# Seminars archive

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 Upcoming Seminars Tue02/27/20184:00pm Colloquium: AlgebraCounting points, counting fields, and heights on stacksJordan Ellenberg, University of Wisconsin-MadisonContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 59.6 kB)Show abstractThe basic objects of algebraic number theory are number fields, and the basic invariant of a number field is its discriminant, which in some sense measures its arithmetic complexity. A basic finiteness result is that there are only finitely many degree-$d$ number fields of discriminant at most $X$; more generally, for any fixed global field $K$, there are only finitely many degree-$d$ extensions $L/K$ whose discriminant has norm at most $X$. (The classical case is where $K = \mathbb{Q}$.) \\ When a set is finite, we greedily ask if we can compute its cardinality. Write $N_d(K,X)$ for the number of degree-$d$ extensions of $K$ with discriminant at most $d$. A folklore conjecture holds that $N_d(K,X)$ is on order $c_d X$. In the case $K = \mathbb{Q}$, this is easy for $d=2$, a theorem of Davenport and Heilbronn for $d=3$, a much harder theorem of Bhargava for $d=4$ and 5, and completely out of reach for $d > 5$. More generally, one can ask about extensions with a specified Galois group $G$; in this case, a conjecture of Malle holds that the asymptotic growth is on order $X^a (\log X)^b$ for specified constants $a,b$. \\ I'll talk about two recent results on this old problem: \\ 1) (joint with TriThang Tran and Craig Westerland) We prove that $N_d(\mathbb{F}_q(t),X)) < c_{\epsilon} X^{1+\epsilon}$ for all $d$, and similarly prove Malle’s conjecture up to epsilon" — this is much more than is known in the number field case, and relies on a new upper bound for the cohomology of Hurwitz spaces coming from quantum shuffle algebras: https://arxiv.org/abs/1701.04541 \\ 2) (joint with Matt Satriano and David Zureick-Brown) The form of Malle's conjecture is very reminiscent of the Batyrev-Manin conjecture, which says that the number of rational points of height at most $X$ on a Batyrev-Manin variety also grows like $X^a (\log X)^b$ for specified constants $a,b$. What’s more, an extension of $\mathbb{Q}$ with Galois group $G$ is a rational point on a Deligne--Mumford stack called $BG$, the classifying stack of $G$. A natural reaction is to say “the two conjectures is the same; to count number fields is just to count points on the stack BG with bounded height?” The problem: there is no definition of the height of a rational point on a stack. I'll explain what we think the right definition is, and explain how it suggests a heuristic which has both the Malle conjecture and the Batyrev--Manin conjecture as special cases. Thu03/01/20183:00pm Defense: DissertationOn Cycles, Chorded Cycles, and Degree ConditionsAriel Keller, Emory UniversityContact: Ariel Keller, ariel.keller@emory.eduVenue: MSC N301Download printable flyer (PDF, 50.4 kB)Show abstractSufficient conditions to imply the existence of certain substructures in a graph are of considerable interest in extremal graph theory, and conditions that guarantee a large set of cycles or chorded cycles are a recurring theme. This dissertation explores different degree sum conditions that are sufficient for finding a large set of vertex-disjoint cycles or a large set of vertex-disjoint chorded cycles in a graph. \vskip.1in For an integer $t\ge 1$, let $\sigma_t (G)$ be the smallest sum of degrees of $t$ independent vertices of $G$. We first prove that if a graph $G$ has order at least $7k+1$ and degree sum condition $\sigma_4(G)\ge 8k-3$, with $k\ge 2$, then $G$ contains $k$ vertex-disjoint cycles. Then, we consider an equivalent condition for chorded cycles, proving that if $G$ has order at least $11k+7$ and $\sigma_4(G)\ge 12k-3$, with $k\ge 2$, then $G$ contains $k$ vertex-disjoint chorded cycles. We prove that the degree sum condition in each result is sharp. Finally, we conjecture generalized degree sum conditions on $\sigma_t(G)$ for $t\ge 2$ sufficient to imply that $G$ contains $k$ vertex-disjoint cycles for $k \ge 2$ and $k$ vertex-disjoint chorded cycles for $k \ge 2$. This is joint work with Ronald J. Gould and Kazuhide Hirohata. Tue03/27/20184:00pm Seminar: AlgebraTitle to be announcedNathan Kaplan, UC IrvineContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 19.2 kB) Mon04/02/20184:00pm Defense: DissertationPatching and local-global principles for gerbes with an application to homogeneous spacesBastian Haase, Emory UniversityContact: Bastian Haase, bastian.haase@emory.eduVenue: Mathematics and Science Center, Room W302Download printable flyer (PDF, 44.9 kB)Show abstractStarting