All Seminars
| Title: The Weierstrass mock modular form and modular elliptic curves |
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| Seminar: Algebra |
| Speaker: Ken Ono of Emory |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2013-09-11 at 4:00PM |
| Venue: W306 |
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| Abstract: No abstract. |
| Title: On a Conjecture of Thomassen |
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| Seminar: Combinatorics |
| Speaker: Domingos Dellamonica of Emory University and the University of Sao Paulo |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2013-09-06 at 4:00PM |
| Venue: W306 |
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| Abstract: About thirty years ago, Thomassen announced the following conjecture in graph theory: for all positive integers k, g there exists some D such that any graph with average degree at least D must contain a subgraph which has average degree at least k AND at the same time does not contain any cycle of length g or smaller. The conjecture is still open but it is known to be true with some additional constraints on the graph or when g < 6. This seminar will present joint work with Daniel Martin, Vaclav Koubek, and Vojta Rodl. |
| Title: A determining condition of Hilbert modular forms |
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| Seminar: Algebra |
| Speaker: Yuuki Takai of Keio University |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2013-09-05 at 4:00PM |
| Venue: W306 |
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| Abstract: The elliptic holomorphic modular forms of weight $k$ and level $\Gamma_1(N)$ are determined by the first $(k/12)[\Gamma_1(1): \Gamma_1(N)]$ Fourier coefficients. The mod $\ell$ analogue of the fact is called Sturm's theorem. In this talk, I will give a generalization of the first fact and Sturm's theorem for Hilbert modular forms. To prove them, I use geometric properties of some compacitifications of Hilbert modular varieties. |
| Title: Mathematical Models and Numerical Methods for Wavefront Reconstruction |
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| Defense: Dissertation |
| Speaker: Qing Chu of Emory University |
| Contact: Qing Chu, chu@emory.edu |
| Date: 2013-08-27 at 11:00AM |
| Venue: MSC W301 |
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| Abstract: Obtaining high resolution images of space objects from ground based telescopes is challenging, and often requires computational post processing methods to remove blur caused by atmospheric turbulence. In order for an image deblurring (deconvolution) algorithm to be effective, it is important to have a good approximation of the blurring operator. In space imaging, the blurring operator is defined in terms of the wavefront of light, and how it is distorted as it propagates through the atmosphere.\\ \\ In this thesis we consider new mathematical models and algorithms to reconstruct the wavefront, which requires solving a large scale ill-posed inverse problem. We show that by exploiting and fusing information from multiple measurements, we are able to obtain better reconstructed wavefronts than existing methods. In addition, we present results of a parallel implementation utilizing the Trilinos project. |
| Title: Maass forms and quantum modular forms |
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| Defense: Dissertation |
| Speaker: Larry Rolen of Emory University |
| Contact: Larry Rolen, lrolen@emory.edu |
| Date: 2013-06-26 at 1:00PM |
| Venue: W304 |
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| Abstract: This thesis describes several new results in the theory of harmonic Maass forms and related objects. Maass forms have recently led to a flood of applications throughout number theory and combinatorics, especially following their development by the work of Bruinier and Funke and the interpretation of Ramanujan's mock theta functions in this framework by Zwegers. Here, we will prove results on integrality of singular moduli and we will revisit Ramanujan's original definition of a mock theat function. Furthermore, we will construct a new example of a quantum modular form using ``strange'' series and sums of tails formulas. |
| Title: The generalized Sato-Tate conjecture |
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| Seminar: Algebra |
| Speaker: Andrew Sutherland of MIT |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2013-05-15 at 3:00PM |
| Venue: W306 |
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| Abstract: The Sato-Tate conjecture is concerned with the statistical distribution of the number of points on the reduction modulo primes of a fixed elliptic curve defined over the rational numbers. It predicts that this distribution can be explained in terms of a random matrix model, using the Haar measure on the special unitary group SU(2). Thanks to recent work by Richard Taylor and others, this conjecture is now a theorem. The Sato-Tate conjecture generalizes naturally to abelian varieties of dimension g, where it associates to each such abelian variety a compact subgroup of the unitary symplectic group USp(2g), the Sato-Tate group, whose Haar measure governs the distribution of certain arithmetic data attached to the abelian variety. While the Sato-Tate conjecture remains open for all g>1, I will present recent work that has culminated in a complete classification of the Sato-Tate groups that can arise when g=2 (and proofs of the Sato-Tate conjecture in some special cases), and highlight some of the ongoing work in dimension 3. I will also present numerical computations that support the conjecture, along with animated visualizations of this data. This is joint work with Francesc Fit\'{e}, Victor Rotger, and Kiran S. Kedlaya, and also with David Harvey. |
