All Seminars
| Title: The fractional version of Hedetniemi's Product Conjecture |
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| Seminar: Combinatorics |
| Speaker: Dwight Duffus of Emory University |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2011-01-28 at 4:00PM |
| Venue: W306 |
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| Abstract: Hedetniemi's Product Conjecture -- that the chromatic number of the categorical product of n-chromatic graphs is n-chromatic -- was made in 1967 and remains open. The fractional version of this has recently been proved by Xuding Zhu. We would like to present this, beginning with an introduction to fractional parameters and past work on the conjecture and related problems. We will also show how Zhu's result settles a conjecture of Burr, Erdos and Lovasz on chromatic Ramsey numbers. It will likely take a few presentations to get through the background material and Zhu's proof, so this will be the first of a series of seminars on this topic. |
| Title: The Euler-Kronecker constant of a number field |
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| Colloquium: N/A |
| Speaker: V. Kumar Murty of University of Toronto |
| Contact: R. Parimala, parimala@mathcs.emory.edu |
| Date: 2011-01-27 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: Following Ihara, we define an invariant $g(K)$ for each algebraic number field which for the rational numbers is Euler's constant and for imaginary quadratic fields is a constant studied by Kronecker. This invariant reflects subtle aspects of the arithmetic of the field, and also is (conjecturally) related to periods of Abelian varieties with complex multiplication. |
| Title: The field of Fourier coefficients of a modular form |
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| Seminar: Algebra and Number Theory |
| Speaker: V. Kumar Murty of University of Toronto |
| Contact: R. Parimala, parimala@mathcs.emory.edu |
| Date: 2011-01-26 at 4:00PM |
| Venue: W306 |
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| Abstract: Let $f$ be a holomorphic cusp form of level $N$ which is a normalized Hecke eigenform. The Fourier coefficients of $f$ generate a number field. If $N$ is squarefree, we show that in fact, this field is generated by a single Fourier coefficient. The result is effective. |
| Title: Gauge Theory in Four Dimensions and Mock Modular Forms |
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| Seminar: Algebra and number theory |
| Speaker: Andreas Malmendier of Colby College |
| Contact: Zachary Kent, kent@mathcs.emory.edu |
| Date: 2011-01-25 at 3:00PM |
| Venue: W306 |
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| Abstract: In physics, the moduli space of vacua for the topological N = 2 supersymmetric pure gauge theories with gauge group SO(3) is the universal elliptic curve for the modular group of level 2. Moreover, the supersymmetric gauge theory associates to each four-manifold a not necessarily holomorphic modular form of level two. I will explain why for the complex projective plane this modular form is a Mock theta function - in fact, it is one of the examples listed in Ramanujan's letter to Hardy to undermine a notoriously obscure definition. In joint work with Ken Ono, we then proved that its cusp contributions are the Donaldson invariants of $CP^2$, a conjecture made by Moore and Witten. Time permitting, I will also sketch how string theory suggests a connection of this construction to a generalized elliptic genus. |
| Title: Integrating Formalism and Pragmatism to Build Real Data Privacy Solutions |
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| Seminar: Computer Science |
| Speaker: Brad Malin of Vanderbilt University |
| Contact: Li Xiong, lxiong@mathcs.emory.edu |
| Date: 2011-01-21 at 3:00PM |
| Venue: MSC W301 |
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| Abstract: Organizations collect personal information while providing an ever-expanding set of services. The collected data can support various “secondary” uses and to protect privacy, various law requires information be rendered “de-identified” before it is repurposed. However, a growing body of evidence suggests data can be “re-identified” and the viability of such laws have been questioned. I will review how and why re-identification occurs, but also push the discussion from a deterministic view (i.e., re-identification can or cannot occur) toward a probabilistic view (i.e., the likelihood of re-identification). In doing so, I will illustrate how to construct efficient de-identification algorithms that mitigate risks without precluding the secondary endeavors. This work will draw upon experience and case studies from with multiple medical centers around the United States. Brad Malin is an Assistant Professor of Biomedical Informatics and Computer Science at Vanderbilt University, where he directs the Health Information Privacy Laboratory (HIPLab). The HIPLab is funded through grants from the NSF and NIH and its research artifacts have received awards of distinction from the American and International Medical Informatics Associations. In 2010, Dr. Malin received the Presidential Early Career Award for Scientists and Engineers (PECASE), the highest honor bestowed by the U.S. government on outstanding scientists and engineers beginning their independent careers. He completed his education at Carnegie Mellon University, where he received a bachelor’s in biological sciences, a master’s in data mining and knowledge discovery, a master’s in public policy and management, and a doctorate in computer science. |
| Title: Adding and Counting |
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| Special Lecture: Number Theory |
| Speaker: Ken Ono of Emory University |
| Contact: Susan Guppy, sguppy@mathcs.emory.edu |
| Date: 2011-01-21 at 8:00PM |
| Venue: Oxford Road Building |
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| Abstract: Professor Ono will explain how the simple task of adding and counting has fascinated many of the world's leading mathematicians. As is typical in number theory, many of the most fundamental questions have remained unsolved. In 2010, Ono, with the support of the AIM and the NSF, assembled an international team of distinguished researchers to attack some of the problems in partition numbers. He will announce their findings: new theories which solve some of these famous questions. |
| 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. |