# 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. Mon03/05/20184:00pm Defense: DissertationOn Spanning Trees with few Branch VerticesWarren Shull, Emory UniversityContact: Warren Shull, warren.edward.shull@emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 53.8 kB)Show abstractHamiltonian paths, which are a special kind of spanning tree, have long been of interest in graph theory and are notoriously hard to compute. One notable feature of a Hamiltonian path is that all its vertices have degree two in the path. In a tree, we call vertices of degree at least three \emph{branch vertices}. If a connected graph has no Hamiltonian path, we can still look for spanning trees that come "close," in particular by having few branch vertices (since a Hamiltonian path would have none). \bigskip A conjecture of Matsuda, Ozeki, and Yamashita posits that, for any positive integer $k$, a connected claw-free $n$-vertex graph $G$ must contain either a spanning tree with at most $k$ branch vertices or an independent set of $2k+3$ vertices whose degrees add up to at most $n-3$. In other words, $G$ has this spanning tree whenever $\sigma_{2k+3}(G)\geq n-2$. We prove this conjecture, which was known to be sharp. 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 Tue11/11/20141:00pm Defense: Masters ThesisAssessing Motor Function in Parkinson’s Disease using a Web-based, Computerized and User-friendly ToolNoah Adler, Emory UniversityContact: Noah Adler, ndadler@emory.eduVenue: Woodruff Library, Rm. 213Download printable flyer (PDF, 45.2 kB)Show abstractAbstract: Parkinson’s disease (PD) is a neurodegenerative disease resulting in motor- and movement-related impairments. A clinical diagnosis of Parkinson’s disease requires clinically detectable motor symptoms, which do not occur until six to eight years after the nigral neurons in the brain begin to degenerate. By detecting PD at an earlier stage, patients can begin therapy sooner, and consequently receive better treatment and care. Therefore, in order to detect motor defects prior to clinical detection, we developed a web-based, user-friendly computer task called Predictive Movement and Trajectory Tracking (PMATT). This task was administered to 23 PD patients and 14 normal controls while recording computer cursor movements. Using machine learning techniques, we calculated fifteen significant motor-related behavioral metrics which strongly distinguish the two groups of patients. By implementing a J48 classifier with these behavioral metrics, over 97% of subjects were correctly classified with an AUC of 0.992. From these results, we conclude that PMATT may be a helpful tool in screening for PD. Since it is easily scalable and automated for individual use, PMATT can be effortlessly administered to the general population. Furthermore, its use in research may help provide insights into the development of motor impairment in pre-clinical PD and help track symptom progression with a higher precision than is currently possible.​ Mon11/10/20144:00pm Seminar: Analysis and Differential GeometryMathematical problems in visual sciencesProfessor Jacob Rubinstein, Israel Institute of Technology - TechnionContact: Vladimir Oliker, oliker@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 35.1 kB)Show abstractThis talk should be of general interest to mathematicians and researchers in visual science and ophthalmology. It will be accessible to graduate students. Tue11/04/20144:00pm Seminar: AlgebraJoint Athens-Atlanta number theory seminar (at Georgia Tech)Arul Shankar and Wei Zhang, Venue: Download printable flyer (PDF, 22.6 kB) Tue11/04/20141:00pm Seminar: CombinatoricsDistinct edge weights on graphsMichael Tait, The University of California, San DiegoContact: Vojtech Rodl, rodl@mathcs.emory.eduVenue: Mathematics and Science Center, Room E408Download printable flyer (PDF, 37.1 kB)Show abstractA Sidon set is a subset of an abelian group which has the property that all of its pairwise sums are distinct. Sidon sets are well-studied objects in combinatorial number theory and have applications in extremal graph theory and finite geometry. Working in the group of integers with multiplication, Erdos showed that one cannot find a Sidon set that is asymptotically denser than the primes. In this talk, we show that one can obtain the same result with a much weaker restriction than requiring a Sidon set. This complements work of Bollobas and Pikhurko from 2004. We also discuss an open problem that they posed, with some ideas for how to attack it. This is joint work with Jacques Verstraete. Mon11/03/20144:00pm Seminar: CombinatoricsSemidefinite programming in extremal graph theoryFlorian Pfender, The University of Colorado, DenverContact: Dwight Duffus, dwight@mathcs.emory.eduVenue: Mathematics and Science Center, Room W302Download printable flyer (PDF, 37.9 kB)Show abstractRazborov developed in 2007 the theory of flag algebras. Within this theory, densities of small substructures in large combinatorial structures can be described and computed. His so called "plain flag algebra method" uses semidefinite programming to optimally combine a large number of true inequalities to get bounds on densities in many contexts.\\ \\ One context the method can be used in is the inducibility of graphs. We are looking to maximize the number of induced copies of a given small graph in a very large graph. Whenever the extremal graph to a problem has a simple blow-up structure, the plain method often works very well. But when the structure is more complicated, the bounds tend to get weaker. We recently expanded the plain method to be able to deal with an iterated blow-up structure, which often appears as extremal construction for inducibility questions. Fri10/31/20143:00pm Defense: DissertationHigh Performance Spatial Query Processing for Large Scale Spatial Data WarehousingAblimit Aji, Emory UniversityContact: James Lu, jlu@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 44.5 kB)Show abstractSupport of high performance queries on large volumes of spatial data have become important in many application domains, including geowspatial problems in numerous fields, location based services, geo-social networks, and emerging scientific applications that are increasingly data- and compute-intensive. There are two major challenges for managing and querying massive spatial data: the explosion of spatial data, and the high computational complexity of spatial queries due to the multi-dimensional nature of spatial