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

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

For (X,d,w) be a space of homogeneous type in the sense of Coifman and Weiss, suppose that u and v are two locally finite positive Borel measures on (X,d,w).  Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderon--Zygmund operator T from L^{2}(u) to L^{2}(v) in terms of the A_{2} condition and two testing conditions. The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.

We also give the two weight quantitative estimates for the commutator of maximal functions and the maximal commutators with respect to the symbol in weighted BMO space on spaces of homogeneous type. These commutators turn out to be controlled by the sparse operators in the setting of space of homogeneous type. The lower bound of the maximal commutator is also obtained.

Zoom link:

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

The writhe of a braid (=#pos crossing - #neg crossings) and the fractional Dehn twist coefficient of a braid (a rational number that measures "how much the braid twists") are the two most prominent examples of what is known as a quasimorphism (a map that fails to be a group homomorphism by at most a bounded amount) from Artin's braid group on n-strands to the reals.
We consider characterizing properties for such quasimorphisms and talk about relations to the study of knot concordance. For the latter, we consider inequalities for quasimorphism modelled after the so-called slice-Bennequin inequality:
writhe(B) ≤ 2g_4(K) - 1 + n for all n-stranded braids B with closure a knot K.
Based on work in progress.