All Seminars
| Title: Fractional perfect matchings in hypergraphs |
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| Seminar: Combinatorics |
| Speaker: Andrzej Rucinski of Adam Mickiewicz University, Poznan, Poland and Emory University, Atlanta |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2010-11-19 at 4:00PM |
| Venue: W306 |
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| Abstract: A perfect matching in a k-uniform hypergraph H=(V,E) on n vertices is a set of n/k disjoint edges of H, while a fractional perfect matching in H is a function w assigning to each edge of H a real number from [0,1] in such a way that for each vertex v the sum of the weights of the edges containing v equals 1. Given n>3 and 2< k< n, let m be the smallest integer such that whenever the minimum vertex degree in H is at least m then H contains a perfect matching, and let m* be defined analogously with respect to fractional perfect matchings. Clearly, m* does not exceed m. We prove that for large n, m and m* are asymptotically equal, and suggest an approach to determine m*, and consequently m, utilizing the Farkas Lemma. This is a joint work with Vojta Rodl. |
| Title: Filling invariants at infinity |
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| Seminar: Topology |
| Speaker: Pallavi Dani of Louisiana State University |
| Contact: Aaron Abrams, abrams@mathcs.emory.edu |
| Date: 2010-11-17 at 2:00PM |
| Venue: MSC E408 |
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| Abstract: The $k$-dimensional isoperimetric function of a space captures the difficulty of filling $k$-spheres with $(k+1)$-balls in the space. Once one understands the isoperimetric functions of a space, it is interesting to study how they change when an obstruction is introduced. In this spirit, Brady and Farb introduced the notion of ``filling invariants at infinity'', by considering the volume required to fill spheres in Hadamard manifolds, provided both the sphere and the filling are far from a fixed basepoint. I will talk about a group theoretic version of this concept, and describe joint work with Aaron Abrams, Noel Brady, Moon Duchin and Robert Young on the case of right-angled Artin groups. |
| Title: Projective modules over an affine algebra |
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| Seminar: Algebra and number theory |
| Speaker: Ravi A. Rao of Tata Institute of Fundamental Research |
| Contact: R. Parimala, parimala@mathcs.emory.edu |
| Date: 2010-11-16 at 3:00PM |
| Venue: MSC E408 |
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| Abstract: The study of finitely generated projective modules along the lines indicated by J-P.~Serre and H.~Bass is outlined; its continuation by the three step Eisenbud-Evans program is then described. The development of the first part of the EE-program by Mohan Kumar, Pavaman Murthy, Madhav Nori's Euler classes vision and Bhatwadekar-Sridharan's completion of it, the functorial French approach of Barge-Morel, and Morel's later ideas are touched. The thesis of Andrei Suslin which touches on the second part of the EE program, his conjecture at his Helsinki talk in 1978, and Jean Fasel's breakthrough work on threefolds, and recent developments of the work of Fasel-Rao-Swan in higher dimension are surveyed. |
| Title: Commensurability classes of hyperbolic knot complements |
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| Seminar: Topology |
| Speaker: Neil Hoffman of University of Texas |
| Contact: Aaron Abrams, abrams@mathcs.emory.edu |
| Date: 2010-11-10 at 2:00PM |
| Venue: MSC E408 |
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| Abstract: Two manifolds are commensurable if they share a common finite sheeted cover. In 2008, Reid and Walsh conjectured that there are at most 3 hyperbolic knot complements in a given commensurability class. Recently, Boileau, Boyer, Cebanu, and Walsh have shown that the conjecture holds in the case where the knot complements do not admit hidden symmetries. After introducing the necessary ideas, we will talk about the case where we assume hidden symmetries exist. |
| Title: Eta-quotients and theta functions |
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| Seminar: Algebra and number theory |
| Speaker: Robert Lemke Oliver of Emory University |
| Contact: Ken Ono, ono@mathcs.emory.edu |
| Date: 2010-11-09 at 3:00PM |
| Venue: MSC E408 |
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| Abstract: The Jacobi Triple Product Identity gives a closed form for many infinite product generating functions that arise naturally in combinatorics and number theory. Of particular interest is its application to Dedekind's eta-function $\eta(z)$, defined via an infinite product, giving it as a certain kind of infinite sum known as a theta function. Using the theory of modular forms, we classify all eta-quotients that are theta functions. |
| Title: Special values of function field Dirichlet L-functions |
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| Seminar: Algebra |
