-knot is an embbeded surface in the \(4\)-dimensional space. It is known that Alexander's theorem in this case also holds: “any surface-knot can be represented as a closure of a surface braid”. In the end of talk, we state recent results for the braid index of a surface-knot, which is one of a scale for complicatedness of surface-knots and related to surface braids.