Events of the Week

The plan: (1) Executive summary of the connection between eigenvalues of random matrices and Fredholm determinants. (2) T random matrix theory laboratory – testing the theory. (3) Behavior of eigenvalues when the size of the matrices grows to $\infty$.

Given a set of points X and a generic point P in a projective 3-space, it is interesting to see the properties of the projection of X with respect to P. In recent years, generic projections have been gathering a lot of attention. Various properties of the generic projection are currently being developed by Luca Chiantini, Łucja Farnik, Giuseppe Favacchio, Brian Harbourne, Juan Migliore, Tomasz Szemberg, and Justyna Szpond. They also showed how this projection is related to finding unexpected hypersurfaces. While doing the same, they asked interesting questions. Many of them are open. In this talk, I will discuss the concept of generic projection and discuss interesting properties, e.g, Hilbert function. I will also discuss some of their open problems.

We study the boundary layer theory for slightly viscous stationary flows forced by an imposed slip velocity at the boundary. According to the theory of Prandtl (1904) and Batchelor (1956), any Euler solution arising in this limit and consisting of a single ``eddy" must have constant vorticity. Feynman and Lagerstrom (1956) gave a procedure to select the value of this vorticity by demanding a necessary condition for the existence of a periodic Prandtl boundary layer description. We will show that this condition is also sufficient for the existence of a periodic boundary layer described by the Prandtl equations. This is joint work with S. Iyer and T. Nguyen.

Classical Interpolation problem of estimating new data from a set of known data is well understood under one variable situation. Here we are more interested in higher dimensional Projective Spaces, where we are trying to find the lowest possible degree of the hyper-surface passing through a given set of points with prescribed multiplicity. There are famous conjectures to tackle such problems: Chudnovsky’s Conjecture, Demailly’s Conjecture. I will be discussing those conjectures and some recent developments in this area.

Title and abstract to be announced

TBA

Homeostasis occurs in a biological system when a chosen output variable remains approximately constant despite changes in an input variable. In this work we specifically focus on biological systems which may be represented as chemical reaction networks and consider their infinitesimal homeostasis, where the derivative of the input-output function is zero. The specific challenge of chemical reaction networks is that they often obey various conservation laws complicating the standard input-output analysis. We derive several results that allow to verify the existence of infinitesimal homeostasis points both in the absence of conservation and under conservation laws where conserved quantities serve as input parameters. In particular, we introduce the notion of infinitesimal concentration robustness, where the output variable remains nearly constant despite fluctuations in the conserved quantities. We provide several examples of chemical networks which illustrate our results both in deterministic and stochastic settings.

TBA

Title and abstract to be announced

Title and abstract to be announced

Title and abstract to be announced