| Speaker: Matt Papanikolas of Texas A\&M University |
| Contact: Zachary A. Kent, kent@mathcs.emory.edu |
| Date: 2010-11-02 at 3:00PM |
| Venue: MSC E408 |
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| Abstract: Similar to Euler's formula that values of the Riemann zeta function at positive even integers are rational multiples of powers of pi, one knows that values of Dirichlet L-functions at positive integers are also expressible in terms of powers of pi and values of polylogarithms at algebraic numbers. In this talk we will focus on finding analogies of these results over function fields of positive characteristic. In particular, we will consider special values of Goss L-functions for Dirichlet characters, which take values in the completion of the rational function field in one variable over a finite field. Building on work of Anderson for the case of L(1,chi), we deduce various power series identities on tensor powers of the Carlitz module that are "log-algebraic" and in turn use these formulas to determine exact values of L(n,chi) for arbitrary n > 0. Moreover, we relate these L-series values to powers of the Carlitz period and values of Carlitz polylogarithms at algebraic points. |
| Title: Celebration of Mind - The Mathematics, Magic and Mystery of Martin Gardner |
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| Seminar: Combinatorics |
| Speaker: Colm Mulcahy of Spelman College |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2010-10-29 at 4:00PM |
| Venue: W306 |
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| Abstract: Martin Gardner (1914-2010) "brought more mathematics to more millions than anyone else," according to Elwyn R. Berlekamp, John H. Conway and Richard K. Guy. Who was this man, how was he so influential, and will his legacy matter in the 22nd century? We'll try to answer these questions. This event is part of a global celebration this month of the life of Martin Gardner. \\ \\ See www.g4g-com.org and www.spelman.edu/~colm |
| Title: Metanumerical computing for partial differential equations: the Sundance project |
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| Seminar: Scientific Computing |
| Speaker: Robert Kirby of Department of Mathematics and Statistics, Texas Tech University |
| Contact: Michele Benzi, benzi@mathcs.emory.edu |
| Date: 2010-10-28 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Metanumerical computing deals with computer programs that use abstract mathematical structure to manipulate, generate, and/or optimize compute-intensive numerical codes. This idea has gained popularity over the last decade in several areas of scientific computing, including numerical linear algebra, signal processing, and partial differential equations. The Sundance project is such an example, using high-level software-based differentiation of variational forms to automatically produce high-performance finite element implementations, all within a C++ library. In addition to automating the discretization of PDEs by finite elements, recent work is demonstrating how to produce block-structured matrices and streamline the implementation of advanced numerical methods. I will conclude with some examples of this for some incompressible flow problems. |
| Title: The special fiber of a parahoric group scheme |
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| Seminar: Algebra and number theory |
| Speaker: George McNinch of Tufts University |
| Contact: Skip Garibaldi, skip@mathcs.emory.edu |
| Date: 2010-10-26 at 3:00PM |
| Venue: MSC E408 |
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| Abstract: Let $G$ be a connected and reductive algebraic group over the field of fractions $K$ of a complete discrete valuation ring $A$ with residue field $k$. Bruhat and Tits have associated with $G$ certain smooth $A$-group schemes $P$ --- called parahoric group schemes --- which have generic fiber $P/K = G$. The special fiber $P/k$ of such a group scheme is a linear algebraic group over $k$, and in general it is not reductive. In some recent work, it was proved that $P/k$ has a Levi factor in case $G$ splits over an unramified extension of $K$. Even more recently, this result was (partially) extended to cover the case where G splits over a tamely ramified extension. The talk will discuss these results and some applications. In particular, it will mention possible applications to the description of the scheme-theoretic centralizer of suitable nilpotent sections in Lie$(P)(A)$. |
| Title: Directed immersions of closed manifolds |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Professor Mohammad Ghomi of Georgia Institute of Technology |
| Contact: Vladimir Oliker, oliker@mathcs.emory.edu |
| Date: 2010-10-26 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Given any finite subset X of the sphere $S^n, n>1$, which includes no pairs of antipodal points, we explicitly construct smooth immersions of closed hypersurfaces in Euclidean space $R^{n+1}$ whose Gauss map misses X. In particular, this answers a question of Gromov. |