All Seminars
| Title: Secondary Terms in the Davenport-Heilbronn Theorems |
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| Seminar: Algebra |
| Speaker: Frank Thorne of Stanford University |
| Contact: Zachary A. Kent, kent@mathcs.emory.edu |
| Date: 2011-01-20 at 3:00PM |
| Venue: MSC E406 |
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| Abstract: In a 1971 paper, Davenport and Heilbronn proved asymptotics for the number of cubic fields of bounded discriminant, and a related result for 3-torsion in class groups of quadratic fields. However, numerical experiments revealed that their formulas were a poor match for the data.\\ \\ In the cubic field case, Roberts conjectured that this poor match is explained by a secondary term of order $X^{5/6}$, and we will prove his conjecture as well as the analogous statement for class groups. We also obtain a surprising non-equidistribution in arithmetic progressions, which appears only in the secondary term.\\ \\ Our work is independent of another proof of Roberts' conjecture by Bhargava, Shankar, and Tsimerman, and uses the theory of Shintani zeta functions. This is joint work with Takashi Taniguchi. |
| Title: The average rank of elliptic curves |
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| Colloquium: N/A |
| Speaker: Manjul Bhargava of Princeton University |
| Contact: R. Parimala, parimala@mathcs.emory.edu |
| Date: 2011-01-14 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: A {\it rational elliptic curve} may be viewed as the set of solutions to an equation of the form $y^2=x^3+Ax+B$, where $A$ and $B$ are rational numbers. It is known that the rational points on this curve possess a natural abelian group structure, and the Mordell-Weil theorem states that this group is always finitely generated. The {\it rank} of a rational elliptic curve measures {\it how many} rational points are needed to generate all the rational points on the curve. There is a standard conjecture---originating in work of Goldfeld and Katz-Sarnak---that states that the {\it average} rank of all elliptic curves should be 1/2; however, it has not previously been known that the average rank is even finite! In this lecture, we describe recent work that shows that the average rank is finite (in fact, we show that the average rank is bounded by 1.5). |
| Title: MAD world and world of torsors |
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| Seminar: Algebra and number theory |
| Speaker: Vladimir Chernousov of University of Alberta |
| Contact: R. Parimala, parimala@mathcs.emory.edu |
| Date: 2010-12-07 at 3:00PM |
| Venue: MSC E408 |
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| Abstract: One of the central theorems of classical Lie theory is that all split Cartan subalgebras of a finite dimensional simple Lie algebra over an algebraically closed field are conjugate, a theorem of Chevalley. This result yields the most elegant proof that the type of the root system of a simple Lie algebra is its invariant. In infinite dimensional Lie theory maximal abelian diagonalizable subalgebras (MADs) plays the role which Cartan subalgebras plays in the classical theory. In the talk we address the problem of conjugacy of MADs in a big class of Lie algebras which are called in the literature by extended affine Lie algebras (EALA). To attack this problem we develop a bridge which connects the world of MADs in infinite dimensional Lie algebras and world of torsors over the Laurent polynomial rings. |
| Title: Loose Hamilton cycles in Hypergraphs |
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| Seminar: Combinatorics |
| Speaker: Mathias Schacht of Hamburg University |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2010-12-03 at 4:00PM |
| Venue: W306 |
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| Abstract: A classic result of G. A. Dirac in graph theory asserts that every n-vertex graph $(n > 2)$ with minimum degree at least n/2 contains a spanning (so-called Hamilton) cycle. G. Y. Katona and H. A. Kierstead suggested a possible extension of this result for k-uniform hypergraphs. There a Hamilton cycle of an n-vertex hypergraph corresponds to an ordering of the vertices such that every k consecutive (modulo n) vertices in the ordering form an edge. Moreover, the minimum degree is the minimum $(k- 1)$-degree, i.e. the minimum number of edges containing a fixed set of k - 1 vertices. V. Rodl, A. Rucinski, and E. Szemeredi verified (approximately) the conjecture of Katona and Kierstead and showed that every n-vertex, k-uniform hypergraph with minimum $(k-1)$-degree $(1/2 + o(1))n$ contains such a tight Hamilton cycle. We study the similar question for Hamilton r-cycles.\\ \\ A Hamilton r-cycle in an n-vertex, k-uniform hypergraph $(1 < r < k)$ is an ordering of the vertices and an ordered subset of the edges such that each such edge corresponds to k consecutive (modulo n) vertices and two consecutive edges intersect in precisely r vertices. We prove sufficient (and approximately best possible) minimum $(k-1)$-degree conditions for Hamilton r-cycles if $r < k/2$ and minimum 1-degree conditions for Hamilton 1-cycles in 3-uniform hypergraphs. This is joint work with E. Buss and H. Han. |
| Title: Harmonic Maass forms and periods |
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| Seminar: Algebra |
| Speaker: Jan Hendrik Bruinier of Technische Universität Darmstadt |
| Contact: Zachary A. Kent, kent@mathcs.emory.edu |
| Date: 2010-12-02 at 1:00PM |
| Venue: MSC E406 |
