Seminar

Abstract:

Irreversible, dissipative processes can be often naturally modeled as gradient flows; under certain assumptions, flow in deformable porous media is such a process. In this talk, we formulate the problem of linear poro-(visco-)elasticity as generalized gradient flow. By exploiting this structure, the analysis of well-posedness and construction of numerical solvers can be performed in a rather straight-forward and abstract manner. For this, we apply results from gradient flow theory and convex optimization.

Hopefully we know that $\displaystyle \sum_{i \geq 0} x^i $ $= \frac{1}{1-x}$. Similarly one computes that $ \displaystyle \sum_{i \leq 1} x^i $ $= \frac{x^2}{x-1}$. Interestingly, $ \frac{1}{1-x} + \frac{x^2}{x-1} = 1 + x$ which is the sum corresponding to the integers in the interval $[0, 1] = [0, \infty] \cap [-\infty, 1]$. We will explain generalization of this (called Brion's theorem) to integer points in convex polytopes of arbitrary dimension.

When a young mathematician told his advisor that he had found a way to turn a sphere inside out (without making any creases), his advisor told him that this was impossible to do so, and gave him a proof. His advisor was wrong. That young mathematician (Smale) went on to win the highest mathematical prize (the Fields Medal). His solution is very non-intuitive. Today, there are several excellent videos on YouTube that show how to turn a sphere inside out, and this talk will explain some of them. 

We will discuss telescoping, Tannery's Theorem and other techniques for calculating infinite sums.

Collisions between two masses $m_{1}$ and $m_{2}$ have two important quantities: momentum and energy, respectively given by

$$ p = m_{1}v_{1} + m_{2}v_{2}$$

and

$$E = \frac{1}{2}m_{1}v_{1}^2 + \frac{1}{2}m_{2}v_{2}^2.$$

In any normal linear algebra class you learn how to solve a system of linear equations using Gaussian elimination methods. But how would solve a system like the following:
$$x^2+y+z=1$$

$$x+y^2+z=1$$

$$x+y+z^2=1$$

In this talk I will go over how one can use Gr\"obner theory to solve a system of nonlinear equations and what assumptions might need to be made to make this possible.