May 4 Elijah

Title: The Pólya-Vinogradov and Burgess inequalities for Character Sums

Abstract: I will be discussing bounds for long and short sums over Dirichlet characters. Towards the end, I will present a result that connects these bounds in an unexpected way.

Apr 27 Dan

Title: Shellability and the Euler-Poincaré formula

Abstract: Given a d-dimensional polytope with f-vector f=(f_-1, f_0, ..., f_d), the Euler-Poincare formula says that the alternating sum of the f_i's is 0. For a 3 dimensional polytope, this is the familiar v-e+f=2. There are many proofs of this fact; in this talk I will discuss one proof using the technique of shelling. Along the way, we will use a rocket ship and a polytopal planet to show that the boundary of a polytope is shellable.

Apr 20 Pratyush

Title: Solving the Poisson equation

Abstract: I will start with motivation for the Poisson equation and then derive the energy formulation. Using techniques from calculus of variations, I will outline one way to find solutions.

Apr 13 Daping

Title: Quantum binomial coefficients, Grassmannian, and an example of cyclic sieving phenomenon

Abstract: I will define quantum binomial coefficients and relate their special values at powers of primes to Grassmannians over finite fields and their special value at some roots of unity to the number of fixed points of some finite set under a finite cyclic group action (a.k.a. cyclic sieving phenomenon). This talk will be completely elementary and does not assume any prerequisite beyond linear algebra.

Apr 5 Fernando

Title: Ends of random manifolds

Abstract: What do Jacob's ladder, the plane, the cylinder, the Cantor tree (in every season of the year) and the infinite Loch Ness monster (aka the infinite prison window) have in common? Answer: they are all possible topological types of noncompact regular covers of a closed surface. Come have some Costa Pizza and learn about a probabilistic proof (due to Biringer-Raimbault) of this fact, and more!

Mar 2 Aaron

Title: The Fault in Our Surfaces: Earthquakes in hyperbolic and projective geometry

Abstract: In this talk, I will give a picture-filled and proof-free survey of how earthquakes play a vital role in understanding deformations of geometric structures on surfaces. We will begin by investigating real earthquakes, before moving onto imaginary (grafting), and then finally complex earthquakes (bending). Along the way we will visit classical Teichmüller theory, projective geometry, and the theory of quasi-Fuchsian manifolds. No prior experience with geometry or geology will be assumed.

Feb 23 Byungmin

Title: LLL, Coppersmith and RSA

Abstract: Can we factorize the public key N = pq of an RSA cryptosystem with a partial knowledge of the secret keys p or q? Don Coppersmith(1996) invented an algorithm for finding small integer roots of a polynomial modulo given integer. This algorithm is based on LLL algorithm for finding a small vector in a lattice. In this talk, we use Coppersmith's idea to give an affirmative answer to the aforementioned question.

Feb 9 Joon-Hyeok

Title: Ulam's problem on the longest increasing subsequence

Abstract: Given a permutation, one can consider its increasing subsequence as long as possible, which is called the "longest increasing subsequence". Ulam's problem concerns its "average" length if the given permutation is chosen randomly. In this talk, I will describe some properties of the longest increasing subsequence of a (random) permutation and show its expectation follows an asymptotic formula.

Feb 2 Elad

Title: On representations of sl_2 C and graph isomorphism

Abstract: Given a positive integer n, denote for k = 0, ..., (n choose 2), by g_k the number of unlabeled graphs with n vertices and k edges. We show using representation theory of the Lie algebra sl_2 C that g_k is symmetric and unimodal.