- February 12, 2008

**Mee Seong Im**(UIUC Math)

**Title:**Geometric Invariants of Blow-Ups and del Pezzo surfaces

**Abstract:**I will first give a brief background on algebraic geometry and then introduce the notion of blow-ups with some examples. I will then give some applications of blow-ups to del Pezzo surfaces which are 2-dimensional Fano Varieties. If time allows, I will discuss how Fano manifolds are obtained by blowing-up non Fano manifolds.

- February 26, 2008

**Supawadee Prugsapitak**(UIUC Math)

**Title:**The Tarry-Escott problem over Z[i]

**Abstract:**The Tarry-Escott problem is an old problem that asks for two different sets $\{a_1,a_2,...,a_n\}$ and $\{b_1,b_2,...,b_n\}$ of integers such that \begin{equation} \sum_{i=1}^n a_i^j = \sum_{i=1}^n b_i^j \quad \text{for any}~~ 1 \leq j \leq k. \end{equation} We call $n$ the \emph{size} of solution and $k$ the \emph{degree}. In 2006, A. Alpers and R. Tijdeman generalized the classical Tarry-Escott problem to multinomials and they ask one to find the solution of Tarry-Escott problem over \mathbb{Z}[i]. In this talk, I will discuss Tarry-Escott problem over \mathbb{Z} and give a solution degree 2 of Tarry-Escott problem over \mathbb{Z}[i].

- March 11, 2008

**Merc Chasman**(UIUC Math)

**Title:**Rayleigh Quotients and the Free Plate

**Abstract:**I discuss Rayleigh Quotients and how they can be used to estimate and characterize eigenvalues of operators. I'll then exploit this characterization to show that for a free plate, the fundamental tone (lowest frequency of vibration) is maximal for the disk. That is, of all metal plates vibrating at their lowest frequency, the disk has the highest pitch.

- April 1, 2008

**Chia-yen Tsai**(UIUC Math)

**Title:**Bounds for a least pseudo-Anosov dilatation

**Abstract:**A current area of interest in hyperbolic two-manifolds is understanding pseudo-Anosov homeomorphisms. First I will explain what pseudo-Anosov dilatations mean. Since dilatations of a surface S form a discrete set, it has a minimal element, which we call the least pseudo-Anosov dilatation of S. I will give you known results of bounds for some surfaces S. After that, I will state my main theorem, bounds for a least pseudo-Anosov dilatation of a genus(>1) surface with punctures, then I will give a proof of the upper bound.

- April 29, 2008

**Melissa Dennison**(UIUC Math)

**Title:**Purely Even Terms in the Bow Sequence

**Abstract:**By modifying the recursion in the Stern sequence, we obtain a new sequence called the bow sequence. I will discuss some properties of the bow sequence modulo 2. In particular, I will show that 1/7 of the terms of the bow sequence are even for all sets of initial conditions.