Georgia Institute of Technology College of Sciences

I will consider a class of mathematical models of decision making. These models are based on dynamics in the neighborhood of unstable equilibria and involve random perturbations due to small noise. I will report results on the vanishing noise limit for these systems, providing precise predictions about the statistics of decision making times and sequences of unstable equilibria visited by the process. Mathematically, the results are based on the analysis of random Poincare maps in the neighborhood of each equilibrium point. I will also discuss some experimental data.

The Scenery Reconstruction Problem consists in trying to reconstruct a coloring of the integers given only the observations made by a random walk. For this we consider a random walk S and a coloring of the integers X. At time $t$ we observe the color $X(S(t))$. The coloring is i.i.d. and we show that given only the sequence of colors $$X(S(0)),X(S(1)),X(S(2)),...$$ it is possible to reconstruct $X$ up to translation and reflection. The solution depends on the property of the random walk and the distribution of the coloring. Longest Common Subsequences (LCS) are widely used in genetics. If we consider two sequences X and Y, then a common subsequence of X and Y is a string which is a subsequence of X and of Y at the same time. A Longest Common Subsequence of X and Y is a common subsequence of X and Y of maximum length. The problem of the asymptotic order of the flucutation for the LCS of independent random strings has been open for decades. We have now been able to make progress on this problem for several important cases. We will also show the connection to the Scenery Reconstruction Problem.

Almost all models for complex fluids can be fitted into the energetic variational framework. The advantage of the approach is the revealing/focus of the competition between the kinetic energy and the internal "elastic" energies. In this talk, I will discuss two very different engineering problems: free interface motion in Newtonian fluids and viscoelastic materials. We will illustrate the underlying connections between the problems and their distinct properties. Moreover, I will present the analytical results concerning the existence of near equilibrium solutions of these problems.

The speaker has combined his two loves to create a dynamic presentation called "Mathemagics," suitable for all audiences, where he demonstrates and explains his secrets for performing rapid mental calculations faster than a calculator. Reader's Digest calls him "America's Best Math Whiz". He has presented his high energy talk for thousands of groups throughout the world. This event is free but reservations are required. The signup form will be available before 5pm on February 25. See details about the speaker.