All Seminars
| Title: On Erdos' conjecture on the number of edges in 5-cycles |
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| Seminar: Combinatorics |
| Speaker: Zoltan Furedi of Renyi Institute of Mathematics, Budapest, Hungary |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2013-12-06 at 4:00PM |
| Venue: W306 |
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| Abstract: Erdos, Faudree, and Rousseau (1992) showed that a graph on n vertices and at least $[n^2/4]+1$ edges has at least $2[n/2]+1$ edges on triangles and this result is sharp. They also considered a conjecture of Erdos that such a graph can have at most $n^2/36$ non-pentagonal edges with an extremal graph having two components, a complete graph on $[2n/3]+1$ vertices and a complete bipartite graph on the rest. This was mentioned in other papers of Erdos and also it is No. 27 in Fan Chung's problem book.\\ \\ In this talk we give a graph of $[n^2/4]+1$ edges with much more, namely $n^2/8(2+\sqrt{2})$ + $O(n) = n^2/27.31…$ pentagonal edges, disproving the original conjecture. We show that this coefficient is asymptotically the best possible.\\ \\ Given graphs H and F let $E_0(H,F)$ denote the set of edges of H which do not appear in a subgraph isomorphic to F, and let $h(n,e,F)$ denote the maximum of $|E_0(H,F)|$ among all graphs H of n vertices and e edges. We asymptotically determine $h(n,cn^2, C_3)$ and $h(n,cn^2, C_5)$ for fixed c, $1/4 < c < 1/2$. For $2k+1\ge$ 7 we establish the conjecture of Erdos et al. that $h(n,cn^2, C_{2k+1})$ is obtained from the above two-component example.\\ \\ One of our main tools (beside Szemeredi's regularity) is a new version of Zykov's symmetrization what we can apply for more graphs, simultaneously. |
| Title: Degree 3 cohomological invariants and quadratic splitting of hermitian forms |
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| Seminar: Algebra |
| Speaker: Jean-Pierre Tignol of Université Catholique de Louvain |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2013-11-26 at 4:00PM |
| Venue: W306 |
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| Abstract: Using variations on the Rost invariant of Spin groups, we define an invariant with values in $H^3(F,\mu_2)$ for hermitian forms with vanishing lower-degree invariants, and discuss the relation between this invariant and the existence of quadratic extensions of the base field that make the hermitian form split hyperbolic. (Joint work in progress with Anne Qu\'eguiner-Mathieu) |
| Title: 3-Coloring and 3-List-Coloring Graphs on Surfaces |
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| Seminar: Combinatorics |
| Speaker: Luke Postle of Emory University |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2013-11-22 at 4:00PM |
| Venue: W306 |
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| Abstract: Grotzsch proved that every triangle-free planar graph is 3-colorable. This theorem follows easily once one proves that every planar graph of girth at least five is 3-colorable. There are now three short proofs of this using three different methods: discharging, list-coloring, and the new potential technique of Kostochka and Yancey. It is natural to wonder how this can be generalized to surfaces; for example whether locally planar graphs of girth at least five are 3-colorable or whether there exists a polynomial-time algorithm to decide 3-coloring on a fixed surface. Both of these results are implied by the following more general result of Thomassen: For every surface, there exist only finitely many 4-critical graphs of girth at least five embeddable on that surface. Dvorak, Kral and Thomas provided a discharging proof of Thomassen's theorem to prove a stronger result that the number of vertices in such graphs is linear in genus. This was a needed ingredient in their proof of Havel's conjecture which states that a planar graph with triangles far apart is 3-colorable. We will discuss two new proofs of this linear bound result, one using list-coloring and one using the potential technique, and their various corollaries, such as the generalization of Thomassen's theorem to 3-list-coloring. |
| Title: The distribution of 2-Selmer ranks and additive functions |
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| Seminar: Algebra |
| Speaker: Robert Lemke Oliver of Stanford University |
| Contact: TBA |
| Date: 2013-11-21 at 5:00PM |
| Venue: W306 |
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| Abstract: The problem of determining the distribution of the 2-Selmer ranks of quadratic twists of an elliptic curve has received a great deal of recent attention, both in works conjecturing distributions and in those providing solutions; in both cases, the nature of the two-torsion of the elliptic curve plays a cruical role. In particular, if $E/\mathbb{Q}$ has full two-torsion, the distribution is known, due to work of Heath-Brown, Swinnerton-Dyer, and Kane, and if $E$ possesses no two-torsion, then, again, the distribution is known, due to work of Klagsbrun, Mazur, and Rubin, though with the caveat that one arranges discriminants in a non-standard way. In stark contrast to these two cases, we show that if $K$ is a number field and $E/K$ is an elliptic curve with partial two-torsion, then no limiting distribution on 2-Selmer ranks exists. We do so by showing that, for any fixed integer $r$, at least half of the twists of $E$ have 2-Selmer rank greater than $r$, and we establish an analogous result for simultaneous twists, either for multiple elliptic curves twisted by the same discriminant or for a single elliptic curve twisted by a tuple of discriminants. These results depend upon connecting the 2-Selmer rank of twists to the values of an additive function and then establishing results analogous to the classical Erd\H{o}s-Kac theorem. This work is joint with Zev Klagsbrun. |
