Seminar

Are you interested in becoming a math major or what happens after graduation for math majors? Come and ask our undergraduate seniors!

If you give one of your friends a puzzle to solve, they will often dive right into a trial-and-error approach to finding a solution.  For instance, if we have a line of lightbulbs, each of which with a switch that affects its state along with the state of its immediate neighbor bulbs, we can ask if it's possible to turn off all of the bulbs from some starting configuration of on/off.  The trial-and-error approach may work for four bulbs, but what if there were 20 bulbs?  Luckily for us, some simple linear algebra tools can give us clarity for all possible cases.

The derivative of $f(x,y)$ in the direction $\vec{v}=(v_1,v_2)$, at point $p$, is:
$$(f_x(p),f_y(p)) \cdot (v_1,v_2) = v_1 f_x(p) + v_2 f_y(p) $$

But wait: $f_x(p)$ and $f_y(p)$ measure how $f$ changes in directions of $x$ and $y$. Then the formula above claims that knowing only this piece of information, we can know how $f$ changes in any other direction as well?! How could this be true? I will answer this, and end with a eulogy of derivatives in general!

In this talk, we will discuss so-called ``Tupper's self-referential formula'', given by

$$\displaystyle \frac{1}{2} < \left\lfloor \bmod\left(\left\lfloor\frac{y}{17}\right\rfloor 2^{-17\lfloor x\rfloor-\bmod(\lfloor y\rfloor, 17)},2\right)\right\rfloor$$

Math gets a bad rap. It's too hard, it's not useful, it's not fun... We hope to change your mind! This introductory talk will discuss the goal(s) of the Undergraduate Math Seminar. We will introduce ourselves (the organizers) and talk about the type of research we all do and how we became interested in it. Further, we will tell you of our experiences with research and how you can become involved in the math community. We hope you join us and tell us a bit about yourself as well!

Muckenhoupt Weights are ubiquitous in the field of harmonic analysis. In particular, they are appropriate weights for weighted $L^p$ bounds for many classical harmonic analysis objects (maximal functions, Riesz transforms, etc.) These weights also play a role in the solvability of the $L^p$ Dirichlet problem for the Laplacian in `rough’ sets. In particular, if the harmonic measure is a Muckenhoupt weight (in some sense) then the $L^p$ Dirichlet problem is solvable for some p.

Abstract: