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Title: Nucleic-Acid-Based Sensors and Logic Gates Speaker: Milan Stojanovic, Department of Medicine Brookhaven National Laboratory Time: 4:00pm‐5:00pm Place: CHE 100

Title: Strictly Hermitian Positive Definite Functions Speaker: Allan Pinkus, Technion University Haifa, Israel Time: 3:00pm‐4:00pm Place: PHY 013

We talk about characterizations of various classes of positive definite and Hermitian positive definite functions. In particular we are interested in when \(f(\langle x,y\rangle)\) is a (Hermitian) positive definite, and strictly (Hermitian) positive definite function for \(x, y\in H\), where \(H\) is an arbitrary (complex) inner product space.

Title: Quasi-Stationary Behavior in a Simple Discrete-Time Population Model Speaker: Göran Högnäs, Åbo Akademi University Åbo, Finland Time: 3:00pm‐4:00pm Place: PHY 013

We discuss some stochastic versions of the classical deterministic Ricker model \(x_{t+1}=x_t\exp(r-\gamma x_t)\), \(t=0,1,2\dotsc\) of the time evolution of the density of a population. Here \(r>0\) models the intrinsic growth rate and \(\gamma>0\) is an inhibitive environmental factor. The introduction of demographic stochasticity leads us to a size-dependent branching process whose quasi-stationary behavior (for some values of \(r\)) tends to concentrate around the attracting period cycle of the deterministic system. When we allow the environment to vary, modeled by an i.i.d. sequence of parameters \(\gamma_t\), the branching process may exhibit growth-catastrophe behavior.

Title: Primes is in \(P\) Speaker: Rani Siromony, Madras Christian College/ Chennai Mathematical Institute India Time: 3:00pm‐4:00pm Place: PHY 013

On August 8, 2002, a polynomial time algorithm for recognizing Primes was given by three young computer scientists, Agrawal, Kayal and Saxena of IIT, Kanpur, India. This is a milestone in centuries-old journey towards understanding prime numbers, solving a longstanding open problem in Computational Number Theory and Complexity Theory.

Title: The Blaschke Conjecture Speaker: Benjamin McKay, University of South Florida, St. Petersburg Time: 3:00pm‐4:00pm Place: PHY 013

On a thin sphere of glass, all light rays leaving one point will focus on the antipodal point. A surface on which all light rays from any single point collide at some other point is called a Blaschke surface. Leon Green (1961) showed that all Blaschke surfaces are spheres; his techniques were very hard. The classification of Blaschke objects in higher dimensions is open. I will present my new results on this old (1921) problem; I employ only elementary calculus of differential forms and elementary projective geometry.

Title: Asymptotics of Orthogonal Polynomials, the Riemann-Hilbert Problem and Universality in Matrix Models Speaker: Alexander Its, Indiana Univesity-Purdue University at Indianapolis Time: 3:00pm‐4:00pm Place: PHY 013

Recent developments in the theory of random matrices and orthogonal polynomials reveal striking connections of the subject to integrable nonlinear differential equations of both the KP and the Painlevé types. These connections, in particular, make it possible to use nontraditional analytical schemes of the theory of integrable systems, such as the Riemann-Hilbert asymptotic method, for proving Dyson's universality conjecture concerning the scaling limit of correlations between eigenvalues for a wide class of exponential weights. In the talk, the essence of the Riemann-Hilbert approach to matrix models will be presented together with an exposition of their occurrence in diverse areas of mathematics and physics.