All Seminars
| Title: Mahler measures of hypergeometric families of Calabi-Yau varieties |
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| Seminar: Algebra |
| Speaker: Detchat Samart of Texas A\&M |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2013-04-10 at 3:00PM |
| Venue: W306 |
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| Abstract: The (logarithmic) Mahler measure of an $n$-variable Laurent polynomial $P$ is defined by $m(P)=\int_0^1\cdots \int_0^1 \log |P(e^{2\pi i \theta_1},\ldots,e^{2\pi i \theta_n})|\,d\theta_1\cdots d\theta_n.$ In some certain cases, Mahler measures are known to be related to special values of $L$-functions. We will present some new results relating the Mahler measures of polynomials whose zero loci define elliptic curves, $K3$ surfaces, and Calabi-Yau threefold of hypergeometric type to $L$-values of elliptic modular forms. A part of the talk is joint work with Matt Papanikolas and Mat Rogers. |
| Title: Automatic Transcription of Polyphonic Musical Signals with Linear Matching Pursuit |
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| Defense: Masters Thesis |
| Speaker: Andrew McLeod of Emory University |
| Contact: Andrew McLeod, apmcleo@emory.edu |
| Date: 2013-04-08 at 3:05PM |
| Venue: W304 |
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| Abstract: The Harmonic Matching Pursuit (HMP) algorithm has ordered promising results in the au- tomatic transcription of audio signals. It works by decomposing the given signal into a set of harmonic atoms, and then grouping those atoms into individual notes. HMP has shown very promising results, but more research has been needed for one case: when multiple notes with rational frequency relation are played simultaneously. This situation is called the overlapping partial problem, and it is very common in music, occurring in intervals such as major thirds, perfect fourths, and perfect fifths. A few solutions have been proposed to handle this over- lapping partial problem by performing post-processing on the output of HMP (notably HMP with Spectral Smoothness (HMP SS)). In this paper, I propose an algorithm called Linear Matching Pursuit (LMP) to solve the overlapping partial problem of automatic note detection, which uses new heuristics to solve the problem with no post-processing required. LMP's run- time is independent of the number of notes present in a given audio signal, unlike HMP. My experiments show that LMP offers an improvement upon the accuracy of the HMP algorithm, though not to the extent of HMP SS, and is very robust in runtime with respect to polyphony. |
| Title: Topics in analytic number theory |
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| Defense: Dissertation |
| Speaker: Robert Lemke Oliver of Emory University |
| Contact: Robert Lemke Oliver, rlemkeo@emory.edu |
| Date: 2013-04-04 at 2:30PM |
| Venue: MSC E408 |
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| Abstract: In this thesis, the author proves results using the circle method, sieve theory and the distribution of primes, character sums, modular forms and Maass forms, and the Granville-Soundararajan theory of pretentiousness. In particular, he proves theorems about partitions and $q$-series, almost-prime values of polynomials, Gauss sums, modular forms, quadratic forms, and multiplicative functions exhibiting extreme cancellation. This includes a proof of the Alder-Andrews conjecture, generalizations of theorems of Iwaniec and Ono and Soundararajan, and answers to questions of Zagier and Serre, as well as questions of the author in the Granville-Soundararajan theory of pretentiousness.\\ \\ The talk will focus on three topics: Gauss sums over finite fields, eta-quotients and theta functions, and the pretentious view of analytic number theory. |
| Title: Homogeneous spaces over function fields of dimension two |
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| Seminar: Algebra and Number Theory |
| Speaker: Yi Zhu of University of Utah |
| Contact: R. Parimala, parimala@mathcs.emory.edu |
| Date: 2013-04-03 at 3:00PM |
| Venue: W306 |
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| Abstract: Let $K$ be either a global function field or a function field of an algebraic surface. Johan de Jong formulated the following principle: a ``rationally simply connected'' $K$-variety admits a rational point if and only if the elementary obstruction vanishes. In this talk, I will discuss how this principle works for projective homogeneous spaces. In particular, it leads to a classification-free result towards the quasi-split case of Serre's Conjecture II over $K$. |
| Title: On derived Witt groups of algebraic varieties |
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| Seminar: Algebra |
| Speaker: Jeremy Jacobson of Fields institute of Toronto |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2013-04-03 at 4:00PM |
| Venue: W306 |
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| Abstract: The Witt group of an algebraic variety is a globalization to varieties of the Witt group of a field. It is a part of a cohomology theory for varieties called the derived Witt groups. After an introduction, we recall two problems about the derived Witt groups--the Gersten conjecture and a finiteness question for varieties over a finite field--and then explain recent progress on them. |
| Title: On Problems in extremal graph theory and Ramsey theory |
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| Defense: Dissertation |
| Speaker: Steven La Fleur of Emory University |
| Contact: Steven La Fleur, slafeu@emory.edu |