analytics. High performance computing capabilities are fundamental to efficiently handling of massive spatial datasets. MapReduce based computing model provides a highly scalable, reliable, elastic and cost effective framework for processing massive data on a cluster or cloud environment. While the MapReduce model fits nicely with large scale problems through data partitioning, spatial queries and analytics are intrinsically complex to fit into the MapReduce model easily. Meanwhile, hybrid systems combining CPUs and GPUs are becoming commonly available in commodity clusters, but the computing capacity of such systems is often underutilized. Providing new spatial querying and analytical methods to run on such architecture requires us to answer several fundamental research questions that are of practical importance. The goal of my dissertation is to create a framework with new systematic methods to support high performance spatial queries for spatial big data on MapReduce and CPU-GPU hybrid platforms, driven by real-world use cases. Towards that end, we have researched multi-level parallelism methods of spatial queries running on these platforms. Specifically, we have conducted following studies: 1) create new spatial data processing methods and pipelines with spatial partition level parallelism through a simple programming model MapReduce and propose multi-level indexing methods to accelerate spatial data processing, 2) develop two critical components to enable data parallelism: effective and scalable spatial partitioning in MapReduce (pre-processing), and query normalization methods for partition effect, 3) integrate GPU-based spatial operations into MapReduce pipelines 4) investigate optimization methods for data skew mitigation, and CPU/GPU resource coordination in MapReduce, and 5) support declarative spatial queries for workload composition, and create a query translator to automatically translate the queries into MapReduce programs. Consequently, we have developed Hadoop-GISb a MapReduce based high performance spatial querying system for spatial data warehousing. The system supports multiple types of spatial queries on MapReduce through spatial partitioning, implicit parallel spatial query execution on MapReduce, and effective methods for amending query results through handling bound- ary objects. Hadoop-GIS utilizes global partition indexing and customizable on demand local spatial indexing to achieve efficient query processing. Hadoop-GIS is integrated into Hive to support declarative spatial queries with an integrated architecture. The systems and developed approaches are released as an open source software package for use. Fri10/31/201412:00pm Seminar: Numerical Analysis and Scientific ComputingRegularization by Krylov-Tikhonov methodsSilvia Gazzola, University of PadovaContact: James Nagy, nagy@mathcs.emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 38.6 kB)Show abstractKrylov subspace methods have always played a central role in the iterative regularization of large-scale linear discrete ill-posed problems, which arise in a variety of scientific and engineering applications; we are particularly interested in image deblurring and denoising issues. In addition to a purely iterative approach to regularization, some "hybrid" Krylov-Tikhonov methods have also been derived, which merge an iterative and a variational (Tikhonov-like) approach to regularization. The purpose of this talk is to survey some classical Krylov and Krylov-Tikhonov methods, and to present some original ones, comparing their performance on some meaningful test problems. Particular emphasis will be posed on the strategies to be employed to set the regularization parameters and matrices in the Krylov-Tikhonov framework. Tue10/28/20145:00pm Seminar: AlgebraEmbeddings of maximal tori in classical groups and explicit Brauer–Manin obstructionEva Bayer, EPFLContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 41.6 kB)Show abstractThis is a joint work with Parimala and Ting–Yu Lee. Embeddings of maximal tori into classical groups over global fields of characteristic $\neq$ 2 are the subject matter of several recent papers (for instance by Prasad and Rapinchuk, Fiori, Lee), with special attention to the Hasse principle. The aim of this talk is to describe a complete criterion for the Hasse principle to hold, and to give necessary and sufficient conditions for a classical group to contain a maximal torus of a given type. The embedding problem will be described in terms of embeddings of \'etale algebras with involution into central simple algebras with involution. Tue10/28/20144:00pm Seminar: AlgebraThe genus of a division algebraIgor Rapinchuk, HarvardContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 38.4 kB)Show abstractIn this talk, I will address the following problem. Suppose D and D' are central division algebras over a field K. What can be said about D and D' if they have the same maximal subfields? I will discuss various motivations for this question and recent results. I will also mention some generalizations to arbitrary absolutely almost simple algebraic groups. This is joint work with V. Chernousov and A. Rapinchuk. Mon10/27/20144:00pm Seminar: CombinatoricsOn the Number of B_h Sets of a Given SizeDomingos Dellamonica, Sao PauloContact: Vojtech Rodl, rodl@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Show abstractFor an integer h bigger or equal to 2, a $B_h$ set is a set of integers with the property that every collection containing h of its elements yield a unique sum (and repetitions are allowed). For $h = 2$, such sets are also called Sidon sets. In this talk we will describe our recent results on estimating $F(n, s, h)$, which we define as the number of $B_h$ sets of cardinality s containing integers from $[n] = {1, 2, ..., n}$. It is not hard to see that for $s > n^(1/h)$, we have $F(n, s, h) = 0$. Indeed, in this case there are more h-sums than possible outcomes for the sums. On the other hand, there are constructions of B-h sets having cardinality $c.n^(1/h)$, (with c depending on h only) hence we shall estimate the behavior of $F(n, s, h)$ for s up to $O( n^(1/h))$. Our counting shows the existence of a surprising threshold function $T(n)$: for values of $s << T(n)$, the B-h sets are abundant while for $s >> T(n)$ the B-h sets become very rare. More precisely, we show that $T(n) ~ n^{(1 + o(1))/(2h - 1)}$ and establish fairly precise estimates of $F(n, s, h)$ for the entire range of s.