Consider a Markov chain on a one-dimensional torus $\mathbb T$, where a moving point jumps from $x$ to $x\pm \alpha$ with probabilities $p(x)$ and $1-p(x)$, respectively, for some fixed function $p\in C^{\infty}(\mathbb T, (0,1))$ and $\alpha\in\mathbb R\setminus \mathbb Q$. Such Markov chains are called random walks in a quasi-periodic environment. It was shown by Ya. Sinai that for Diophantine $\alpha$ the corresponding random walk has an absolutely continuous invariant measure, and the distribution of any point after $n$ steps converges to this measure. Moreover, the Central Limit Theorem with linear drift and variance holds.

In contrast to these results, we show that random walks with a Liouvillian frequency $\alpha$ generically exhibit an erratic statistical behavior. In particular, for a generic $p$, the corresponding random walk does not have an absolutely continuous invariant measure, both drift and variance exhibit wild oscillations (being logarithmic at some times and almost linear at other times), Central Limit Theorem does not hold.

These results are obtained in a joint work with Dmitry Dolgopyat and Bassam Fayad.

Data are prevalent in all aspects of society today. Data-driven decision making is important in education, business, politics, psychology, and many more disciplines. In this talk, I will discuss three different data types and research that surrounds them. How can data be visualized affectively? What are statistical techniques for analyzing different data types? How do the analysis techniques vary depending on the research questions one is interested in answering? Through this, I will introduce my different areas of research and also discuss my pathway as an academic.
 

A reaction network is a graphical configuration that describes an interaction between species (molecules). If the abundances of the network system is small, then the randomness inherent in the molecular
interactions is important to the system dynamics, and the abundances are modeled stochastically as a jump by jump fashion continuous-time Markov chain. One of challenging issues facing researchers who study biological

systems is the often extraordinarily complicated structure of their interaction networks. Thus, how to characterize network structures that induce characteristic behaviors of the system dynamics is one of the major open questions in this literature.

In this talk, I will provide an analytic approach to find a class of reaction networks whose associated Markov process has a stationary distribution. Moreover I will talk about the convergence rate for the process to its stationary distribution with the mixing time.

This talk concerns a twenty-thousand-year old mistake: The natural numbers record only the result of counting and not the process of counting. As algebra is rooted in the natural numbers, the higher algebra of Joyal and Lurie is rooted in a more basic notion of number which also records the process of counting. Long advocated by Waldhausen, the arithmetic of these more basic numbers should eliminate denominators. Notable manifestations of this vision include the Bökstedt-Hsiang-Madsen topological cyclic homology, which receives a denominator-free Chern character, and the related Bhatt-Morrow-Scholze integral p-adic Hodge theory, which makes it possible to exploit torsion cohomology classes in arithmetic geometry. Moreover, for schemes smooth and proper over a finite field, the analogue of de Rham cohomology in this setting naturally gives rise to a cohomological interpretation of the Hasse-Weil zeta function by regularized determinants, as envisioned by Deninger.

This is the seventh in a series of lectures on naive descriptive set theory based on an expository paper by Matt Foreman. We will continue discussing the Borel hierarchy.