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| Abstract: The Fourier coefficients of automorphic forms often encode important arithmetic information, such as for instance representation numbers of quadratic forms, divisor sums, and numbers of points on elliptic curves over finite fields. In our talk we consider the coeficients of harmonic Maass forms of weight 1/2. We show that their coefficients are given by the periods of certain algebraic differentials on modular curves. As an example we consider rational elliptic curves. |
| Title: Questions on Serre's open image theorem |
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| Seminar: Algebra |
| Speaker: David Zywina of University of Pennsylvania |
| Contact: Zachary A. Kent, kent@mathcs.emory.edu |
| Date: 2010-12-02 at 3:00PM |
| Venue: MSC E408 |
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| Abstract: Elliptic curves (smooth curves of genus 1 with a fixed point) are fundamental objects in number theory and one fruitful way to study them is through their Galois representations. These representations arise by considering the natural Galois action on the torsion points of the curve. For a non-CM elliptic curve defined over a number field, a famous theorem of Serre says that the Galois action on the torsion points is "almost as large as possible". After some review and motivation, we will state a precise version of Serre's theorem. This will lead us to a series of natural questions; for example, how large/small can this action be, and what are the possible actions? We will explain some recent results that give partial answers and, time permitting, give some speculation on what one might hope to be true. |
| Title: Iterative linear solvers for PDE-constrained Optimization problems |
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| Colloquium: Numerical Analysis and Scientific Computing |
| Speaker: Andrew J. Wathen of Oxford University |
| Contact: Michele Benzi, benzi@mathcs.emory.edu |
| Date: 2010-12-02 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: The numerical approximation of Partial Differential Equation (PDE) problems leads typically to large dimensional linear or linearised systems of equations. For problems where such PDEs provide only a constraint on an Optimization problem (so-called PDE-constrained Optimization problems), the systems are many times larger in dimension. We will discuss the solution of such problems by preconditioned iterative techniques in particular where the PDEs in question are the steady Stokes equations describing incompressible fluid flow and some very recent work on the time-dependent diffusion equation. |
| Title: Tropical and Berkovich analytic curves |
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| Seminar: Algebra and Number Theory |
| Speaker: Matt Baker of Georgia Institute of Technology |
| Contact: Zachary A. Kent, kent@mathcs.emory.edu |
| Date: 2010-11-30 at 3:00PM |
| Venue: MSC E408 |
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| Abstract: We will discuss the relationship between a Berkovich analytic curve over a complete and algebraically closed non-Archimedean field and its tropicalizations, paying special attention to the natural metric structure on both sides. This is joint work with Sam Payne and Joe Rabinoff. |
| Title: Latex for Dummies |
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| Type: General interest, Math and CS |
| Speaker: Hernando Bermudez of Emory University |
| Contact: Veronica Bustamante, vmejia@emory.edu |
| Date: 2010-11-22 at 4:15PM |
| Venue: MSC W201 |
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| Abstract: A beginner/intermediate level seminar on the basics of latex for mathematics document and article writing. Graduate and undergraduates are especially welcomed! |
| Title: From Kinsey to Anonymization: Approaches to Preserving the Privacy of Survey Participants |
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| Seminar: Computer Science |
| Speaker: Raquel Hill of Indiana University |
| Contact: Li Xiong, lxiong@mathcs.emory.edu |
| Date: 2010-11-19 at 3:00PM |
| Venue: MSC W301 |
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| Abstract: Preserving the privacy of medical related data becomes even more challenging when the data is obtained from longitudinal studies that were designed to create unique profiles of individual participants. These studies may create participant profiles where each corresponding record is so unique that traditional anonymization techniques cannot be used to generalize and de-identify the record. Therefore sharing of this data with external parties becomes a lengthy process of negotiating specific use agreements. In some cases, sharing of the data among researchers within the organization that owns the data also risks privacy. Even when traditional identifiers are removed, the uniqueness of these records makes re-identification probable for anyone who has access to the complete record. During this talk, I will present a case study of a Kinsey dataset and discuss the challenges of protecting high dimensional data.\\ \\ Bio: Raquel Hill is an Assistant Professor of Computer Science in the School of Informatics and Computing. Her primary research interests are in the areas of trust and security for distributed and pervasive computing environment and privacy of medical related data. Dr. Hill’s research is funded by the National Science Foundation and the Center for Applied CyberSecurity Research (CACR). She holds B.S. and M.S. degrees in Computer Science from Georgia Tech and a Ph.D. in Computer Science from Harvard University. |