Topological data analysis (TDA) emerged as a branch of applied topology about twenty years ago and has produced some of the most valuable tools for the study of Big Data since then. TDA can be used to detect topologically significant features (holes, connected components, etc) of high-dimensional data sets, without any a priori knowledge of the data. It’s even been used to analyze basketball games, and found that basketball players displayed thirteen statistically distinct playing styles, despite there being only five official positions.

In this talk I will introduce the basics of persistent homology, a fundamental tool for TDA, and describe some of the recent achievements in this field. I'll also touch on some current areas of active research and why people like NASA are trying to get in on the action.

While the long-time behavior of small amplitude solutions to nonlinear dispersive and wave equations on Euclidean spaces $(R^n)$ is relatively well-understood, the situation is marked different on bounded domains. Due to the absence of dispersive decay, a very rich set of dynamics can be witnessed even starting from arbitrary small initial data. In particular, the dynamics in this setting is characterized by out-of-equilibrium behavior, in the sense that solutions typically do not exhibit long-time stability near equilibrium configurations. This is even true for equations posed on very large domains (e.g. water waves on the ocean) where the equation exhibits very different behaviors at various time-scales.

In this talk, we shall consider the nonlinear Schr\"odinger equation posed on a large box of size $L$. We will analyze the various dynamics exhibited by this equation when $L$ is very large

Initial data in general relativity must satisfy certain underdetermined differential equations called the constraint equations. A natural problem is to find a parameterization of all possible initial data. A standard method for this is called the conformal method. In the relatively simple ``constant mean curvature" (CMC) case, this method provides a good parameterization of initial data. However, the far-from-CMC case has resisted analysis. In part this is because researchers were trying to prove theorems that are false. In this talk, I'll introduce the problem and known results, and talk about our numerical results that show that the standard conjectures about solvability were all wrong. Numerical investigations can play an important part in informing conjectures about purely analytical questions.

Research mathematics is about asking questions. When's the last time you did? Bring your cell phone or computer, and be prepared to change the way you approach everything in mathematics, and life.

Mori introduced a program for classifying projective varieties by using algebraic surgeries, Jian Song and Gang Tian
introduced Analytical Minimal Model Program for the classification of Kahler varieties by using PDE surgeries. For the intermediate
Kodaira dimension they proved that there exists a unique generalized Kahler-Einstein metric which twisted with Weil-Petersson metric.
I extended their result in my PhD thesis on pair $(X,D)$ where $D$ is a snc divisor with conic singularities and I showed that there exists a generalized Kahler-Einstein metric which twisted with logarithmic Weil-Petersson metric plus additional term which we can find such additional term by using higher canonical bundle formula of Fujino and Mori. Moreover I extended Song-Tian program for Sasakian varieties in my PhD thesis. In fact when the basic first Chern class of a
Sasakian variety is not definite then the question is how can we find generalized Kahler-Einstein metric for such varieties. I gave a positive answer to this question in my thesis. Moreover I will explain how the Lei Ni method which later improved by V.Tosatti for the classification of the solution of Kahler-Ricci flow could be extended to conical Kahler-Ricci flow and I finally will explain how the classification of the solutions of relative Kahler-Ricci flow is related to the Gromov invariant of Ruan-Tian.

In a series of papers, E. Hecke described the representation of the group
$SL(2,p)$ on the regular differentials of the modular curve $X$ of level $p$. This was one of the
first applications of character theory outside of finite group theory, and one of the first
constructions of representations using cohomology. I will review Hecke's results, and
interpret them in the modern language of automorphic representations.

We prove that cohomology classes in $H^{2,2}(S_1\times S_2)$
of Hodge isometries
$$\psi \colon H^2(S_1,\mathbb Q)\rightarrow H^2(S_2,\mathbb Q)$$ between
any two
projective complex $K3$ surfaces $S_1$ and $S_2$
are polynomials in Chern classes of coherent analytic sheaves.

Consequently, the cohomology class of $\psi$ is algebraic
This proves a conjecture of Shafarevich announced at ICM in 1970.