This paper studies the polynomial optimization problem whose feasible set is a union of several basic closed semialgebraic sets. We propose a unified hierarchy of Moment-SOS relaxations to solve it globally. Under some assumptions, we prove the asymptotic or finite convergence of the unified hierarchy. Special properties for the univariate case are discussed. The numerical experiments demonstrate that solving this unified hierarchy takes less computational time than optimizing the objective over each individual constraining subset separately. The application for computing (p,q)-norms of matrices is also presented.

The notion of von Neumann dimension for a tracial von Neumann algebra $(M,\tau)$ has been used extensively throughout the theory, particularly in defining numerical invariants from Jones' index of a subfactor to Connes and Shlyakhtenko's $\ell^2$-Betti numbers of von Neumann algebras. The latter relies on work of Lück showing that Murray and von Neumann's original definition could be extended to purely algebraic $M$-modules, and more recently Petersen further extended von Neumann dimension to pairs $(M,\tau)$ where $\tau$ is a faithful normal semifinite tracial weight. In this talk, I will introduce a yet further extension of this theory to pairs $(M,\varphi)$ where $\varphi$ is a faithful normal strictly semifinite weight. Here strict semifiniteness means the restriction of $\varphi$ to the centralizer subalgebra $M^\varphi \subset M$ is still semifinite (note this condition is automatic for faithful normal states), and by work of Takesaki this is equivalent to the existence of a $\varphi$-invariant faithful normal conditional expectation $\mathcal{E}_\varphi\colon M \to M^\varphi$. Consequently, one can consider the Jones basic construction for the inclusion $M^\varphi \subset M$, and this is the key ingredient in our definition of von Neumann dimension for the pair $(M,\varphi)$. I will discuss properties of this dimension and how it can be used to recover the index for certain inclusions of factors. This is based on joint work with Aldo Garcia Guinto and Matthew Lorentz.

The celebrated result of Catlin on global regularity of the $\bar\partial$-Neumann operator for pseudoconvex domains of finite type links local algebraic- and analytic geometric invariants through potential theory with estimates for $\bar\partial$-equation. Yet despite their importance, there seems to be a major lack of understanding of Catlin's techniques, resulting in a notable absence of an alternative proof, exposition or simplification.

The goal of my talk will be to present an alternative proof based on a new notion of a ''tower multi-type''. The finiteness of the tower multi-type is an intrinsic geometric condition that is more general than the finiteness of the regular type, which in turn is more general than the finite type. Under that condition, we obtain a generalized stratification of the boundary into countably many level sets of the tower multi-type, each covered locally by strongly pseudoconvex submanifolds of the boundary. The existence of such stratification implies Catlin's celebrated potential-theoretic ''Property (P)'', which, in turn, is known to imply global regularity via compactness estimate. Notable applications of global regularity include Condition R by Bell and Ligocka and its applications to boundary smoothness of proper holomorphic maps generalizing a celebrated theorem by Fefferman.

Louis Solomon observed in the 1970s that, within the group algebra of the symmetric group, there is an interesting subalgebra spanned linearly by sums of permutations with the same sets of descents.  Later work of several authors showed that this contains a further subalgebra spanned by sums of permutations with the same numbers of descents, and that this has a connection with the topology of configuration spaces of n labeled distinct points in odd dimensional Euclidean spaces.

We review some of this story, as well as an analogous story that replaces descent sets with peak sets of permutations.  We then report on the connection to topology, which is new. Joint work with Marcelo Aguiar and Sarah Brauner.

I will discuss in detail some work of Joe Kohn involving subelliptic estimates. I hope to provide an understandable account of some of the technical matters from his 1979 Acta paper, and I will discuss some of my own work on points of finite type.  Although there is no new theorem to present, I will provide several new approaches to these ideas. To prevent the talk from being too technical, I will also include several elementary interludes that can be understood by graduate students.

he generalized Tate diagram developed by Greenlees and May is a fundamental tool in equivariant homotopy theory. In this talk, we will discuss an integration of the generalized Tate diagram with the equivariant slice filtration of Hill—Hopkins—Ravenel, resulting in a generalized Tate diagram for equivariant spectral sequences. This new diagram provides valuable insights into various equivariant spectral sequences and allows us to extract information about isomorphism regions between these equivariant filtrations.

As an application, we will begin by proving a stratification theorem for the negative cone of the slice spectral sequence. Building upon the work of Meier—Shi—Zeng, we will then utilize this stratification to establish shearing isomorphisms, explore transchromatic phenomena, and analyze vanishing lines within the negative cone of the slice spectral sequences associated with periodic Hill—Hopkins—Ravenel and Lubin—Tate theories.  This is joint work of Yutao Liu, XiaoLin Danny Shi and Guoqi Yan.

