Logarithmic Sobolev inequalities, first introduced by Gross in 70s, have rich connections to probability, geometry, as well as information theory. In recent years, logarithmic Sobolev inequalities for quantum Markov semigroups attracted a lot of attentions for its applications in quantum information theory and quantum many-body systems. In this talk, I'll present a simple, information-theoretic approach to modified logarithmic Sobolev inequalities for both quantum Markov semigroup on matrices, and classical Markov semigroup on matrix-valued functions. In the classical setting, our results implies every sub-Laplacian of a Hörmander system admits a uniform  modified logarithmic Sobolev constant for all its matrix valued functions. For quantum Markov semigroups, we improve a previous result of Gao and Rouzé by replacing the dimension constant by its logarithm. This talk is based on a joint work with Marius Junge, Nicholas, LaRacunte, and Haojian Li.

A fundamental question in classical stable homotopy theory is to understand the stable homotopy groups of the spheres. A relatively new method is via the motivic approach. Motivic stable homotopy theory has an algebro-geometric root and closely connects to questions in number theory. Besides, it relates to the classical and the equivariant theories. The motivic category has good properties and allows different computational tools. I will talk about some of these properties and computations, and will show how it relates to the classical and equivariant categories.

We apply Quantum Linear System Algorithms (QLSAs) to Newton systems within IPMs to gain quantum speedup in solving LO problems. Due to their inexact nature, QLSAs can be applied only to inexact variants of IPMs, which are inexact infeasible methods due to the inexact nature f their computations. We propose Inexact-Feasible IPMs (IF-IPM) for LO and SDO problems, using novel Newton systems to generate inexact but feasible steps. We show that this method enjoys the to-date best iteration complexity. Further, we explore how QLSAs can be used efficiently in iterative refinement schemes to find an exact optimal solution without excessive calls to QLSAs. Finally, we experiment with the proposed IF-IPM’s efficiency using IBMs QISKIT environment.

Bio of the Speaker:

Dr. Terlaky is a George N. and Soteria Kledaras ’87 Endowed Chair Professor Department of Industrial and Systems Engineering, Lehigh University, and Director of the Quantum Computing Optimization Laboratory.

Dr. Terlaky has published four books, edited over ten books and journal special issues and published over 200 research papers. Topics include theoretical and algorithmic foundations of operations research, computational optimization, nuclear reactor core reloading optimization, oil refinery and VLSI design optimization, robust radiation therapy treatment optimization, inmate assignment optimization, quantum computing.

His research interest includes high performance optimization methods, optimization modeling, optimization problems in engineering sciences and service systems, and quantum computing optimization.

Dr. Terlaky is Editor-in-Chief of the Journal of Optimization Theory and Applications. He has served as associate editor of ten journals and has served as conference chair, conference organizer, and distinguished invited speaker at conferences all over the world. He was general Chair of the INFORMS 2015 Annual Meeting, a former Chair of INFORMS’ Optimization Society, Chair of the ICCOPT Steering Committee of the Mathematical Optimization Society, Chair of the SIAM AG Optimization. He received the MITACS Mentorship Award; Award of Merit of the Canadian Operational Society, Egerváry Award of the Hungarian Operations Research Society, H.G. Wagner Prize of INFOMRS, Outstanding Innovation in Service Science Engineering Award of IISE. He is Fellow of INFORMS, SIAM, IFORS, The Fields Institute, and elected Fellow of the Canadian Academy of Engineering. Currently he is serving as Vice President of INFORMS.

 In 2020, more than a hundred years after Einstein's publication of his theory of gravitation, half of the Nobel prize in Physics was awarded to Sir Roger Penrose "for the discovery that black hole formation is a robust prediction of the general theory of relativity". In this talk, I will present the basic mathematical formalism of general relativity, black holes, and their connections with modern analysis.

Stationary symmetric $\alpha$-stable ($S \alpha S$) processes is an important class of stochastic processes, including Gaussian processes, Cauchy processes and Lévy processes. In an analogy to that the ergodicity of a Gaussian process is determined by its spectral measure, it was shown by Rosinski and Samorodnitsky that the ergodicity of a stationary $S \alpha S$ process is characterized by its spectral representation. While this result is known when the process is indexed by $\mathbb{Z}$ or $\mathbb{R}$, the classical techniques fail when it comes to processes indexed by non-Abelian groups and it was an open question whether the ergodicity of stationary $S \alpha S$ processes indexed by amenable groups admits a similar characterization.

In this talk I will introduce the fundamentals of stable processes, the ergodic theory behind their spectral representation, and the key ideas of the characterization of the ergodicity for processes indexed by amenable groups. If time permits, I will explain how to use a recent construction of A. Danilenko in order to prove the existence of weakly-mixing but not strongly-mixing stable processes indexed by many groups (Abelian groups, Heisenberg group).

Let $E/\mathbb{Q}$ be an imaginary quadratic extension and
suppose $G$ is the unitary group attached to hermitian space over $E$ of
signature $(2,n)$. The symmetric domain $X$ attached to $G$ is a
quaternionic Kahler manifold in the sense of differential geometry. In
the late nineties N. Wallach studied harmonic analysis on $X$ in the
context of this quaternionic structure. He established a multiplicity
one theorem for spaces of generalized Whittaker periods appearing in the
cohomology of certain quaternionic $G$-bundles on $X$.

We prove new cases of Wallach's multiplicity one statement for some
degenerate generalized Whittaker periods and give explicit formulas for
these periods in terms of modified K-Bessel functions. Our results can
be interpreted as giving a refined Fourier expansion for automorphic
forms on $G$ which are quaternionic discrete series at infinity. As an
application we study the cusp forms on $G$ which arise as theta lifts of
holomorphic modular forms on quasi-split $\mathrm{U}(1,1)$. We show that
these cusp forms can be normalized so that all their Fourier
coefficients are algebraic numbers. (joint with Anton Hilado and Pan Yan)

We will discuss Einstein manifolds which are invariant under an isometric Lie group action. Our main goal is to explain the proof of the 1975 Alekseevskii Conjecture on non-compact homogeneous Einstein spaces, recently obtained in collaboration with Christoph Böhm (Münster). To that end, we will also present new structure results for Einstein metrics on principal bundles. The talk will conclude with open questions and future directions.

Differential item functioning (DIF) refers to the situation where responses to a given question on an exam (or survey or similar) differ between several groups. For several decades now, social scientists and education researchers have employed a standard battery of statistical tools to detect DIF from sample data, but essentially all of these standard tools rely on theoretical asymptotic results and presuppose sample sizes that are rarely achieved by real data sets. In this talk, we'll discuss how ideas dating back to Diaconis and Sturmfels, in which techniques from computational algebra are brought to bear in statistics, provide an alternative method to detect DIF which avoids asymptotics and is more robust with smaller sample sizes. This is joint work with Luis David Garcia-Puente, Minho Kim, and Flavia Sancier-Barbosa.

In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative within the "forced" category, by exhibiting a one-parameter family of distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo.