In this talk, I will discuss a convex formulation of the clustering problem and its generalization to biclustering of matrices and more broadly to co-clustering of multiway arrays or tensor data. The key advantage in formulating clustering as a convex program is that doing so addresses well-known issues of instability and parameter selection that plague…
Categorification lifts natural numbers to vector spaces and integers to complexes. Natural number n becomes a vector space of dimension n, and an integer becomes the Euler characteristic of a complex of vector spaces. A well-known example of categorification is lifting the Euler characteristic of a topological space to its homology or cohomology groups. The…
Gerrymandering, the act of drawing political maps to achieve a desirable election outcome, has been increasingly in the news as cases wind their way to the Supreme Court and as the country approaches a new census in 2020. In this talk we’ll look at some of the mathematical strategies and problems that arise in gerrymandering…
Classical Schur Q-functions describe characters of a queer Lie superalgebra, projective representations of a symmetric group and provide solutions of a BKP hierarchy. This talk is devoted to properties of a generalization of Schur Q-functions - factorial Q-functions, including a particular important case of shifted Schur Q-functions.
Triangle Math Teachers' Circle Workshop (sponsored by American Institute of Mathematics) will be held at NCSU, Math Department, on Saturday February 10. This event is aimed at current and future teachers, math education students and anyone who is interested in creative approaches to teaching mathematics through problem solving to K-12 students. The presenters are Natasha Rozhkovskaya (Kansas State University) and Hector Rosario (South Gwinnet High…
Recently, there has been an increasing interest from the community in real-life complex system modeling. The most popular example is provided by systems where the number of agents is so large, that only a statistical description (reminiscent to the statistical mechanics description of systems in thermodynamics) turns out to be viable. The usual way to…
I will discuss formulations and algorithms for computing sparse optimal controls in systems governed by PDEs. These sparse solutions can guide the placement of control devices in applications. After reviewing results for elliptic and parabolic PDEs, I will focus on recent work on sparse optimal control governed by linear PDEs with uncertain coefficients. Here, we aim at finding stochastic controls that…
Suppose you want to convince someone that you know the solution to a problem, but you don’t want them to learn any- thing about the solution. How can you do it? Such a protocol is called a zero-knowledge proof. In this talk, we’ll define what it means to be a zero-knowledge proof, show several ex-…
We consider the action of the group SL_n x SL_n on the space of m-tuples of n x n matrices by simultaneous left-right multiplication. Visu Makam and the speaker recently proved that invariants of degree at most mn^4 generate the invariant ring. This result has interesting applications in algebraic complexity theory and is related to the notion of non-commutative rank.…
Several topics will be presented in this talk. In the first part, we consider some portfolio optimization problems with stochastic dividends, stochastic volatility or delays. The Hamilton-Jacobi-Bellman (HJB) equations are derived, which are second order nonlinear PDEs. We then establish the existence results of the HJB equations and prove the verification theorems. In the second…
An increasing number of scientific and enterprise data sets are multidimensional, where data is gathered for every configuration of three or more parameters. For example, physical simulations often track a set of variables in two or three spatial dimensions over time, yielding 4D or 5D data sets. Tensor decompositions are structured representations of multidimensional data…
Historically, the study of the collection of concordance classes of knots has focused on understanding its group structure while devoting relatively little attention to the natural metric induced by the 4-genus. Cochran and Harvey investigated the metric properties of the maps on concordance induced by satellite operators, asking when two patterns P and Q are of bounded distance in their…
We propose a model of sovereign debt management, where a state trade some bonds to service the debt with a pool of risk-neutral competitive foreign investors. At each instant of time, the government decides which fraction of the GDP (subject to random fluctuations) must be used to repay the debt, by paying a social cost.…
There is an increased interest in alternatives to the traditional lecture-based instruction in math courses. Inquiry-Based Learning (IBL) is a teaching philosophy that challenges students to think more like mathematicians by posing questions rather than presenting established facts. In this talk, I will start by introducing some of the core tenets of IBL. I will…
Quite a large polygon can squeeze between the integer points in the plane, but what if it has to be symmetric bout the origin (and avoid all other integer points)? In this talk, I’ll discuss Minkowski’s theorem, which bounds the area of such shapes, and a surprising consequence for the problem of writing integers as sums of squares of other…
Phylogenetics is the branch of mathematical biology concerned with constructing evolutionary relationships between collections of species. These lectures will introduce these models, in particular emphasizing the ways that algebraic statistics can be used to analyze properties of the models. Viewed from the perspective of algebraic statistics, the corresponding algebraic varieties that arise are often familiar…