Georgia Institute of Technology College of Sciences

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

There are two purposes of this talk: 1. to give an example of representation theory in algebraic combinatorics and 2. to explain some of the early work on unimodal/symmetric sequences in combinatorics related to recent work on Hodge theory in combinatorics. We will investigate the structure of graded vector spaces $\bigoplus V_j$ with two "shifting" operators $V_j \to V_{j+1}$ and $V_j → V_{j-1}$. We will see that this leads to a very rich theory of unimodal and symmetric sequences with several interesting connections (e.g. the Edge-Reconstruction Conjecture and Hard Lefschetz). The majority of this talk should be accessible to anyone with a solid knowledge of linear algebra.

https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1611606555671

If we can write a (homogeneous) polynomial as a sum of squares(SOS), the polynomial is guaranteed to be a non-negative polynomial. However, every non-negative forms does not have to be written as sums of squares in general. This implies that set of sums of square is strictly contained in set of non-negative forms in general. We want to discuss about one way to describe the gaps between the two sets. Since the sets have cone structures, we can consider dual cones of each cones. In particular, the description of dual cone of non-negative polynomials is simple: convex hull of point evaluations. Therefore, we are interested in positive semi-definite quadratic forms that is not point evaluations. We call "Hankel index" the minimal rank of quadratic form (on extreme ray of the dual cone of SOS) which is not a point evaluation. In this talk, we introduce the Hankel index of variety and will discuss about a criterion to obtain the Hankel index of projected rational curves.

Teams Link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1600608874868?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%223eebc7e2-37e7-4146-9038-a57e56c92d31%22%7d

Locally PSD matrices are a generalization of PSD matrices which appear in sparse semidefinite programming. We will try to explore some connections of extreme rays of this type of matrix with algebraic topology.