The importance of price impact in HFT is outlined via well known phenomena of micro-price. We look at price impact as macro phenomena and for its explanation turn to micro phenomena, specifically to order book dynamics modeled via multi-dimensional Hawkes processes. We then show that on the level of first-moment simplification of the model that this type of order book implies well known phenomena of price impact observed by Almgren and more precise one proposed by Gatheral.

We study the monodromy of a certain class of semistable hyperelliptic curves over the rationals that was introduced by Shigefumi Mori forty years ago (before his Minimal Model Program). Using ideas of Chris Hall, we prove that the corresponding $\ell$-adic monodromy groups are (almost) ``as large as possible". We also discuss an explicit construction of two-dimensional families of hyperelliptic curves over an arbitrary global field with big monodromy.

Imagine that every vertex of a graph contains a sleeping frog. At time 0, the frog at some designated vertex wakes up and begins a simple random walk. When it lands on a vertex, the sleeping frog there wakes up and begins its own simple random walk, which in turn wakes up any sleeping frogs it lands on, and so on. This process is called the frog model.

Algebraic complexity theory is the study of the computational difficulty of infinite families of polynomials -- a generalization of the computational complexity of decision problems. In this theory, there are analogues of the usual complexity classes P and NP, as well as different NP-complete problems. The main aim is to find complexity lower bounds for certain specific families, such as the families for matrix multiplication and the permanent family. There are different approaches using algebraic geometry and representation theory to attack such lower bound problems. This talk will be a brief introduction to this area and its central problems.