| Title: Deficient, abundant, perfect, and all that |
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| Seminar: Algebra |
| Speaker: Paul Pollack of UGA |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2013-05-08 at 3:00PM |
| Venue: W306 |
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| Abstract: Let $\sigma(n)$ be the usual sum-of-divisors function. The ancient Greeks put the natural numbers into three categories: deficient numbers, for which $\sigma(n) < 2n$, abundant numbers, for which $\sigma(n) > 2n$, and perfect numbers, for which $\sigma(n)=2n$. While early discussion of these numbers has more in common with numerology than with number theory, the 20th century saw great progress in understanding how these numbers were distributed within the sequence of natural numbers. I will survey the problems, the known methods and results, and the (numerous!) still unresolved questions in this area. Some of this represents joint work with Mits Kobayashi and Carl Pomerance. |
| Title: Visual Thinking in Autism, Psychometrics, and AI: The Case of the Raven's Progressive Matrices Test |
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| Seminar: Computer Science |
| Speaker: Maithilee Kunda of Georgia Tech |
| Contact: Eugene Agichtein, eugene@mathcs.emory.edu |
| Date: 2013-05-02 at 12:00PM |
| Venue: MSC W301 |
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| Abstract: How do humans perform high-level reasoning and problem solving? Much of cognitive science research and almost all of AI research into problem solving has focused on the use of verbal or amodal propositional representations, despite the growth of evidence from neuroscience showing that many mental representations function as modal perceptual symbols. In this talk, I will discuss the role of iconic mental representations in high-level problem-solving tasks. I will first examine the notion of whether certain individuals with autism may have a bias towards "thinking visually." I will then focus on one problem-solving domain in particular: the Raven's Progressive Matrices test, which represents one of the single best psychometric measures of general intelligence that has yet been developed. I will describe previous computational theories of problem solving on the Raven's test, which have all been propositional in nature, and then present a new computational model, the ASTI model, which uses purely visual operations akin to those used in mental imagery. I will end by discussing implications of the model for our evolving understanding of cognition in autism, general human cognition, and computational views of intelligence. |
| Title: Symbol length over $C_r$ fields |
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| Seminar: Algebra and Number Theory |
| Speaker: Eli Matzri of University of Virginia |
| Contact: Skip Garibaldi, skip@mathcs.emory.edu |
| Date: 2013-05-01 at 3:00PM |
| Venue: W306 |
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| Abstract: A field $F$ is called $C_r$ if every homogenous form of degree $n$ in more then $n^r$ variables has a non-trivial solution. Consider a central simple algebra $A$ of exponent $n$ over a field $F$. By the Merkurjev-Suslin theorem assuming $F$ contains a primitive $n$-th root of unity, $A$ is similar to the product of symbol algebras. The smallest number of symbols required is called the \emph{length} of $A$ and is denoted $l(A)$. If $F$ is $C_r$ we prove $l(A) \leq n^{r-1}-1$. In particular the length is independent of the index of $A$. |
| Title: On Erdos-Ko-Rado-type theorems |
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| Colloquium: N/A |
| Speaker: Peter Frankl of The Hungarian Academy of Sciences |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2013-05-01 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: The lecture is going to focus on extremal set theory. The general problem is concerned with the maximum possible size of a subset of the power set of a finite set $X$ of $n$ elements subject to some conditions. The simplest result is probably the following.\\ \\ Proposition 0. If $F$ is a subset of $2^X$, such that any two sets in $F$ have non-empty intersection then $|F| \leq 2^(n-1)$.\\ \\ One way to achieve equality is by taking all subsets containing a fixed element.\\ \\ Erdös-Ko-Rado Theorem. If $F$ is a collection of $k$-element subsets of $X$ such that any two sets in $F$ have non-empty intersection and $2k < n$ , then $|F| \leq {n-1 \choose k-1}$ with equality holding only if all subsets in $F$ contain a fixed element. We are going to discuss various generalizations and extensions of this result, some of which are still unsolved. |