# Faculty Activities

*Abstract:*

We extend Gour et al's characterization of quantum majorization via conditional min-entropy to the context of semifinite von Neumann algebras. Our method relies on a connection between conditional min-entropy and operator space projective tensor norm for injective von Neumann algebras. This approach also connects the tracial Hahn-Banach theorem of Helton, Klep and McCullough to noncommutative vector-valued L1-space.

This will be the first in a series of 2 talks.

*Abstract:*

We extend Gour et al's characterization of quantum majorization via conditional min-entropy to the context of semifinite von Neumann algebras. Our method relies on a connection between conditional min-entropy and operator space projective tensor norm for injective von Neumann algebras. This approach also connects the tracial Hahn-Banach theorem of Helton, Klep and McCullough to noncommutative vector-valued L1-space.

This will be the second in a series of 2 talks.

*Abstract:*

In this talk, the speaker aims to present his current investigation of the ``Moebius-invariant Willmore flow'', which is a conformally invariant modification of the classical Willmore-flow, i.e. of the gradient flow of the Willmore functional. Since the Willmore functional - applied to immersions f of some fixed closed surface into Euclidean space - can be defined as the integral over the squared modulus of the trace-free part of the second fundamental form of f plus 2 pi times the Euler-characteristic of the underlying surface, it can be geometrically interpreted as a global conformal invariant which measures the deviation of immersions from being totally umbilic. In 2001--2004, Kuwert and Schaetzle proved in 3 consecutive articles that the classical Willmore flow moving immersions of some fixed 2-sphere into Euclidean space exists globally and actually converges smoothly to a round sphere, if the initial Willmore energy is smaller than 8 pi. Only this year, the speaker invented a new ``descent-technique'' in order to prove that the Moebius-invariant Willmore flow moving immersions of a fixed torus into the 3-sphere exists globally if the start immersion maps this torus onto a Hopf-torus in the 3-sphere, and that it actually converges smoothly to some conformal image of the Clifford-torus in the 3-sphere, if the initial Willmore energy is smaller than a certain number between 8 pi and 9 pi. In order to precisely compute this concrete threshold, the speaker reduced the Euler-Lagrange equation of the Willmore functional for Hopf-tori to a non-linear ODE of first order which can be integrated in terms of elliptic functions - an adaption of Einstein's computation of the annual shift of the perihelion of a relativistic planetary orbit - and then continued to apply classical methods and formulas of Arithmetic Geometry and Analytic Number Theory.

*Abstract:*

The theory of Markov chains has applications in diverse areas of research such as group theory, dynamical systems, electrical networks and information theory. These days, connections with operator algebras seem to manifest mostly in quantum information theory, where Markov chains are generalized to quantum channels.

In this talk we will show how questions about operator algebras constructed from stochastic matrices, studied by Markiewicz and myself, still motivate new problems in classical Markov chain theory. More precisely, we characterize coincidence of conditional probabilities in terms of generalized Doob transforms, which then leads to stronger classification results for the associated operator algebras. This turns out to be intimately related to determining positive harmonic functions for the stochastic matrix. Time permitting,I will explain how non-commutative peak points of the associated operator algebra can be completely characterized in terms of the stochastic matrix.

*This is based on joint work with Xinxin Chen, Langwen Hui, Christopher Linden and Yifan Zhang, conducted as part of an undergraduate research project in Illinois Geometry Lab at UIUC.

*Abstract:*

An aperiodic tiling of R^d has a hull, which typically is a compact foliated space X. The hull comes equipped with an invariant transverse measure, corresponding to a Z^d-invariant measure on a totally disconnected transversal N. Mathematical physicists attach two types of invariants to this situation. On the one hand, the invariant transverse measure gives rise to a real-valued map on H^d(X; R) (Cech cohomology) which corresponds to the diffraction pattern aspect of tilings. On the other hand. it also gives rise to a real-valued map on K_0(C^*G(X)), the topological K-theory of the C^*-algebra of the holonomy groupoid of the tiling, which corresponds to the spectral (of Hamiltonians defined on such tilings) aspect of tilings. We use the Index Theorem for foliated spaces to prove that under some general assumptions that these points of view are equivalent.

This is joint work with Eric Akkermans (physics, the Technion) and Jonathan Rosenberg (math, U. Maryland).

*Abstract:*

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*Abstract:*

Many problems in statistical learning, imaging and computer vision involve the optimization of a non-convex objective function with singularities at the boundary of the feasible set. For such challenging instances, we develop a new interior-point technique building on the Hessian-barrier algorithm introduced by Bomze, Mertikopoulos, Schachinger and Staudigl [SIAM J. Opt. 29(3) (2019), 2100-2127], where the Riemannian metric is induced by a generalized self-concordant function. This class of functions is sufficiently general to include most of the commonly used barrier functions in the literature of interior point methods. We prove global convergence to an approximate stationary point of the method, and in cases where the feasible set admits an easily computable self-concordant barrier, we verify optimal iteration complexity of the method. Applications in non-convex statistical estimation and $L_{p}$-minimization are discussed to illustrate the efficiency of the method. This is joint work with Pavel Dvurechensky (WIAS) and Cesar A. Uribe (MIT).