Starting from Total Variation, this talk will overview some mathematical approaches for image processing, such as removing noise. We will also consider numerical application to data understanding. A few more application maybe presented.
This is a general audience Geometry-Topology talk where I will give a broad overview of my research interests and techniques I use in my work. My research concerns the study of link concordance using groups, both extracting concordance data from group theoretic invariants and determining the properties of group structures on links modulo concordance. Milnor's invariants are one of the more fundamental link concordance invariants; they are thought of as higher order linking numbers and can be computed using both Massey products (due to Turaev and Porter) and higher order intersections (due to Cochran). In my work, I have generalized Milnor's invariants to knots inside a closed, oriented 3-manifold M. I call this the Dwyer number of a knot and show methods to compute it for null-homologous knots inside a family of 3-manifolds with free fundamental group. I further show Dwyer number provides the weight of the first non-vanishing Massey product in the knot complement in the ambient manifold. Additionally, I proved the Dwyer number detects knots K in M bounding smoothly embedded disks in specific 4-manifolds with boundary M which are not concordant to the unknot in M x I. This result further motivates my definition of a new link concordance group in joint work with Matthew Hedden using the knotification construction of Ozsv'ath and Szab'o. Finally, I will briefly discuss my recent result that the string link concordance group modulo its pure braid subgroup is non-abelian.
We will define Newton polygons for polynomials over a valued field and prove a couple theorems using them. For example, relating the valuations of the roots of the polynomial to the slopes of the Newton polygon and proving the algebraic closure of the Puiseux series in characteristic 0.
For the first time in 2020, the US Census Bureau will apply a differentially private algorithm before publicly releasing decennial census data. Recently, the Bureau publicly released their code and end-to-end tests on the 1940 census data at various privacylevels. We will outline the DP algorithm (which is still being developed) and discuss the accuracy of these end-to-end tests. In particular, we focus on the bias and variance of the reported population counts. Finally, we discuss the choices the Bureau has yet to make that will affect the balance between privacy and accuracy. This talk is based on joint work with Abraham Flaxman.
Please Note:
In 1999, Alon proved the “Combinatorial Nullstellensatz” which resembles Hilbert’s Nullstellensatz and gives combinatorial structure on the roots of a multivariate polynomial. This method has numerous applications, most notably in additive number theory, but also in many other areas of combinatorics. We will prove the Combinatorial Nullstellensatz and give some of its applications in graph theory.
We study the effect of sparsity on the appearance of outliers in the semi-circular law. As a corollary of our main results, we show that, for the Erdos-Renyi random graph with parameter p, the second largest eigenvalue is (asymptotically almost surely) detached from the bulk of the spectrum if and only if pn
In this talk I will briefly talk about coding theory and introduce a specific family of codes called Quasi-quadratic residue (QQR) codes. These codes have large minimum distances, which means they have good error-correcting capabilities. The weights of their codewords are directly related to the number of points on corresponding hyperelliptic curves. I will show a heuristic model to count the number of points on hyperelliptic curves using a coin-toss model, which in turn casts light on the relation between efficiency and the error-correcting capabilities of QQR codes. I will also show an interesting phenomenon we found about the weight enumerator of QQR codes. Lastly, using the bridge between QQR codes and hyperelliptic curves again, we derive the asymptotic behavior of point distribution of a family of hyperelliptic curves using results from coding theory.
A 2-knot is a smooth embedding of S^2 in S^4, and a 0-concordance of 2-knots is a concordance with the property that every regular level set of the concordance is just a collection of S^2's. In his thesis, Paul Melvin proved that if two 2-knots are 0-concordant, then a Gluck twist along one will result in the same smooth 4-manifold as a Gluck twist on the other. He asked the following question: Are all 2-knots 0-slice (i.e. 0-concordant to the unknot)? I will explain all relevant definitions, and mostly follow the paper by Nathan Sunukjian on this topic.
Mathematicians have long been trying to understand which domains admit an orthogonal (or, sometimes, not) basis of exponentials of the form , for some set of frequencies (this is the spectrum of the domain). It is well known that we can do so for the cube, for instance (just take ), but can we find such a basis for the ball? The answer is no, if we demand orthogonality, but this problem is still open when, instead of orthogonality, we demand just a Riesz basis of exponentials.
We will explore some of the basic notions in quantum topology. Our focus will be on introducing some of the foundations of diagrammatic algebra through the lens of the Temperley-Lieb algebra. We will attempt to show how these diagrammatic techniques can be applied to low dimensional topology. Every effort will be made to make this as self-contained as possible. If time permits we will also discuss some applications to topological quantum computing.
