All Seminars
| Title: Field Patching and Galois Cohomology -- Indecomposable and Noncrossed Product Division Algebras over Curves |
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| Defense: Dissertation |
| Speaker: Feng Chen of Emory University |
| Contact: Feng Chen, fchen@emory.edu |
| Date: 2010-07-09 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Let $T$ be a complete discrete valuation ring and let $\hat X$ be a smooth projective $T$-curve. In this talk I will talk about construction of indecomposable and noncrossed product division algebras over $F$, which is the function field of $\hat X$.\\ \\ The construction is based on the technique "patching over fields", which was proposed by Harbater and Hartmann. In this talk I will recall the technique and present its application to Galois cohomology. In particular, I will apply this patching technique to construct an index preserving section ${\mathrm Br}(\hat F)\to{\mathrm Br}(F)$ (where $\hat F$ is the completion of $F$ with respect to the valuation induced by the closed fibre), which splits the restriction and use this section to lift indecomposable and noncrossed product division algebras over $\hat F$ to $F$. |
| Title: On $K_t$-Saturated Graphs |
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| Defense: Dissertation |
| Speaker: Kinnari Amin of Emory University |
| Contact: Kinnari Amin, kinnari.amin@emory.edu |
| Date: 2010-07-07 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Let $G$ be a graph on $n$ vertices. Let $H$ be a graph. Any $H$-free graph $G$ is called $H$-saturated if the addition of any edge $e \notin E(G)$ results in $H$ as a subgraph of $G$. The minimum size of an $H$-saturated graph on $n$ vertices is denoted by $sat(n, H)$. The edge spectrum for the family of graphs with property $P$ is the set of all sizes of graphs with property $P$.\\ \\ In this talk, I will present the results about the edge spectrum of $K_4$-saturated graphs. I will show that there is a $K_{4}$-saturated graph $G$ if and only if either $G$ is complete tripartite graph or $3n -8 \leq |E(G)| \leq \lfloor \frac{n^2 - n + 4}{3} \rfloor$. I will also classify all $K_{4}$-saturated graph with $\kappa(G)=2$ and $\kappa(G)=3$. I will present the result on the edge spectrum of $K_t$-saturated graphs for $t \geq 5$. I will show that, for $n \geq 5t-7$, there is an $(n, m) \mbox{ } K_t$-saturated graph $G$ if and only if $G$ is complete $(t-1)$-partite graph or $(t-1)(n-\frac{t}{2}) - 2 \leq m \leq \lfloor \frac{(t-2)n^2 - 2n + (t-2)}{2(t-1)} \rfloor + 1$. |
| Title: Combinatorial Analysis of Go Endgame Positions |
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| Defense: Master's Thesis |
| Speaker: Jacob McMillen of Emory University |
| Contact: Jake McMillen, jamcmil@emory.edu |
| Date: 2010-06-10 at 3:00PM |
| Venue: W302 |
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| Abstract: Go is a two player skill game of Chinese origin. Although Go is praised for having a simple rule set, the game generates tremendous complexity. As such, programmers have been unable to design Go AI programs that exceed the level of intermediate human players. By using techniques of combinatorial theory, mathematicians have recently developed methods of determining optimal play on certain classes of Go positions. It is our goal to present a survey of these analytic methods. We will first provide an introduction on the rules of Go as well as relevant concepts of combinatorial game theory. We will then proceed to solve several categories of small Go positions and finally show how to determine perfect play on a full size Go endgame by way of partitioning into solvable subgames. |
| Title: Generic Galois Extensions for Families of Finite Groups |
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| Job Talk: N/A |
| Speaker: Shuvra Gupta of University of Pennsylvania |
| Contact: Susan Guppy, sguppy@emory.edu |
| Date: 2010-05-19 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: In order to answer a question of Noether about the rationality of invariant spaces, David Saltman defined the notion of generic Galois extensions. A generic Galois extension for a group G over a field k is a versal G-torsor over a rational base and the existence of generic Galois extensions is known for certain families of finite groups due to the work of Saltman and others. We shall talk about the existence of generic Galois extensions for central extensions of symmetric groups and also for dihedral groups with certain conditions on the base field. |
| Title: The O-Z Conjecture and Ono's Fundamental Problem Number 1 |
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| Job Talk: N/A |
| Speaker: Zachary Kent of The University of Hawaii at Manoa |
| Contact: Susan Guppy, sguppy@emory.edu |
| Date: 2010-05-18 at 1:00PM |
| Venue: MSC W303 |
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| Abstract: The study of mock modular forms and mock theta functions is one of the hottest areas in number theory with important works by Bringmann, Ono, Zagier, and Zwegers, among many others. The theory, which is still in its infancy, has many applications: additive number theory, elliptic curves, mathematical physics, representation theory, etc. Despite this high level of activity, many fundamental problems remain open. The first of Ono's "Fundamental Problems" is: Find a direct method for relating the coefficients of Zagier shadows and mock modular forms. Here I will announce a solution to this problem, and along the way will announce a proof of the O-Z Conjecture. |
