Events

Title: Online Unrelated-Machine Load Balancing and Generalized Flow with Recourse

Abstract: I will present the online algorithms for unrelated-machine load balancing problem with recourse.  First, we shall present a (2+\epsilon)-competitive algorithm for the problem with O_\epsilon(\log n) amortized recourse per job.

Title: L-systems and the Lovasz number

Abstract: For positive integers n and k, an L-system is a collection of k-uniform subsets of a set of size n whose pairwise intersection sizes all lie in in the set L. The maximum size of an L-system is equal to the independence number of a certain union of graphs in the Johnson scheme. The Lovasz number is a semidefinite programming approximation of the independence number of a graph. In this talk, we survey the relationship between the maximum size of an L-system and the Lovasz number, illustrating examples both where the Lovasz number is a good approximation and where it is a bad approximation.

Title: Research in Applications

Abstract: In this talk I will present my personal experiences in doing research involving applications. I will go over some of my work, presenting some of the key aspects that are involved, and trying to take stock of a few lessons learned.

Title: Monomial ideals, Galois closures, and Hilbert schemes of points

Abstract: Manjul Bhargava and the speaker introduced a functorial Galois closure operation for finite-rank ring extensions, generalizing constructions of Grothendieck and Katz-Mazur. In this talk, we use Galois closures to construct new components of Hilbert schemes of points, which are fundamental objects in algebraic geometry whose component structure is largely mysterious. We answer a 35 year old open problem posed by Iarrobino by constructing an infinite family of low dimensional components. This talk is based on joint work with Andrew Staal. No prior knowledge of Hilbert schemes will be assumed.

Title: To be announced

Title: Error bounds for conic feasibility problems: case studies on the exponential cone

Abstract: Conic feasibility problems naturally arise from linear conic programming problems. An understanding of error bounds for these problems is instrumental in the design of termination criteria for conic solvers and the study of convergence rate of algorithms.

Title: Partial geometric designs, directed strongly regular graphs, and association scheme

Abstract: A partial geometric design with parameters $(v, b, k, r; \alpha, \beta)$ is a tactical configuration $(P, \mathcal{B})$ (with $|P|=v$, $|\mathcal{B}|=b$, every point $p\in P$ belonging to $r$ blocks, and every block $B\in\mathcal{B}$ consisting of $k$ points) satisfying the property:

{for any pair $(p, B)\in P\times \mathcal{B}$, the number of flags $(q, C)$ with $q\in B$ and $C\ni p$ equals to $\alpha  \mbox{ if } p\notin B$ and to $\beta  \mbox{ if } p\in B$.}

Neumaier studied partial geometric designs in detail in his article, ``$t\frac12$-designs," [JCT A {\bf 28}, 226-248 (1980)]. He investigated their connection with strongly-regular graphs and gave various characterizations of partial geometries, bipartite graphs, symmetric 2-designs, and transversal designs in terms of partial geometric designs.

Title: A Primal-Dual Extension of the Goemans--Williamson Algorithm for the Weighted Fractional Cut Covering Problem

Abstract:

A cut in a graph \(G = (V, E)\) is a set of edges which has precisely one endpoint in \(S\), for a given subset \(S\) of \(V\). The fractional cut-covering number is the optimal value of a linear programming relaxation for the problem of covering each edge by a set of cuts. We define a semidefinite programming relaxation of fractional cut covering whose approximate optimal solutions may be rounded into a fractional cut cover via a randomized algorithm.

Title: An invitation to monotone operators and their applications in optimization

Abstract: In this talk, I give an overview of the theory of monotone operators and its connection to optimization algorithms. This talk is a good introduction to how abstract theoretical results serve as bases for successful algorithms in practice.  

Title: Poset subHopf algebras from growth models in causal set theory and quantum field theory

Abstract: In a story some of you have heard from me before, we get subHopf algebras of the Connes-Kreimer Hopf algebra of rooted trees from certain simple tree classes which correspond to solutions to combinatorial analogues of Dyson-Schwinger equations in quantum field theory.  Another important subHopf algebra of the Connes-Kreimer Hopf algebra is the Connes-Moscovici Hopf algebra which can be viewed as coming from rooted trees grown by adding leaves.