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Events for November 17, 2017 from General and Seminar calendars

Mathematical Physics and Harmonic Analysis Seminar

Time: 1:50PM - 2:50PM

Location: BLOC 220

Speaker: Fritz Gesztesy, Baylor University

Title: On factorizations of differential operators and Hardy-Rellich-type inequalities

Abstract: We will illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisely, using this factorization method, we will derive a general (and, apparently, new) inequality and demonstrate how particular choices of the parameters contained in this inequality yield well-known inequalities, such as the classical Hardy and Rellich inequalities as special cases. Actually, other special cases yield additional and apparently less well-known inequalities.

We will indicate that our method, in addition to being elementary, is quite flexible when it comes to a variety of generalized situations involving the inclusion of remainder terms and higher-order operators.

This is based on various joint work with Lance Littlejohn, I. Michael, M. Pang, and R. Wellman.

Algebra and Combinatorics Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 117

Speaker: X. Shawn Cui, Stanford University

Title: Topological quantum computation and compilation

Abstract: Topological quantum computation is a fault tolerant protocol for quantum computing using non-abelian topological phases of matter. Information is encoded in states of multi-quasiparticle excitations(anyons), and quantum gates are realized by braiding of anyons. The mathematical foundation of anyon systems is described by unitary modular tensor categories. We will show one can encode a qutrit in four anyons in the SU(2)_4 anyon system, and universal qutrit computation is achieved by braiding of anyons and one projective measurement which checks whether the total charge of two anyons is trivial. We will also give an algorithm to approximate an arbitrary quantum gate with the ones from the anyon system. The algorithm produces more efficient circuits than the Solovay-Kitaev algorithm. Time allowed, applications in quantum complexity classes will also be addressed.

Mathematical Physics and Harmonic Analysis Seminar

Time: 3:00PM - 3:50PM

Location: BLOC 220

Speaker: Maxim Zinchenko, University of New Mexico

Title: Chebyshev Polynomials on Subsets of the Real Line

Abstract: Chebyshev polynomials are the unique monic polynomials that minimize the sup-norm on a given compact set. These polynomials have important applications in approximation theory and numerical analysis. H. Widom in his 1969 influential work initiated a study of Szego-type asymptotics of Chebyshev polynomials on compact sets given by finite unions of disjoint arcs in the complex plane. He obtained several partial results on the norm and pointwise asymptotics of the polynomials and made several long lasting conjectures. In this talk I will present some of the classical results on Chebyshev polynomials as well as recent progress on Widom's conjecture on the large n asymptotics of Chebyshev polynomials on finite and infinite gap subsets of the real line.

Based on Asymptotics of Chebyshev Polynomials, I. Subsets of R with J. Christiansen and B. Simon. Invent. Math. 208 (2017), 217-245, and Asymptotics of Chebyshev Polynomials, II. DCT Subsets of R with J. Christiansen, B. Simon, and P. Yuditskii (preprint arXiv:1709.06707).

Geometry Seminar

Time: 4:00PM - 5:00PM

Location: BLOC 628

Speaker: Taylor Brysiewicz, TAMU

Title: Counting polynomially parametrized interpolants via Necklaces

Abstract: We consider the problem of locally approximating an analytic curve in the complex plane plane by a polynomial parametrization t -> (x_1(t),x_2(t)) of bidegree (d_1,d_2). Contrary to Taylor approximations, these parametrizations can achieve a higher order of contact at the cost of losing uniqueness and possibly the reality of the solution. We study the extent to which uniqueness fails by counting the number of such curves as the number of aperiodic combinatorial necklaces on d_1 white beads and d_2 black beads. We analyze when this count is odd as an initial step in studying when real solutions exist.

Linear Analysis Seminar

Time: 4:00PM - 5:00PM

Location: BLOC 220

Speaker: Anna Skripka, University of New Mexico

Title: Schur multipliers in perturbation theory.

Abstract: Schur multipliers and their generalizations have been actively studied for over a century. Classical linear Schur multipliers act on matrices as entrywise multiplications; multilinear Schur multipliers act by some intricate products. After recalling finite dimensional Schur multipliers, we will concentrate on their generalizations to multilinear transformations arising in infinite dimensional perturbation theory and consider an application to approximation of operator functions. In particular, we will discuss sharp conditions for Schatten class membership of remainders of approximations [1]. The affirmative case relies on the approach to Schur multipliers without separation of variables emerging from [3] and addressed in the nonself-adjoint case in [2]. [1] "Multilinear Schur multipliers and applications to operator Taylor remainders", with D. Potapov, F. Sukochev, and A. Tomskova, Adv. Math., 320 (2017), 1063-1098. [2] "Estimates and trace formulas for unitary and resolvent comparable perturbations", Adv. Math., 311 (2017), 481-509. [3] "Spectral shift function of higher order", with D. Potapov and F. Sukochev, Invent. Math., 193 (2013), no. 3, 501-538.