Georgia Institute of Technology College of Sciences

Weighted Bipartite Edge Coloring problem is a generalization of two classical optimization problems: Bipartite Edge Coloring and Bin Packing. Given an edge-weighted bipartite multi-graph G, the goal is to color all edges with minimum colors such that the sum of the edges incident to any vertex of any color is at most one. Chung and Ross conjectured that given an instance of the weighted bipartite edge coloring problem, there is a proper weighted coloring using at most 2n-1 colors where n denotes the maximum over all the vertices of the number of unit-sized bins needed to pack the weights of edges incident at the vertex. In this talk I will present an algorithm that gives a proper weighted coloring using $20n/9$ colors and improved results for some special cases. I will also present an alternative proof of Konig's edge coloring theorem using skew-supermodular functions. The talk will have all three components of ACO: Approximation Algorithms, Graph Theory and Supermodular Optimization.

The immune system must simultaneously mount a response against foreign antigens while tolerating self. How this happens is still unclear as many mechanisms of immune tolerance are antigen non-specific. Antigen specific immune cells called T-cells must first bind to Immunogenic Dendritic Cells (iDCs) before activating and proliferating. These iDCs present both self and foreign antigens during infection, so it is unclear how the immune response can be limited to primarily foreign reactive T-cells. Regulatory T-cells (Tregs) are known to play a key role in self-tolerance. Although they are antigen specific, they also act in an antigen non-specific manner by competing for space and growth factors as well as modifying DC behaviorto help kill or deactivate other T-cells. In prior models, the lack of antigen specific control has made simultaneous foreign-immunity and self-tolerance extremely unlikely. We include a heterogeneous DC population, in which different DCs present antigens at different levels. In addition, we include Tolerogenic DC (tDCs) which can delete self-reactive T-cells under normal physiological conditions. We compare different mathematical models of immune tolerance with and without Tregs and heterogenous antigen presentation.For each model, we compute the final number of foreign-reactive and self-reactive T-cells, under a variety of different situations.We find that even if iDCs present more self antigen than foreign antigen, the immune response will be primarily foreign-reactive as long as there is sufficient presentation of self antigen on tDCs. Tregs are required primarily for rare or cryptic self-antigens that do not appear frequently on tDCs. We also find that Tregs can onlybe effective when we include heterogenous antigen presentation, as this allows Tregs and T-cells of the same antigen-specificity to colocalize to the same set of DCs. Tregs better aid immune tolerance when they can both compete forspace and growth factors and directly eliminate other T-cells. Our results show the importance of the structure of the DC population in immune tolerance as well as the relative contribution of different cellular mechanisms.

Here is a classical theorem. Consider a bijection (just a set map!) from the Euclidean plane to itself that takes 0 to 0 and takes the points on an arbitrary line to the points on a (possibly different line). The theorem is that such a bijection always comes from a linear map. I'll discuss various generalizations of this theorem in geometry, topology, and algebra, ending with a discussion of some recent, related research on the topology of surfaces.

A discussion of case studies on the making, giving, grading, etc. of exams, followed by course group meetings for 2401 and 2403.

Canceled due to speaker illness, date will be moved forward.

I will give an introduction to the method of Chabauty-Coleman for finding rational points on curves.

Let (X_k)_{k \geq 1} and (Y_k)_{k\geq1} be two independent sequences of independent identically distributed random variables having the same law and taking their values in a finite alphabet \mathcal{A}_m. Let LC_n be the length of the longest common subsequence of the random words X_1\cdots X_n and Y_1\cdots Y_n. Under assumptions on the distribution of X_1, LC_n is shown to satisfy a central limit theorem. This is in contrast to the Bernoulli matching problem or to the random permutations case, where the limiting law is the Tracy-Widom one. (Joint with Umit Islak)