Eco-Evolutionary dynamics are at the core of carcinogenesis. Mathematical methods can be used to study ecological and evolutionary processes, and to shed light into cancer origins, progression, and mechanisms of treatment. I will present two broad approaches to cancer modeling that we have developed. One is concerned with near-equilibrium dynamics of stem cells, with the goal of figuring out how tissue cell turnover is orchestrated, and how control networks prevent “selfish” cell growth. The other direction is studying evolutionary dynamics in random environments.  The presence of temporal or spatial randomness significantly affects the competition dynamics in populations and gives rise to some counterintuitive observations. Applications to biomedical problems will be discussed.

Quantum Interior Point Methods (QIPMs) build on classic polynomial time IPMs. With the emergence of quantum computing we apply Quantum Linear System Algorithms (QLSAs) to Newton systems within IPMs to gain quantum speedup in solving Linear Optimization (LO) and Semidefinite Optimization (SDO) problems. Due to their inexact nature, QLSAs mandate the development of inexact variants of IPMs which, due to the inexact nature f their computations, by default are inexact infeasible methods. We propose “quantum inspired‘’ Inexact-Feasible IPMs (IF-IPM) for LO and SDO problems, using novel Newton systems to generate inexact but feasible steps. We show that IF-QIPMs enjoys the to-date best iteration complexity. Further, we explore how QLSAs can be used efficiently in iterative refinement schemes to find optimal solutions without excessive calls to QLSAs. Finally, we experiment with the proposed IF-IPM’s efficiency using IBMs QISKIT environment.

Speaker Bio:

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 mathematical optimization; nuclear reactor core reloading, oil refinery, VLSI design, radiation therapy treatment, and inmate assignment optimization; quantum computing.

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, and Vice President of INFORMS. 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. He will be a Plenary Speaker at ISMP’2024 in Montreal.

Originally, quantum ergodicity concerned equidistribution properties of Laplacian eigenfunctions with large eigenvalue on manifolds for which the geodesic flow is ergodic. More recently, several authors have investigated quantum ergodicity for sequences of spaces which "converge" to their common universal cover and when one restricts to eigenfunctions with eigenvalues in a fixed range. Previous authors have considered this type of quantum ergodicity in the settings of regular graphs, rank one symmetric spaces, and some higher rank symmetric spaces. We prove analogous results in the case when the underlying common universal cover is the Bruhat-Tits building associated to PGL(3, F) where F is a non-archimedean local field. This may be seen as both a higher rank analogue of the regular graphs setting as well as a non-archimedean analogue of the symmetric space setting.

Locally analytic representations were introduced by Peter Schneider and Jeremy Teitelbaum as a tool to understand $p$-adic Langlands program. From the very beginning the theory of $p$-valued groups played an instrumental role in the study of locally analytic representations. In a previous work we attached a rigid analytic group to a  $\textit{$p$-saturated group}$ (a class of $p$-valued groups that contains uniform pro-$p$ groups and pro-$p$ Iwahori subgroups as examples). In this talk I will outline how to elevate the rigid group to a $\textit{dagger group}$, a group object in the category of dagger spaces as introduced by Elmar Grosse-Klönne. I will further introduce the space of $\textit{overconvergent functions}$ and its strong dual the $\textit{overconvergent distribution algebra}$ on such a group. Finally I will show that in analogy to the locally analytic distribution algebra of compact $p$-adic groups, in the case of uniform pro-$p$ groups the overconvergent distribution algebra is a Fr´echet-Stein algebra, i.e., it is equipped with a desirable algebraic structure. This is joint work with Claus Sorensen and Matthias Strauch.

The Quantum Computing (QC) revolution is spreading fast and has the potential of disrupting all industries. It is widely expected that QC can revolutionize the way we perform and think about computation and optimization, and QC will be the backbone of thrilling new technologies and products. Governments and private investors are already investing billions of dollars annually to accelerate developments in QC technologies and to explore a myriad of potential applications. The focus of this presentation will be on the impact of Quantum Computing on optimization sciences, the potential of making optimized decisions faster and better, let it be engineering design, systems performance, supply chain, or finance. Specifically, in the mathematical optimization area, Quantum Computing has the potential to speed up problem solving tremendously and solve very large-scale problems that are not solvable to date. Just to mention, almost all results and claims about “Quantum Supremacy” are about solving optimization problems. Projected trends of QC hardware development with challenges ahead are discussed. Computing, algorithmic, and software stack developments, along with actual and potential applications of QC Optimization, and related areas will be discussed.

Speaker Bio:

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 mathematical optimization; nuclear reactor core reloading, oil refinery, VLSI design, radiation therapy treatment, and inmate assignment optimization; quantum computing.

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, and Vice President of INFORMS. 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. He will be a Plenary Speaker at ISMP’2024 in Montreal.