In this presentation, we will give some recent progress on the geometry of compact complex manifolds with RC-positive tangent bundles including confirming a long-standing open problem of Yau on rational connectedness and various rigidity theorems of harmonic maps and holomorphic maps.

For a complex Hilbert space H, the d-copy tensor product of H is denoted by H^⊗d. For a class of tensor products of operators on H^⊗d which are invariant under a subgroup of the permutation group of d element, we identify their reducing subspaces. These reducing subspaces are formally (or implicitly) known through Schur-Weyl duality in the group representation theory where finite dimensional vectors spaces and the invertible similarity are general used. We state these results in the operator theoretic framework which deals with infinite dimensional complex Hilbert spaces and uses the unitary similarity. We explicitly display some of these reducing subspaces. Most importantly we initiate the investigation of the question for which operator these reducing subspaces are minimal.

We study the asymptotic behavior of small eigenvalues of Riemann surfaces for large genus. We show that for any integer k>0, as the genus g goes to infinity, the smallest k-th eigenvalue of a Riemann surface in any thick part of moduli space of Riemann surfaces of genus g is uniformly comparable to 1/g^2 in g. This is a joint work with Yuhao Xue.

We first discuss some relations between positivity, geometric PDEs and Hodge-index type theorems. It is noted that using complex Hessian equations and the concavity inequalities for elementary symmetric polynomials inspires a generalized form of Hodge index inequality. Inspired by this result, we obtain a mixed Hodge-Riemann bilinear relation. The new feature is that this Hodge-Riemann bilinear relation holds with respect to classes in which some satisfy particular positivity condition, but could be degenerate along some directions. We also discuss its some connnections with the Hodge theory of algebraic maps.

The dual Brunn-Minkowski theory, initiated by Lutwak, provides powerful tools to solve the long-standing Busemann-Petty problem in the 1990's. Among those deep results, the dual Brunn-Minkowski and dual Minkowski inequalities are the most important. In this talk, I will discuss the newly introduced dual Orlicz-Brunn-Minkowski theory, a nontrivial extension of the dual Brunn-Minkowski theory. In particular, I will talk about the dual Orlicz-Minkowski and dual Orlicz-Brunn-Minkowski inequalities. These inequalities are based on the Orlicz-addition of star bodies, and are thought to

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.