Georgia Institute of Technology College of Sciences

In this talk we will discuss epidemic modeling in the context of COVID-19.  We will review the basics of classical epidemic models, and present joint work with Maria Chikina on the use of age-targeted strategies in the context of a COVID-19-like epidemic.  We will also discuss the broader roles epidemic modeling has played over the past year, and the limitations it as presented as a primary lens through which to understand the pandemic.

In this talk, we will discuss the minimum positive value of the stationary distribution of a random walk on a directed random graph with given degrees. While for undirected graphs the stationary distribution is simply determined by the degrees, the graph geometry plays a major role in the directed case. Understanding typical stationary values is key to determining the mixing time of the walk, as shown by Bordenave, Caputo, and Salez. However, typical results provide no information on the minimum value, which is important for many applications. Recently, Caputo and Quattropani showed that the stationary distribution exhibits logarithmic fluctuations provided that the minimum degree is at least 2. In this talk, we show that dropping the minimum degree condition may yield polynomially smaller stationary values of the form n^{-(1+C+o(1))}, for a constant C determined by the degree distribution. In particular, C is the combination of two factors: (1) the contribution of atypically thin in-neighborhoods, controlled by subcritical branching processes; and (2) the contribution of atypically "light" trajectories, controlled by large deviation rate functions. As a by-product of our proof, we also determine the hitting and cover time in random digraphs. This is joint work with Xing Shi Cai.

Even though it is not easy to determine global non-negativity of a polynomial, if the polynomial can be written as a sum of squares(SOS), we certainly see that it must be non-negative(PSD). Representability of polynomials in terms of sums of squares is a good certification for global non-negativity in the sense that any non-negative polynomials is just a sum of squares in some cases. However, there are some non-negative polynomials which cannot be written as sum of squares in general. So, one can ask about when the set of sums of squares is same as the set of non-negative polynomials or describing gap between set of sums of squares and non-negative polynomials if they are different.

In this talk, we will introduce an algebraic invariant (of variety) which can tell us when the two sets are same (or not). Moreover, we will discuss about cases that we can exactly describe structural gaps between the two sets.

URL: Microsoft Teams

We are motivated by invertible matrix based constructions for expressing the coefficients of ordinary generating functions of special convolution type sums. The sum types we consider typically arise in classical number theoretic applications such as in expressing the Dirichlet convolutions $f \ast 1$ for any arithmetic function $f$. The starting point for this perspective is to consider the so-termed Lambert series generating function (LGF) factorization theorems that have been published over the past few years in work by Merca, Mousavi and Schmidt (collectively). In the LGF case, we are able to connect functions and constructions like divisor sums from multiplicative number theory to standard functions in the more additive theory of partitions. A natural question is to ask how we can replicate this type of unique "best possible", or most expressive expansion relating the generating functions of more general classes of convolution sums? In the talk, we start by summarizing the published results and work on this topic, and then move on to exploring how to define the notion of a "canonically best" factorization theorem to characterize this type of sum in more generality.

BlueJeans link: https://bluejeans.com/936847924