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Farrell-Jones Conjecture plays an important role in manifold topology. It implies for example the Borel Conjecture and the Novikov Conjecture. In the first part of this talk I will give an overview of the conjecture with special emphasis on the K-theory version. I will first give an gentle introduction to algebraic K-theory and discuss the connections with manifold topology. We then formulate the conjecture and discuss the status of it. In the second part of the talk, I will discuss some methods that can be used to prove the conjecture. If time permits, I will also discuss some of my recent work.

Global warming is drastically altering rainfall patterns, increasing the risk of expensive floods and water scarcity.These contingencies are further accentuated by the rapid conversion of natural landscapes to urbanized areas that are continuallygrowing in population. Sustainable management strategies are becoming critical to ensure the optimum use of an essential but limited water resource on the planet. Often, river infrastructures such as dams, gates and reservoirs are built to satisfy human societies’ water and energy demand. However, the maintenance of costly engineered rivers’ infrastructures is still posing crucial problems related to reservoirs sedimentation and ecosystems preservation.

This presentation introduces the design and development of hand gesture recognition systems for robot control. Topics include an overview of hand gesture recognition technologies, key concepts and the training and testing processes with deep learning neural networks, neural network modeling with Tensorflow and Keras, a gesture recognition system with input from Leap Motion Controllers, a gesture recognition system with input from web cams, and demo of experiments. Control targets of the designed systems are robots and drones.

We study efficient quantum error correction schemes for the fully correlated channel on an n-qubit system with error operators that assume the form . In particular, when is odd, we have a quantum error correction scheme using one arbitrary qubit to protect the data state in the -qubit system. When is even, we have a hybrid quantum error correction scheme that protects a -qubit state and 2-classical bits. The scheme was implemented using Matlab, Mathematica and the IBM's quantum computing framework qiskit.

We discuss a novel multi-phase upscaling technique, which uses rigorous multiscale concepts based on Constraint Energy Minimization (CEM-GMsFEM). CEM-GMsFEM concepts use local spectral problems to design multiscale basis functions, which are supported in oversampled regions. The coarse-grid solution using these basis functions provides first-order accuracy with respect to the coarse-mesh size independent of high permeability contrast. The degrees of freedom in multiscale methods represent the coordinates of the solution in the multiscale space. To design an upscaled model, we modify these basis functions such that the degrees of freedom have physical meanings, in particular, the averages of the solution in each continua.

The dual Minkowski problem asks for the necessary and sufficient conditions on a given measure $/mu$ so that it can be realized as the dual curvature measure of a convex body. Just like the classical Minkowski problem, the dual Minkowski problem reduces to Monge-Ampere type equation. However, in this talk, we shall solve the dual Minkowski problem for $o$-symmetric convex bodies directly for measures using variation of calculus. The solution is closely connected to a measure concentration condition.

Kazhdan-Lusztig cells in the weighted Coxeter groups was first introduced systematically by Lusztig in 2003. He propose a bundle of conjectures on the cells in order to generalize some results from the equal parameter case to the unequal parameter case. In this talk, I intend to give a brief introduction on the topic and report some recent achievements concerning the description of cells.

We present some results concerning the solutions of $$ /Delta u +K(x) e^{2u}=0 /quad{/rm in}/;/; /mathbb{R}^2 $$ with $K/le 0$. We introduce the following quantity: $$/alpha_p(K)=/sup/left/{/alpha /in /R:/, /int_{/R^2} |K(x)|^p(1+|x|)^{2/alpha p+2(p-1)} dx<+/infty/right/}, /quad /forall/; p /ge 1.$$ Under the assumption $({/mathbb H}_1)$: $/alpha_p(K)> -/infty$ for some $p>1$ and $/alpha_1(K) > 0$, we show that for any $0 < /alpha < /alpha_1(K)$, there is a unique solution $u_/alpha$ with $u_/alpha(x) = /alpha /ln |x|+ c_/alpha+o/big(|x|^{-/frac{2/beta}{1+2/beta}} /big)$ at infinity and $/beta/in (0,/,/alpha_1(K)-/alpha)$.Furthermore, we show an example $K_0 /leq 0$ such that $/alpha_p(K_0) = -/infty$ for any $p>1$ and $/alpha_1(K_0) > 0$, for which we prove the existence of a solution $u_*$ such that $u_* -/alpha_*/ln|x| = O(1)$ at infinity for some $/alpha_* > 0$, but does not converge to a constant at infinity.