Consider the vanishing viscosity limit for the 2D steady Navier-Stokes equations in the region $0\leq x \leq L$ and $0 \leq y<\infty$ with no slip boundary conditions at $y=0.$ For $L<<1,$ we justify the validity of the steady Prandtl layer expansion for scaled Prandtl layers, including the celebrated Blasius boundary layer. This is joint work with Yan Guo.

We are currently witnessing an explosion of intelligent systems. The year 2012 is often cited as the beginning of the deep learning revolution which transformed the artificial intelligence industry. That year, many notable achievements were accomplished using deep neural networks, large training datasets, and powerful GPUs. In 2015, machine learning finally beat humans in classifying images after attempting for many years in the annual ImageNet competition. In this talk, we explain how deep neural networks work and show the progression of intelligent systems from the year 2000 until the present. We will focus on convolutional neural networks and image recognition tasks.

In this talk, we will explore the problem of the largest singular value of a random sign matrix. We will use the method of $\epsilon$-nets to show that there exists a $C>0$ so that the largest singular value of a random sign matrix of size $n$ is at least $C \sqrt{n}$ with exponentially high probability. While this is a highly combinatorial problem, the method of $\epsilon$-nets could of interest to those in other fields. This talk is based off Tao's book on Random Matrix theory and a recent talk he gave at the 27th Annual PCMI Summer Session on Random Matrices.

To each graded poset one can associate two sequences of numbers: the Whitney numbers of the first kind and the Whitney numbers of the second kind. One sequence keeps track of the M$\ddot{\text{o}}$bius function at each rank level and the other keeps track of the number of elements at each rank level. We say if $P$ and $Q$ are Whitney duals if the Whitney numbers of the first kind of $P$ are the Whitney numbers of the second kind of $Q$ and vice-versa. In this talk, we will discuss a method to construct Whitney duals. This method uses a new type of edge labeling as well as quotient posets. For posets which have this type of labeling, one can construct a simplicity complex whose $f$-vector encodes the Whitney numbers of the second kind of this poset. Time permitting, we will discuss this complex. This is joint work with Rafael S. Gonz\'alez D'Le\'on.

Gaussian mixture model(GMM) is a fundamental tool in applied statistics and machine learning given data from a weighted sum of several Gaussian distributions. The current practice for learning mixture of Gaussians inevitably has high computational and sample complexity which is exponential in the number of Gaussian components. It has been shown in recent work that such estimation can be reduced to the problem of decomposing a symmetric tensor derived from the moments. The decomposition of these specially structured tensors can be solved efficiently by several methods.

Parabolic flows are smoothing for short time however, over long time, singularities are typically unavoidable and can be very nasty. The key to understand such flows is to understand their singularities and the set where those singularities occur. We begin with discussing mean curvature flow and will explain which singularities are generic and what one can say about the short and long time dynamics near singularities. After that we turn to the question of optimal regularity of geometric flows in general. We will see that these seemingly different questions turn out to be related. The ideas draws inspiration from an number of different fields, including Geometry, Analysis, Dynamical Systems and Real Algebraic Geometry.

The polynomial method is a recent trend in combinatorics which draws from methods of algebraic geometry over finite fields. Instances of the theory have been known for some time, and include Stepanov's method for counting points on curves over finite fields or Alon's combinatorial nullstellensatz. In this talk we will follow an expository article of Tao [1] to present basic ideas behind the polynomial method, as well as several applications. Following Tao, ``we will assume as little prior knowledge of algebraic geometry as possible.''