| Title: Set families with a forbidden induced subposet |
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| Seminar: Combinatorics |
| Speaker: Tao Jiang of Miami university |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2013-11-15 at 4:00PM |
| Venue: W306 |
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| Abstract: Sperner’s theorem asserts that the largest antichain in a Boolean lattice Bn has size n n/2 . A few years ago, Bukh obtained a substantial asymptotic extension of Sperner’s theorem by proving that for any poset H whose Hasse diagram is a tree of height k, the largest size of a n subfamily of Bn not containing H is asymptotic to (k − 1) n/2 . We establish an induced version of Bukh’s result, namely that the largest size of a subfamily of Bn not containing H as an induced n subposet, is also asymptotic to (k −1) n/2 . This is an old result (2012). I will focus on presenting the ideas of the proof. This is joint work with Ed Boenhlein. |
| Title: The minimum number of nonnegative edges in hypergraphs |
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| Seminar: Combinatorics |
| Speaker: Hao Huang of Institute for Advanced Study and DIMACS |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2013-11-14 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: An r-unform n-vertex hypergraph H is said to have the Manickam-Miklos-Singhi (MMS) property if for every assignment of weights to its vertices with nonnegative sum, the number of edges whose total weight is nonnegative is at least the minimum degree of H. In this talk I will show that for $n>10r^3$, every r-uniform n-vertex hypergraph with equal codegrees has the MMS property, and the bound on n is essentially tight up to a constant factor. An immediate corollary of this result is the vector space Manickam-Miklos-Singhi conjecture which states that for $n \geq 4k$ and any weighting on the 1-dimensional subspaces of $F_q^n$ with nonnegative sum, the number of nonnegative k-dimensional subspaces is at least ${n-1 \brack k-1}_q$. I will also discuss two additional generalizations, which can be regarded as analogues of the Erdos-Ko-Rado theorem on k-intersecting families. This is joint work with Benny Sudakov. |
| Title: Unramified Brauer classes on cyclic covers of the projective plane |
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| Seminar: Algebra |
| Speaker: Andrew Obus of University of Virginia |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2013-11-12 at 4:00PM |
| Venue: W306 |
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| Abstract: Brauer groups on schemes have many applications, for example, giving obstructions to existence of rational points. After reviewing different ways of presenting Brauer groups of fields and schemes, we will give a method to exhibit p-torsion Brauer classes on a p-cyclic cover of the projective plane, branched over a smooth curve of degree divisible by d. This extends earlier work of van Geemen for degree 2 K3 surfaces. In the case p=2, our method gives all the 2-torsion classes, and is equivalent to another (more geometric) method of Catanese. This is joint work with Colin Ingalls, Ekin Ozman, and Bianca Viray. |
| Title: Characterization of Quasiconformal Mapping and Extremal Length Decomposition and Its Application |
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| Defense: Dissertation |
| Speaker: Wenfei Zou of Emory University |
| Contact: Wenfei Zou, wzou3@emory.edu |
| Date: 2013-11-12 at 4:00PM |
| Venue: W302 |
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| Abstract: My defense includes two parts. The first part is about the characterization of Quasiconformal mapping. It is known that Conformal mapping preserves the measure of angle. The Quasiconformal mapping is a natural generalization of the conformal mapping. Some measure of angle named topological angle could be defined to characterize Quasiconformal mappings. I will discuss these results in higher dimensional Euclidean space.\\ \\ The second part is about extremal length decomposition and its application. Quasiextremal distance domains (QED) are a class of domains introduced by Gehring and Martio in connection with Quasiconformal mapping theories. I will discuss a decomposition theorem about the extremal length of a curve family within the finitely connected QED domain. Moreover, I will discuss its application, a result of sharp upper bound for QED constant of finitely connected domain on the complex plane. |
| Title: SOUTHEAST GEOMETRY SEMINAR (SGS XXIII) |
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| Type: N/A |
| Speaker: . of . |
| Contact: Vladimir Oliker, oliker@mathcs.emory.edu |
| Date: 2013-11-10 at 8:00AM |
| Venue: MSC W201 |
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| Abstract: |
| Title: Thresholds for Random Geometric k-SAT |
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| Seminar: Combinatorics |
| Speaker: Will Perkins of Georgia Tech |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2013-11-08 at 4:00PM |
| Venue: W306 |
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| Abstract: Random $k-SAT$ is a distribution over boolean formulas studied widely in combinatorics, statistical physics, and theoretical computer science for its intriguing behavior at its phase transition. I will present results on the satisfiability threshold in a geometric model of random $k-SAT$: labeled boolean literals are placed uniformly at random in a d-dimensional cube, and for each set of k contained in a ball of radius r, a k-clause is added to the random formula. For all $k$ we show that the satisfiability threshold is sharp, and for $k=2$ we find the location of the threshold as well. I will also discuss connections between this model and the random geometric graph. |