Wednesday, December 7, 2011 - 12:05 , Location: Skiles 005. , Josephine Yu , Georgia Tech , Organizer:
A polytope is a convex hull of a finite set of points in a vector space. The set of polytopes in a fixed vector space generate an algebra where addition is formal and multiplication is the Minkowski sum, modulo some relations. The algebra of polytopes were used to solve some variations of Hilbert's third problem about subdivision of polytopes and to give a combinatorial proof of Stanley's g-Theorem that characterizes face numbers of simplicial polytopes. In this talk, we will introduce McMullen's version of polytope algebra and show that it is isomorphic to the algebra of tropical cycles which are balanced weighted polyhedral fans. The tropical cycles can be used to do explicit computations and examples in polytope algebra.
Wednesday, November 30, 2011 - 12:05 , Location: Skiles 005. , Farbod Shokrieh , Georgia Tech. , Organizer:
I will discuss the theory of chip-firing games, focusing on the interplay between chip-firing games and potential theory on graphs. To motivate the discussion, I will give a new proof of "the pentagon game". I will discuss the concept of reduced divisors and various related algorithmic aspects of the theory. If time permits I will also give some applications, including an "efficient bijective" proof of Kirchhoff's matrix-tree theorem.
Wednesday, November 16, 2011 - 12:05 , Location: Skiles 005. , Manwah Lilian Wong , Georgia Tech , Organizer:
We will discuss the discrete Schroedinger problem on the integer line and on graphs. Starting from the definition of the discrete Laplacian on the integer line, I will explain why the problem is interesting, how the discrete case relates to the continuous case, and what the open problems are. Recent results by the speaker (joint with Evans Harrell) will be presented.The talk will be accessible to anyone who knows arithmetic and matrix multiplications.
Wednesday, November 9, 2011 - 12:05 , Location: Skiles 005. , Andrzej Swiech , Georgia Tech. , Organizer:
I will give a brief introduction to the theory ofviscosity solutions of second order PDE. In particular, I will discussHamilton-Jacobi-Bellman-Isaacs equations and their connections withstochastic optimal control and stochastic differentialgames problems. I will also present extensions of viscositysolutions to integro-PDE.
Wednesday, November 2, 2011 - 12:05 , Location: Skiles 005 , Evans Harrell , School of Mathematics, Georgia Tech , Organizer:
Eigenvalues of linear operators often correspond to physical observables; for example they determine the energy levels in quantum mechanics and the frequencies of vibration in acoustics. Properties such as the shape of a system are encoded in the the set of eigenvalues, known as the "spectrum," but in subtle ways. I'll talk about some classic theorems about how geometry and topology show up in the spectrum of differential operators, and then I'll present some recent work, with connections to physical models such as quantum waveguides, wires, and graphs.
Wednesday, October 26, 2011 - 12:05 , Location: Skiles 005 , Silas Alben , School of Mathematics, Georgia Tech , Organizer:
Vortex methods are an efficient and versatile way to simulate high Reynolds number flows. We have developed vortex sheet methods for a variety of flows past deforming bodies, many of which are biologically inspired. In this talk we will present simulations and asymptotic analysis of selected problems. The first is a study of oscillated and freely-swimming flexible foils. We analyze the damped resonances that determine propulsive performance. The second problem involves multiple passive flapping ``flags" which interact through their vortex wakes. The third problem is a study of flexible falling sheets. Here the flag-flapping instability helps us determine the terminal falling speeds.
Wednesday, October 12, 2011 - 12:05 , Location: Skiles 005 , Greg Blekherman , Georgia Tech , Organizer:
A multivariate real polynomial p(x) is nonnegative if p(x) is at least 0 for all x in R^n. I will review the history and motivation behind the problem of representing nonnegative polynomials as sums of squares. Such representations are of interest for both theoretical and practical computational reasons, with many applications some of which I will present. I will explain how the problem of describing nonnegative polynomials fits into convex algebraic geometry: the study of convex sets with underlying algebraic structure, that brings together ideas of optimization, convex geometry and algebraic geometry. I will end by presenting current research problems in this area.
Wednesday, September 21, 2011 - 12:05 , Location: Skiles 005 , Amit Einav , Georgia Tech , Organizer:
Sharp trace inequalities play a major role in the world of Mathematics. Not only do they give a connection between 'boundary values' of the trace and 'interior values' of the function, but also the truest form of it, many times with a complete classification of when equality is attained. The result presented here, motivated by such inequality proved by Jose' Escobar, is a new trace inequality, connecting between the fractional laplacian of a function and its restriction to the intersection of the hyperplanes x_(n)=0, x_(n-1)=0, ..., x_(n-j+1)=0 where 1<=j<=n. We will show that the inequality is sharp and discussed the natural space for it, along with the functions who attain equality in it. The above result is based on a joint work with Prof. Michael Loss.
Wednesday, September 14, 2011 - 12:05 , Location: Skiles 005. , Rafael De La Llave , Georgia Tech. , Organizer:
Starting in the 30's Physicists were concerned with the problem of motion of dislocations or the problem of deposition of materials over a periodic structure. This leads naturally to a variational problem (minimizing the energy). One wants to know very delicate properties of the minimizers, which was a problem that Morse was studying at the same time. The systematic mathematical study of these problems started in the 80's with the work of Aubry and Mather who developed the basis to deal with very subtle problems. The mathematics that have become useful include dynamical systems, partial differential equations, calculus of variations and numerical analysis. Physical intuition also helps. I plan to explain some of the basic questions and, perhaps illustrate some of the results.
Wednesday, April 20, 2011 - 12:00 , Location: Skiles 006 , Chris Heil , School of Mathematics - Georgia Institute of Technology , Organizer:
Hosts: Amey Kaloti and Ricardo Restrepo
Fourier series provide a way of writing almost any signal as a superposition of pure tones, or musical notes. But this representation is not local, and does not reflect the way that music is actually generated by instruments playing individual notes at different times. We will discuss time-frequency representations, which are a type of local Fourier representation of signals. This gives us a mathematical model for representing music. While the model is crude for music, it is in fact apowerful mathematical representation that has appeared widely throughout mathematics (e.g., partial differential equations), physics (e.g., quantum mechanics), and engineering (e.g., time-varying filtering). We ask one very basic question: are the notes in this representation linearly independent? This seemingly trivial question leads to surprising mathematical difficulties.