Geometric mechanics involves the use of differential geometry and symmetry techniques to study mechanical systems. In particular, it deals with global invariants of the motion, and how they can be used to describe and understand the qualitative properties of complicated dynamical systems, without necessarily explicitly solving the equations of motion. This approach parallels the development of geometric numerical methods in numerical analysis, wherein numerical algorithms for the solution of differential equations are constructed so as to exactly conserve the invariants of motion of the continuous dynamical system.

This talk will provide a gentle introduction to the role of geometric methods in understanding nonlinear dynamical systems, and why it is important to develop numerical methods that have good global properties, as opposed to just good local behavior.

First we focus on the size-Ramsey number of a path $P_n$ on $n$
vertices. In particular, we show that $5n/2 - 15/2 \le \hat{r}(P_n)\le
74n$ for $n$ sufficiently large. This improves the previous lower bound
due to Bollob\'{a}s, and the upper bound due to Letzter.

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Next we study long monochromatic paths in edge-colored random graph
$G(n,p)$ with $pn\to\infty$. Recently, Letzter showed that a.a.s. any
2-edge coloring of $G(n, p)$ yields a monochromatic path of length
$(2/3-o(1))n$, which is optimal. Extending this result, we show that
a.a.s. any 3-edge coloring of $G(n, p)$ yields a monochromatic path of
length $(1/2-o(1))n$, which is also optimal. We also consider a related
problem and show that for any $r\ge 2$, a.a.s. any $r$-edge coloring of
$G(n, p)$ yields a monochromatic connected subgraph on $(1/(r-1)-o(1))n$
vertices, which is also tight.

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Finally, we discuss some extensions of the above results for random
hypergraphs. In particular, we obtain a random analog of a result of
Gy\'arf\'as, S\'ark\'ozy, and Szemer\'edi on the longest monochromatic
loose cycle in $2$-colored complete $k$-uniform hypergraphs.

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This is joint work with Pawel Pralat and also with Patrick
Bennett, Louis DeBiasio, and Sean English.

We prove a quenched invariance principle and local limit theorem
for a random walk in an ergodic balanced time dependent environment
on the lattice. Our proof relies on the parabolic Harnack inequality
for the adjoint operator. This is joint work with X. Guo.

Lots of progress have been made in the recent years on the birational geometry of surfaces and 3-folds in positive characteristic over algebraically closed field. The same can not be said about the varieties over imperfect fields. These varieties appear naturally in positive characteristic while studying fibrations (as a generic fiber). Recently the minimal model program (MMP) for surfaces over excellent base scheme was successfully carried out by Tanaka. He also showed that the abundance conjecture holds for surfaces over imperfect fields. His results have become one of main tools for studying fibrations in positive characteristic. One of the things that is not covered in Tanaka's papers is the del Pezzo surfaces (a regular surface with -$K_X$ ample) over imperfect fields. One interesting feature of del Pezzo surfaces is that over an algebraically closed field they satisfy the Kodaira vanishing theorem. This makes the theory of del Pezzo surfaces quite interesting. However, over imperfect fields it was known for a while that in char 2, Kodaira vanishing fails for del Pezzo surfaces, due to (Schroer and Maddock). It is only very recently that some positive results started to show up. In a recent paper by Patakfalvi and Waldron it was shown that the Kodaira vanishing theorem holds for del Pezzo surfaces over imperfect fields in char $p>3$. In this talk I will show that in fact the Kawamata-Viehweg vanishing theorem holds for del Pezzo surfaces over imperfect fields in char $p>3$. I will also report on a project which is a work in progress (joint with Joe Waldron) on the minimal model program for 3-folds over imperfect fields and the BAB conjecture for del Pezzo surfaces over imperfect fields.