Georgia Institute of Technology College of Sciences

There are many interesting classes of polynomials in real algebraic geometry that are of modern interest. A polynomial is nonnegative if it only takes nonnegative values on R^n. A univariate polynomial is real-rooted if all of its complex roots are real, and a hyperbolic polynomial is a multivariate generalization of a real-rooted polynomial. We will discuss connections between these two classes of polynomials. In particular, we will discuss recent ideas of Saunderson giving new ways of proving that a polynomial is nonnegative beyond showing that it is sum-of-squares.

Teams link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1646885419648?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%2206706002-23ff-4989-8721-b078835bae91%22%7d

We will review how divisors on abstract algebraic curves are connected with projective embeddings and then see how that language translates to tropical curves and tropicalization. This talk aims to explain some of the connections between tropical curves and algebraic curves that was not discussed during the seminar on tropical Brill-Noether theory.

Microsoft Teams Link

Algebra Student Seminar homepage

We will introduce the machinery of hyperbolic polynomial, and see how it can help us generalize classical linear algebra theorems and inequalities on symmetric matrices, including Hadamard-Fischer inequality, Koteljanskii's inequality and Schur-Horn theorem (last one is conjectured but not proved). Joint work with Greg Blekherman, Mario Kummer, Raman Sanyal and Kevin Shu.

Team link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1643388106130?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%2206706002-23ff-4989-8721-b078835bae91%22%7d

Recall that, for a variety $X$ in a projective space $\mathbb{P}^d$, the $X$-rank of a point $p\in \mathbb{P}^d$ is the least number of points of $X$ whose span contains the point $p$. Studies about $X$-ranks include some well-known and important results about various tensor ranks. For example, 

• the rank of tensors is the rank with respect to Segre varieties,
• the rank of symmetric tensors, i.e. Waring rank, is the rank with respect to Veronese embeddings, and
• the rank of anti-symmetric tensors is the rank with respect to Grassmannians in its Plücker embedding.  

In this talk, we focus on ranks with respect to Veronese embeddings of a projective line $\mathbb{P}^1$. i.e. symmetric tensor ranks of binary forms. We will discuss how to associate points in $\mathbb{P}^d$ with binary forms and I will introduce apolarity for binary forms which gives an effective method to study Waring ranks of binary forms. We will discuss various ranks on the Veronese embedding and some results on the ranks.