in 2009, Harbater and Hartmann introduced a new patching setup for semi-global fields, establishing a patching framework for vector spaces, central simple algebras, quadratic forms and other algebraic structures. In subsequent work with Krashen, the patching framework was refined and extended to torsors and certain Galois cohomology groups. After describing this framework, we will discuss an extension of the patching equivalence to bitorsors and gerbes. Building up on these results, we then proceed to derive a characterisation of a local- global principle for gerbes and bitorsors in terms of factorization. These results can be expressed in the form of a Mayer-Vietoris sequence in non-abelian hypercohomology with values in the crossed-module $G->Aut(G)$. After proving the local-global principle for certain bitorsors and gerbes using the characterization mentioned above, we conclude with an application on rational points for homogeneous spaces. Tue04/03/20184:00pm Seminar: AlgebraTitle to be announcedJennifer Berg, RiceContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 19.4 kB) Thu04/05/20184:00pm ColloquiumTitle to be announcedSherry Li, Lawrence Berkeley National LabContact: Lar Ruthotto, lruthotto@emory.eduVenue: Mathematics and Science Center, Room W201Download printable flyer (PDF, 19.3 kB) Thu04/12/20184:00pm Colloquium: AlgebraTitle to be announcedK. Soundararajan, Stanford UniversityContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 19.7 kB) Tue04/17/20184:00pm Seminar: AlgebraTitle to be announcedBrandon William, UC BerkeleyContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 19.1 kB) Tue04/24/20184:00pm Seminar: AlgebraTitle to be announcedFrank Thorne, University of South CarolinaContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 19.3 kB) Past Seminars Wed04/10/20133:00pm Seminar: AlgebraMahler measures of hypergeometric families of Calabi-Yau varietiesDetchat Samart, Texas A\&MContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 58.6 kB)Show abstractThe (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. Mon04/08/20133:05pm Defense: Masters ThesisAutomatic Transcription of Polyphonic Musical Signals with Linear Matching PursuitAndrew McLeod, Emory UniversityContact: Andrew McLeod, apmcleo@emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 40.8 kB)Show abstractThe 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. Thu04/04/20132:30pm Defense: DissertationTopics in analytic number theoryRobert Lemke Oliver, Emory UniversityContact: Robert Lemke Oliver, rlemkeo@emory.eduVenue: Mathematics and Science Center, Room E408Download printable flyer (PDF, 40.2 kB)Show abstractIn 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. Wed04/03/20134:00pm Seminar: AlgebraOn derived Witt groups of algebraic varietiesJeremy Jacobson, Fields institute of TorontoContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 36.4 kB)Show abstractThe 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. Wed04/03/20134:00pm Defense: DissertationOn Problems in extremal graph theory and Ramsey theorySteven La Fleur, Emory UniversityContact: Steven La Fleur, slafeu@emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 64.3 kB)Show abstractExtremal 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. Wed04/03/20133:00pm Seminar: Algebra and Number TheoryHomogeneous spaces over function fields of dimension twoYi Zhu, University of UtahContact: R. Parimala, parimala@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 40.4 kB)Show abstractLet $K$ 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 $K$. Mon04/01/20134:00pm Seminar: CombinatoricsA Double Exponential Bound on Folkman NumbersAndrzej Rucinski, Emory University and Adam Michiwicz UniversityContact: Dwight Duffus, dwight@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Download attached abstract (PDF, 135 kB)Download printable flyer (PDF, 23.4 kB) Mon04/01/20131:00pm Defense: DissertationIterative Polyenergetic Digital Tomosynthesis Reconstructions for Breast Cancer ScreeningVeronica Mejia Bustamante, Emory UniversityContact: Veronica Mejia Bustamante, vmejia@emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 42.3 kB)Show abstractIn 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. Fri03/29/20134:00pm Seminar: CombinatoricsHypergraph Turán and Ramsey results on linear cyclesTao Jiang, Miami UniversityContact: Dwight Duffus, dwight@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 22.2 kB) Wed03/27/20133:00pm Seminar: AlgebraReductions of CM j-invariants modulo pBianca Viray, Brown UniversityContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 44.6 kB)Show abstractThe 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.