All Seminars
| Title: A new filtration of the Magnus kernel of the Torelli group - CANCELLED |
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| Colloquium: N/A |
| Speaker: Taylor McNeill of Rice University |
| Contact: Steve Batterson, sb@mathcs.emory.edu |
| Date: 2013-02-07 at 4:00PM |
| Venue: TBA |
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| Abstract: For a surface $S$, the Torelli group is the group of orientation preserving homeomorphisms of $S$ that induce the identity on homology. The Magnus representation represents the action on $F/F''$ where $F$ is the fundamental group of $S$ and $F''$ is the second term of the derived series. For many years it was unknown whether the Magnus representation of the Torelli group is faithful. In recent years there have been many developments on this front including the result of Church and Farb that the kernel of the Magnus representation, denoted $Mag(S)$, is infinitely generated. I show that, not only is $Mag(S)$ highly non-trivial but that it also has a rich structure as a group. Specifically, I define an infinite filtration of $Mag(S)$ by subgroups, called the higher order Magnus subgroups, $M_n$. I show that for each n the quotient $M_n/M_n+1$ is infinitely generated. To do this, I define a Johnson type homomorphism on each higher order Magnus subgroup quotient and show it has a highly non-trivial image. |
| Title: Fokker-Planck Equation Method for Predicting Viral Signal Propagation in Social Networks |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Xiaojing Ye of Georgia Institute of Technology |
| Contact: James Nagy, nagy@math.cs.emory.edu |
| Date: 2013-02-06 at 12:50PM |
| Venue: W306 |
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| Abstract: We consider the modeling and computations of random dynamical processes of viral signals propagating over time in social networks. The viral signals of interests can be popular tweets on trendy topics in social media, or computer malware on the Internet, or infectious diseases spreading between human or animal hosts. The viral signal propagations can be modeled as diffusion processes with various dynamical properties on graphs or networks, which are essentially different from the classical diffusions carried out in continuous spaces. We address a critical computational problem in predicting influences of such signal propagations, and develop a discrete Fokker-Planck equation method to solve this problem in an efficient and effective manner. We show that the solution can be integrated to search for the optimal source node set that maximizes the influences in any prescribed time period. This is a joint work with Profs. Shui-Nee Chow (GT-MATH), Hongyuan Zha (GT-CSE), and Haomin Zhou (GT-MATH). |
| Title: The degrees of divisors of $x^n-1$ |
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| Seminar: Number Theory |
| Speaker: Lola Thompson of University of Georgia |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2013-02-06 at 3:00PM |
| Venue: W306 |
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| Abstract: We discuss what is known about the following questions concerning the degrees of divisors of $x^n-1 in Z[x]$, as n ranges over the natural numbers:\\ \\ 1. How often does $x^n-1$ have AT LEAST ONE divisor of every degree between 1 and n?\\ \\ 2. How often does $x^n-1$ have AT MOST ONE divisor of every degree between 1 and n?\\ \\ 3. How often does $x^n-1$ have EXACTLY ONE divisor of every degree between 1 and n?\\ \\ 4. For a given m, how often does $x^n-1$ have a divisor of degree m?\\ \\ We will also discuss what changes when Z is replaced by the finite field $F_p$. A portion of this talk is based on joint work with Paul Pollack. |
| Title: Knot Polynomials in the Melvin-Morton-Rozansky Expansion of the Colored Jones Polynomial |
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| Colloquium: N/A |
| Speaker: Andrea Overbay of University of North Carolina |
| Contact: TBA |
| Date: 2013-02-05 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Both the Alexander polynomial and the Jones polynomial are two well-known knot invariants. The Melvin-Morton conjecture, proved by Bar-Natan and Garoufalidis and further generalized by Rozansky, provides a relationship between these two invariants. The relationship appears when expanding the colored Jones polynomial in a certain way. Within this expansion, we get more polynomial invariants of the knot. During this talk, we will discuss some polynomial knot invariants including the Alexander polynomial and the colored Jones polynomial. Then we will describe the polynomial invariants appearing in the Melvin-Morton-Rozansky expansion for some simple knots and outline a method for computing them. |
| Title: Superimposed Codes |
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| Colloquium: N/A |
| Speaker: Zoltan Furedi of University of Illinois at Urbana-Champaign and Renyi Institute of Mathematics of the Hungarian Academy of Sciences |
| Contact: Andrzej Rucinski, andrzej@mathcs.emory.edu |
