All Seminars
| 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. |
| 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. |