Quantum cellular automata (QCA) are models of discrete-time unitary dynamics of quantum spin systems. They can be characterized algebraically as certain automorphisms of the associative algebra generated by local observables of a spin system. We will give a gentle introduction to this topic, and explain some of our recent contributions to the problem of classification…
The talk is about oscillatory integral operators on manifolds. Manifolds of constant sectional curvatures are particularly interesting, and we will see that very good estimates on these manifolds can be expected. We will also discuss Kakeya and Nikodym problems on general manifolds, in particular, manifolds satisfying Sogge’s chaotic curvatures.
In this interactive seminar, we will examine how data (qualitative and quantitative) can be leveraged to interrogate, disrupt, and enact changes in introductory math programs. In particular, I will share insights from the ACT UP Math project, which is studying the role and impact of research-practice partnerships between mathematics education experts and mathematics department faculty…
Khovanov homology is a link invariant which categorifies Jones polynomial. In this talk we present several results concerning Khovanov homology of fibered positive links; in particular, we extend the result by Stosic stating that braid positive links have vanishing Khovanov homology in homological grading 1. We also explore Khovanov homology of certain cable links and…
The formulation of Bayesian inverse problems involves choosing prior distributions; choices that seem equally reasonable may lead to significantly different conclusions. We develop a computational approach to better understand the impact of the hyperparameters defining the prior on the posterior statistics of the quantities of interest. Our approach relies on global sensitivity analysis (GSA) of…
Permutations $w$ in $S_n$ for which the (type-A) Schubert variety $\Omega_w$ is smooth are characterized by avoidance of the patterns 3412 and 4231. The smaller family of codominant permutations, those avoiding the pattern 312, seems to explain a lot about character evaluations at Kazhdan-Lusztig basis elements $C'_w(q)$ of the (type-A) Hecke algebra. In particular, for…
In this talk, I will talk about some recent well-posedness and stability results for several fluid models in 2D. More precisely, I will discuss the global well-posedness for the 2D Boussinesq equations with fractional dissipation. For the Oldroyd-B model, we show that small smooth data lead to global and stable solutions. When Navier-Stokes is coupled…
In volume-constrained capillarity problems, minimizers may have free boundaries adhering to the container. Recent work in the study of capillarity problems has utilized stability theory for the volume-constrained isoperimetric problem to classify the shape of global minimizers and (in the case without free boundary) critical points. In this talk, I will discuss joint work with…
In this talk, we discuss second-order optimality conditions for optimal control problems. This analysis is very important when we study the stability of the solution to the control problem with respect to small perturbations of the data. It is also crucial for proving superlinear or quadratic convergence of numerical algorithms for solving the problem, as…
Over the past three decades, a fruitful approach for analysis and data-driven modeling of dynamical systems has been to consider the action of (nonlinear) dynamics in state space on linear spaces of observables. These methods leverage the linearity of the associated evolution operators, namely the Koopman and transfer operators, to carry out tasks such as…
Zoom Meeting: https://ncsu.zoom.us/j/93046132033?pwd=dkZiTjlKazgzK2Q3aXJra1g2R1Q0dz09 Meeting ID: 930 4613 2033 Passcode: 075251
In several shape optimization problems one has to deal with cost functionals of the form ${\cal F}(\Omega)=F(\Omega)+kG(\Omega)$, where $F$ and $G$ are two shape functionals with a different monotonicity behavior and $\Omega$ varies in the class of domains with prescribed measure. In particular, the cost functional ${\cal F}(\Omega)$ is not monotone with respect to $\Omega$…
Linear complementarity problems (LCP) generalizes some fundamental problems of mathematical optimization like linear programming (LP) problem, linearly constrained quadratic programming (LQP) problem and some others. It admits an enormous number of applications in economics, engineering, science, and many other fields. After all these, it is not surprising that LCPs are usually NP-complete problems (S.J. Chung,…
Inverse problems are an expanding field with many practical applications in scientific computing and engineering. Their Bayesian enhancement encodes prior knowledge and data uncertainties into a posterior. This is an important tool in uncertainty quantification. However, performing uncertainty quantification tasks on top of this posterior is difficult to formulate and often computationally intractable. Hence, for…
Simulating the time evolution of a Hamiltonian system on a classical computer is hard—the computational power required to even describe a quantum system scales exponentially with the number of its constituents, let alone integrating its equations of motion. Hamiltonian simulation on a quantum machine is a possible solution to this challenge. Assuming that a quantum…
Recently, single-cell transcriptomics has provided a powerful approach to investigate cellular properties in unprecedented resolution. However, given a small number of temporal snapshots of single-cell transcriptomics, how to connect them to obtain their collective dynamical information remains an unexplored area. One major challenge to connecting temporal snapshots is that cells measured at one temporal point…