All Seminars
| Title: Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube |
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| Seminar: Combinatorics |
| Speaker: Bernard Lidicky of The University of Illinois at Urbana-Champaign |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2013-03-22 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: |
| Title: A structured QZ method for colleague matrix pencils |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Paola Boito of Universite` de Limoges - CNRS |
| Contact: MIchele Benzi, benzi@mathcs.emory.edu |
| Date: 2013-03-21 at 4:00PM |
| Venue: W306 |
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| Abstract: In this work we present a fast structured version of the QZ algorithm designed to compute the generalized eigenvalues of a class of matrix pencils. In particular, this class includes colleague pencils arising from the zero-finding problem for polynomials expressed in the Chebyshev basis. The method relies on quasiseparable matrix structure and it is based on the representation of the relevant matrices as low rank perturbations of Hermitian or unitary matrices. The complexity for an $N\times N$ pencil is $\mathcal{O}(N^2)$, with $\mathcal{O}(N)$ memory. Numerical experiments confirm the effectiveness and practical stability of the method. |
| Title: Projected Krylov Methods for Saddle-Point Systems |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Dominique Orban of Mathematics and Industrial Engineering Department \linebreak Ecole Polytechnique de Montreal |
| Contact: Michele Benzi, benzi@mathcs.emory.edu |
| Date: 2013-03-20 at 12:50PM |
| Venue: W306 |
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| Abstract: Projected Krylov methods are full-space formulations of Krylov methods that take place in a nullspace. Provided projections into the nullspace can be computed accurately, those methods only require products between an operator and vectors lying in the nullspace. We provide systematic principles for obtaining the projected form of any well-defined Krylov method. We illustrate typical behavior on a few simple problems arising from the discretization of the Stokes and Navier-Stokes equations and describe a convenient object-oriented Matlab implementation. |
| Title: Sparse numerical linear algebra and interpolation spaces |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Mario Arioli of Rutherford Appleton Laboratory, UK |
| Contact: Michele Benzi, benzi@mathcs.emory.edu |
| Date: 2013-03-06 at 12:50PM |
| Venue: W306 |
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| Abstract: We derive discrete norm representations associated with projections of interpolation spaces onto finite dimensional subspaces. These norms are products of integer and noninteger powers of the Gramian matrices associated with the generating pair of spaces for the interpolation space. We include a brief description of some of the algorithms which allow the efficient computation of matrix powers. We consider in some detail the case of fractional Sobolev spaces both for positive and negative indices together with applications arising in preconditioning techniques. Several other applications are described. |
| Title: Selmer groups of elliptic curves |
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| Seminar: Algebra |
| Speaker: Zev Klagsbrun of University of Wisconsin-Madison |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2013-03-06 at 3:00PM |
| Venue: W306 |
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| Abstract: Recently, Poonen and Rains proposed heuristics for the distributions of Selmer groups of elliptic curves. Work dating back to Heath-Brown in 1994 suggests that these heuristics should be satisfied within quadratic twist families of elliptic curves as well. I will be presenting results of Mazur, Rubin, and myself showing that, after a parity adjustment, these heuristics are satisfied within quadratic twist families of elliptic curves with an $S_3$ 2-division field. I will also explain how our work challenges the conventional wisdom of Goldfeld’s conjecture about how ranks are distributed within quadratic twist families of elliptic curves over general number fields. |
| Title: Canonical Representatives for divisor classes on tropical curves |
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| Seminar: Algebra |
| Speaker: Farbod Shokrieh of Georgia Tech |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2013-02-27 at 3:00PM |
| Venue: W306 |
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| Abstract: Tropical curves are algebraic curves defined over the tropical semi-ring. They essentially carry the same information as metric graphs. There is a reasonable theory of divisors in this setting. For example, there is a tropical analogue of the Riemann-Roch theorem. The main technique in studying divisors on tropical curves is often to look for nice canonical representatives in linear equivalence classes. In this talk, we will describe various canonical representatives for divisor classes on tropical curves. We first revisit the concept of "reduced divisors" (which is the main ingredient needed to prove the Riemann-Roch theorem) and explain their various interpretations. We then discuss "break divisors" from multiple points of view. If time permits we discuss the classical analogues of these representatives and give some applications. No prior knowledge in the subject will be assumed. This talk is based on joint works with M. Baker, with M. Baker, G. Kuperberg, A. Yang, and with Ye Luo. |
| Title: Asymptotic distribution for the birthday problem with multiple coincidences |
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| Seminar: Combinatorics |
| Speaker: Skip Garibaldi of Emory University |
| Contact: Dwight Duffus, dwight@mathcs.meory.edu |
| Date: 2013-02-22 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: This talk is about joint work with Richard Arratia and Joe Kilian on a version of the birthday problem. We study the random variable $\mathbf{B}(c, n)$, which counts the number of balls that must be thrown into $n$ equally-sized bins in order to obtain $c$ collisions. We determine the limiting distribution for $(\mathbf{B}(c,n))^2/(2n)$ where $c$ is a function of $n$ that is $o(\sqrt{n})$, among other results. The basis for this result is a coupling. |
| Title: The non-Abelian Whitney theorem and the Higher Pairing on Graphs |
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| Seminar: Algebra |
| Speaker: Eric Katz of University of Waterloo |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2013-02-20 at 3:00PM |
| Venue: W306 |
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| Abstract: For a connected graph G with no 1-valent vertices, the set of based reduced graphs is sufficient to recover the graph. This is non-commutative invariant of the graph. Its abelianization, the cycle space of the graph is sufficient to recover the graph up to two moves by Whitney's 2-isomorphism theorem. In this talk, we will consider a unipotent invariants that interpolates between the set of paths and its abelianization. There is a related isomorphism theorem that lets you recover the graph from the analogous unipotent invariant. In the same spirit, we will introduce a unipotent pairing between paths and ordered n-tuples of cycles which was inspired by Chen's theory of iterated integrals and which generalizes the length pairing between cycles. We conjecture a higher Picard-Lefschetz theorem relating this pairing to the asymptotics of iterated integrals on degenerating families of curves, and state a sort of Torelli theorem relating the asymptotics to the dual graph of a degeneration. |
| Title: Power series expansions for modular forms |
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| Seminar: Number Theory |
| Speaker: John Voight of University of Vermont |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2013-02-13 at 3:00PM |
| Venue: W306 |
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| Abstract: We exhibit a method to numerically compute power series expansions of modular forms on a cocompact Fuchsian group, using the explicit computation a fundamental domain and linear algebra. As applications, we compute Shimura curve parametrizations of elliptic curves over a totally real field including the image of CM points, and equations for Shimura curves. |
| Title: Phase Transitions in Ramsey-Turán Theory |
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| Seminar: Combinatorics |
| Speaker: Jozsef Balogh of The University of Illinois at Urbana-Champaign |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2013-02-08 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Denote by $RT(n,L,f(n))$ the maximum number of edges of an $L$-free graph with independence number at most $f(n)$. This concept was defined by Erd\H{o}s and S\'{o}s in 1970. In this talk I will survey some of the recent progress on studying $RT(n,L,f(n))$ and some related questions. The newer results are partially joint with Hu, Lenz and Simonovits. |