| Date: 2013-02-04 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: There are many instances in Coding Theory when codewords must be restored from partial information, like defected data (error correcting codes), or some superposition of the strings. These lead to superimposed codes, a close relative of group testing problems. There are lots of versions and related problems, like Sidon sets, sum-free sets, union-free families, locally thin families, cover-free codes and families, etc. We discuss two cases {\it cancellative} and {\it union-free} codes. A family of sets $\mathcal F$ (and the corresponding code of 0-1 vectors) is called {\bf union-free} if $A\cup B\neq C\cup D$ and $A,B,C,D\in \mathcal F$ imply $\{ A,B\}=\{ C, D \}$. $\mathcal F$ is called $t$-{\bf cancellative} if for all distict $t+2$ members $A_1, \dots, A_t$ and $B,C\in \mathcal F$ $$A_1\cup\dots \cup A_t\cup B \neq A_1\cup \dots A_t \cup C. $$ Let $c_t(n)$ be the size of the largest $t$-cancellative code on $n$ elements. We significantly improve the previous upper bounds of K\"orner and Sinaimeri, e.g., we show $c_2(n)\leq 2^{0.322n}$ (for $n> n_0$). We introduce a method to deal with such problems, namely we first investigate the constant weight case (i.e., uniform hypergraphs). |
| Title: Faces of Weight Polyhedra and some related algebraic structures |
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| Colloquium: N/A |
| Speaker: Tim Ridenour of Northwestern University |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2013-01-31 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: The set of weights which lie on a face of the weight polyhedron associated to a highest weight module can be used to construct Koszul algebras. I will provide a classification of the set of weights on a face of a weight poylhedron and discuss the construction of the related Koszul algebras. |
| Title: Group presentations: history and recent results |
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| Seminar: Algebra and Number Theory |
| Speaker: William Kantor of University of Oregon |
| Contact: Skip Garibaldi, skip@mathcs.emory.edu |
| Date: 2013-01-30 at 3:00PM |
| Venue: W306 |
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| Abstract: Group presentations somewhat started with Hamilton. I'll review 19th century history before getting to a recent result: almost all finite simple groups have presentations requiring surprisingly few relations (with Guralnick, Kassabov and Lubotzky). For example, all alternating (and symmetric) groups have presentations using only 3 generators and 7 relations. The proofs are relatively elementary but too long for more than a few small hints. |
| Title: The Geometry of Random (Closed) Space Polygons |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Professor Jason Cantarella of University of Georgia |
| Contact: Vladimir Oliker, oliker@mathcs.emory.edu |
| Date: 2013-01-29 at 4:00PM |
| Venue: W306 |
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| Abstract: Here are some natural questions from statistical physics: What is the expected shape of a ring polymer with $n$ monomers in solution? What is the expected knot type of the polymer? The radius of gyration? Numerical simulations are absolutely required to make progress on these questions, but pose some interesting challenges for geometers. We wish to sample the space of closed $n$-gons in 3-space, which is a nonlinear submanifold of the larger space of open $n$-gons. To sample equilateral polygon space, current algorithms use a Markov process which randomly "folds'' polygons while preserving closure and edgelengths. Such algorithms are expected to converge in O$(n^3)$ time. The main point of this talk is that a much better sampling algorithm is available if we widen our view to the space of $n$-gons in three dimensional space of fixed total length (rather than fixed edgelengths). We describe a natural probability measure on length 2$n$-gon space pushed forward from the standard measure on the Stiefel manifold of 2-frames in complex $n$-space using methods from algebraic geometry. We can directly sample the Stiefel manifold in O(n) time, allowing us to generate closed polygons using the pushforward map. We will give some explicit computations of expected values for geometric properties for such random polygons, discuss their topology, and compare our theorems to numerical experiments using our sampling algorithm. The talk describes joint work with Malcolm Adams (University of Georgia, USA), Tetsuo Deguchi (Ochanomizu University, Japan), and Clay Shonkwiler (University of Georgia, USA). |
| Title: A statistical-numerical approach for data fitting over two-dimensional manifolds |
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| Colloquium: N/A |
| Speaker: Bree Ettinger of |
| Contact: Steve Batterson, sb@mathcs.emory.edu |
| Date: 2013-01-29 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: In this talk, a new approach for fitting data over two-dimensional manifolds will be presented. In particular, a generalized additive model with a suitable regularizing term is proposed. The estimation problem is recast over a planar domain via a conformal map. The conformal map and the resulting planar estimation problem are computed by a finite element approximation. The estimators are linear in the observed data values and classical inferential tools are derived. The application driving this research is the study of hemodynamic forces exerted by blood-flow over the wall of an internal carotid artery affected by an aneurysm. Based on joint work with Simona Perotto and Laura Sangalli (MOX, Department of Mathematics, Politecnico di Milano, Italy) and Tiziano Passerini (Mathematics and Computer Science, Emory University) |
| Title: Adaptive spline approximation: error analysis and construction of the partitions |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Yuliya Babenko of Kennesaw State University, GA |
| Contact: Alessandro Veneziani, ale@mathcs.emory.edu |
| Date: 2013-01-28 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: |