| Title: Combinatorial approach to an interpolation method and scaling limits for sparse random graphs |
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| Seminar: Combinatorics |
| Speaker: Prasad Tetali of Georgia Institute of Techonology |
| Contact: Vojtech Rodl, rodl@mathcs.emory.edu |
| Date: 2010-05-06 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: We establish the existence of free energy limits for several sparse random hypergraph models corresponding to certain combinatorial models on Erdos-Renyi graph G(N,c/N) and random r-regular graph G(N,r). For a variety of models, including independent sets, MAX-CUT, Coloring and K-SAT, we prove that the free energy both at a positive and zero temperature, appropriately rescaled, converges to a limit as the size of the underlying graph diverges to infinity. In the zero temperature case, this is interpreted as the existence of the scaling limit for the corresponding combinatorial optimization problem. For example, as a special case we prove that the size of a largest independent set in these graphs, normalized by the number of nodes converges to a limit w.h.p., thus resolving an open problem mentioned by several experts: Wormald '99, Aldous-Steele 2003, Bollobas-Riordan 2008, as well as Janson-Thomason 2008. Our approach is based on extending and simplifying the Guerra-Toninelli interpolation method, as well as extending the work of Franz-Leone and Montanari. We provide a simpler combinatorial approach and work with the zero temperature case (optimization) directly both in the case of Erdos-Renyi graph G(N,c/N) and random regular graph G(N,r), while the previous authors handled the zero temperature case (for other models) by taking limits of positive temperature models. In addition we establish the large deviations principle for the satisfiability property for constraint satisfaction problems such as Coloring, K-SAT and NAE-K-SAT. This is joint work with D. Gamarnik (MIT) and M. Bayati (Stanford). |
| Title: Georges de Rham's work and life before 1945 |
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| Colloquium: N/A |
| Speaker: Manuel Ojanguren of Swiss Federal Institute of Technology, Lausanne |
| Contact: R. Parimala, parimala@mathcs.emory.edu |
| Date: 2010-04-30 at 3:00PM |
| Venue: MSC W303 |
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| Abstract: The talk will deal with the following aspects of de Rham's life and mathematics: \begin{enumerate} \item ``How a duel in Strasbourg changed mathematics in Lausanne" \item Studies in Lausanne \item Thesis in Paris, de Rham's theorems in 1930 \item The idea of currents \item Rotations and Reidemeister torsion \item The war years \end{enumerate} |
| Title: Solar Concentration at the Interface between Optics and Thermodynamics |
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| Colloquium: Joint MathCS and Physics |
| Speaker: Professor Roland Winston of University of California, Merced |
| Contact: Vladimir Oliker, oliker@mathcs.emory.edu |
| Date: 2010-04-27 at 3:00PM |
| Venue: MSC Center, Planetarium |
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| Abstract: I show how the interface between optics and thermodynamics organizes and informs the design of solar concentrators. This extends from ultra-high concentrating systems of ~ 100,000 suns to stationary systems of ~ 4 X concentrations. Luminescent systems which rely on a Stokes shift to concentrate even diffuse sunlight are discussed |
| Title: Mathematical Modeling in Biology and a Problem in Phylogenetics |
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| Lecture Series: Evans/Hall |
| Speaker: Elizabeth Housworth of Indiana University |
| Contact: Steve Batterson, sb@mathcs.emory.edu |
| Date: 2010-04-27 at 4:00PM |
| Venue: Mathematics and Science Center E208 |
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| Abstract: The study of planetary motion leading to Newton's theory of gravity is the epitome of the success of mathematics in the sciences. How does the role of mathematics in biology compare to the role of mathematics in physics?\\ \\ The theory of evolution is arguably of the same caliber as the theory of planetary motion. What is the role of mathematics in this theory? We will discuss an open mathematical problem that should be accessible to most members of the audience. |
| Title: Enumerating Tree Orbits - an Application to Viral Caspid Assembly |
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| Seminar: Combinatorics |
| Speaker: Andrew Vince of University of Florida |
| Contact: Andrzej Rucinski, andrzej@mathcs.emory.edu |
| Date: 2010-04-26 at 4:00PM |
| Venue: W302 |
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| Abstract: Combinatorial methods (permutations, Mobius inversion, generating functions) are used to answer questions about the assembly of icosahedral viral shells. Although the geometric structure of the shell is fairly well understood in terms of its constituent subunits, the assembly process is not. Our approach involves the enumeration of certain labeled trees called assembly trees. |