Georgia Institute of Technology College of Sciences

Several experts from the M2internals group will give tutorials and lead the discussion. This is a part of Meeting on Applied Algebraic Geometry.

(Macaulay2 is a software system devoted to supporting research in algebraic geometry and commutative algebra.)  

Zoom Link: Link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Abstract: Linear response refers to the smooth change in the statistics of an observable in a dynamical system in response to a smooth parameter change in the dynamics. The computation of linear response has been a challenge, despite work pioneered by Ruelle giving a rigorous formula in Anosov systems. This is because typical linear perturbation-based methods are not applicable due to their instability in chaotic systems. Here, we give a new differentiable splitting of the parameter perturbation vector field, which leaves the resulting split Ruelle's formula amenable to efficient computation. A key ingredient of the overall algorithm, called space-split sensitivity, is a new recursive method to differentiate quantities along the unstable manifold.

In the second part, we discuss a new KAM method-inspired construction of transport maps. Transport maps are transformations between the sample space of a source (which is generally easy to sample) and a target (typically non-Gaussian) probability distribution. The new construction arises from an infinite-dimensional generalization of a Newton method to find the zero of a "score operator". We define such a score operator that gives the difference of the score -- gradient of logarithm of density -- of a transported distribution from the target score. The new construction is iterative, enjoys fast convergence under smoothness assumptions, and does not make a parametric ansatz on the transport map.

Lefschetz fibrations are very useful in the sense that they have one-one correspondence with the relations in the Mapping Class Groups and they can be used to construct exotic (homeomorphic but not diffeomorphic) 4-manifolds. In this series of talks, we will first introduce Lefschetz fibrations and Mapping Class Groups and give examples. Then, we will dive more into 4-manifold world. More specifically, we will talk about the history of  exotic 4-manifolds and we will define the nice tools used to construct exotic 4-manifolds, like symplectic normal connect sum, Rational Blow-Down, Luttinger Surgery, Branch Covers, and Knot Surgery. Finally, we will provide various constructions of exotic 4-manifolds.

The Laplacian of a graph is a real symmetric matrix given by $L=D-A$, where $D$ is the degree matrix of the graph and $A$ is the adjacency matrix. The Laplacian is a central object in spectral graph theory, and the spectrum of $L$ contains information on the graph. In the case of a random graph the Laplacian will be a random real symmetric matrix with dependent entries. These random Laplacian matrices can be generalized by taking $A$ to be a random real symmetric matrix and $D$ to be a diagonal matrix with entries equal to the row sums of $A$. We will consider the eigenvalues of general random Laplacian matrices, and the challenges raised by the dependence between $D$ and $A$. After discussing the bulk global eigenvalue behavior of general random Laplacian matrices, we will focus in detail on fluctuations of the largest eigenvalue of $L$ when $A$ is a matrix of independent Gaussian random variables. The asymptotic behavior of these Gaussian Laplacian matrices has a particularly nice free probabilistic interpretation, which can be exploited in the study of their eigenvalues. We will see how this interpretation can locate the largest eigenvalue of $L$ with respect to the largest entry of $D$. This talk is based on joint work with Kyle Luh and Sean O'Rourke.