| Date: 2013-04-03 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Extremal graph theory and Ramsey theory are two large subjects in graph theory. Both subjects involve finding substructures within graphs, or generalize graphs, under certain conditions. This dissertation investigates the following problems in each of these subjects.\\ \\ We consider an extremal problem regarding multigraphs with edge multiplicity bounded by a positive integer $q$. The number $a$, $0 \leq a < q$ is a jump for $q$ if, for any positive $e$, any integer $m$, and any $q$-multigraph on $n > n_0(e,a)$ vertices and at least $(a + e)(n(n-1)/2)$ edges, counting multiplicity, there is a subgraph on $m$ vertices and at least $(a+ c)(m(m-1)/2)$ edges, where $c = c(a)$ does not depend on $e$ or $m$. The Erd\H{o}s-Stone theorem implies that for $q=1$ every $a \in [0,1)$ is a jump. The problem of determining the set of jumps for $q \geq 2$ appears to be much harder. In a sequence of papers by Erd\H{o}s, Brown, Simonovits and separately Sidorenko, the authors established that every $a$ is a jump for $q = 2$ leaving the question whether the same is true for $q \geq 3$ unresolved. A later result of R\"{o}dl and Sidorenko gave a negative answer, establishing that for $q \geq 4$ some values of $a$ are not jumps. The problem of whether or not every $a \in [0,3)$ is a jump for $q = 3$ has remained open. We give a partial positive result in this dissertation, proving that every $a \in [0,2)$ is a jump for all $q \geq 3$. Additionally, we extend the results of R\"{o}dl and Sidorenko by showing that, given any rational number $r$ with $0 < r \leq 1$, that $(q - r)$ is not a jump for any $q$ sufficiently large. This is joint work with Paul Horn and Vojt\v{e}ch R\"{o}dl.\\ \\ Given two (hyper)graphs $T$ and $S$, the Ramsey number $r(T,S)$ is the smallest integer $n$ such that, for any two-coloring of the edges of $K_n$ with red and blue, we can find a red copy of $T$ or a blue copy of $S$. Similarly, the induced Ramsey number, $r_{\mathrm{ind}}(T,S)$, is defined to be the smallest integer $N$ such that there exists a (hyper)graph $R$ with the following property: In any two-coloring of the edges of $R$ with red and blue, we can always find a red \emph{induced} copy of $T$ or a blue \emph{induced} copy of $S$. In this dissertation we will discuss bounds for $r(K^{(k)}_{t,\dots,t}, K_s^{(k)})$ where $K^{(k)}_{t,\dots,t}$ is the complete $k$-partite $k$-graph with partition classes of size $t$. We also present new upper bounds for $r_{\mathrm{ind}}(S, T)$, where $T \subseteq K^{(k)}_{t,\dots,t}$ and $S \subseteq K_s^{(k)}$. This is based on joint work with D.~Dellamonica and V.~R\"odl. |
| Title: Iterative Polyenergetic Digital Tomosynthesis Reconstructions for Breast Cancer Screening |
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| Defense: Dissertation |
| Speaker: Veronica Mejia Bustamante of Emory University |
| Contact: Veronica Mejia Bustamante, vmejia@emory.edu |
| Date: 2013-04-01 at 1:00PM |
| Venue: W306 |
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| Abstract: In digital tomosynthesis imaging, multiple projections of an object are obtained along a small range of incident angles in order to reconstruct a pseudo 3D representation of the object. This technique is of relevant interest in breast cancer screening since it eliminates the problem of tissue superposition that reduces clinical performance in standard mammography. The challenge of this technique is that it is computationally and memory intensive, as it deals with millions of input pixels in order to produce a reconstruction composed of billions of voxels. Standard approaches to solve this large-scale inverse problem have relied on simplifying the physics of the image acquisition model by considering the x-ray beam to be monoenergetic, thus decreasing the number of degrees of freedom and the computational complexity of the solution. However, this approach has been shown to introduce beam hardening artifacts to the reconstructed volume. Beam hardening occurs when there is preferential absorption of low-energy photons from the x-ray by the object, thus changing the average energy of the x-ray beam.\\ \\ This thesis presents an interdisciplinary collaboration to overcome the mathematical, computational, and physical constraints of standard reconstruction methods in digital tomosynthesis imaging. We begin by developing an accurate polyenergetic mathematical model for the image acquisition process and propose a stable numerical framework to iteratively solve the nonlinear inverse problem arising from this model. We provide an efficient and fast implementation of the volume reconstruction process that exploits the parallelism available on the GPU architecture. Under our framework, a full size clinical data set can be reconstructed in under five minutes. The implementation presented reduces storage and communication costs by implicitly storing operators and increasing kernel functionality. We show that our reconstructed volume has no beam hardening artifacts and has better image quality than standard reconstruction methods. Our reconstructions also provide a quantitative measure for each voxel of the volume, allowing the physician to see and measure the contrast between materials present inside the breast. The research presented in this thesis shows that large-scale medical imaging reconstructions can be done using physically accurate models by effectively harnessing the multi-threading power of GPUs. |
| Title: A Double Exponential Bound on Folkman Numbers |
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| Seminar: Combinatorics |
| Speaker: Andrzej Rucinski of Emory University and Adam Michiwicz University |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2013-04-01 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: |
| Title: Hypergraph Turán and Ramsey results on linear cycles |
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| Seminar: Combinatorics |
| Speaker: Tao Jiang of Miami University |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2013-03-29 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: |
| Title: Reductions of CM j-invariants modulo p |
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| Seminar: Algebra |
| Speaker: Bianca Viray of Brown University |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2013-03-27 at 3:00PM |
| Venue: W306 |
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| Abstract: The moduli space of elliptic curves contains infinitely many algebraic points that correspond to curves with complex multiplication. In 1985, Gross and Zagier proved that the $\mathfrak{p}$-adic valuation of the difference of two CM j-invariants is exactly half the sum (over n) of the number of isomorphisms between the corresponding elliptic curves modulo $\mathfrak{p}^n$. Using this relation, Gross and Zagier proved an elegant formula for the factorization of the norm of differences of CM j-invariants, assuming that the CM orders are maximal and have relatively prime discriminants. We generalize their result to the case where one order has squarefree discriminant and the other order is arbitrary. This is joint work with Kristin Lauter. |