# 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 Thu02/20/201412:00pm Colloquium: Computer ScienceHarnessing the Power of Crowd for On-Demand Geographical Data CollectionCyrus Shahabi, University of Southern CaliforniaContact: Li Xiong, lxiong@emory.eduVenue: Mathematics and Science Center, Room W303Download attached abstract (PDF, 105 kB)Download printable flyer (PDF, 105 kB)Show abstractGeoCrowd is an online spatial crowdsourcing market (similar to Amazon’s Mechanical Turk) that matches geo tasks (i.e., tasks associated with a specific location and time such as “Take pictures of Tommy Trojan during 2012 USC-UCLA game”) to human workers. Every person with mobile devices can now act as a multi-modal sensor collecting various types of data instantaneously (e.g., picture, video). With GeoCrowd, subscribers can publish tasks with specific space and time attributes. Subsequently, the workers (with GeoCrowd mobile app) can perform the tasks if they are at the right time and at the right place and upload the results to the GeoCrowd server(s). In this talk, I first introduce our generic framework for GeoCrowd and discuss various techniques for optimal assignment of spatiotemporal tasks to human workers. Next, I show how we can extend this framework to incorporate trust in GeoCrowd in order to ensure workers satisfy a confidence value given by the task requester. Finally, I will show an application of the GeoCrowd framework in a commercial domain. Wed02/19/20144:00pm ColloquiumApplications of Flag Algebras in Hypercubes and PermutationsBernard Lidicky, The University of Illinois at Urbana-ChampaignContact: Dwight Duffus, dwight@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 38.7 kB)Show abstractFlag algebras provide a method, recently developed by Razborov, designed for attacking problems in extremal graph theory. There are recent applications of the method also in discrete geometry or permutation patterns. The aim of talk is to give a gentle introduction to the method and show some of its applications to hypercubes and permutations. The talk is based on joint work with J. Balogh, P. Hu, H. Liu, O. Pikhurko, B. Udvari, and J. Volec. Tue02/18/20144:00pm Seminar: AlgebraLifting Tropical Curves and Linear Systems on GraphsEric Katz, University of WaterlooContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W302Download printable flyer (PDF, 38.4 kB)Show abstractTropicalization is a procedure for associating a polyhedral complex to a subvariety of an algebraic torus. We explain the method of tropicalization and study the question of which graphs arise from tropicalizing algebraic curves. By applying Matthew Baker's technique of specialization of linear systems from curves to graphs, we are able to give a necessary condition for a balanced weighted graph to be the tropicalization of a curve. Our condition is phrased in terms of the harmonic theory of graphs, reproduces the known necessary conditions, and also gives new conditions. Moreover, our method gives a combinatorial way of thinking about the deformation theory of algebraic varieties. Mon02/17/20144:00pm ColloquiumRandomized Block Coordinate Gradient Methods for a Class of Structured Nonlinear ProgrammingZhaosong Lu, Simon Fraser UniversityContact: James Nagy, nagy@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 41.5 kB)Show abstractNowadays a class of huge-scale structured optimization problems arise in some emerging areas such as machine learning. They can be reformulated as minimizing the sum of a smooth and block separable nonsmooth functions. For these problems, it is prohibitive to evaluate the full gradient of the smooth component of the objective function due to huge dimensionality and hence the usual gradient methods cannot be efficiently applied. Nevertheless, its partial gradients can often be computed much more cheaply. In this talk we study a randomized block coordinate gradient (RBCG) method for solving this class of problems. At each iteration this method randomly picks a block, and solves a proximal gradient subproblem over the subspace defined by the block that only uses a partial gradient and usually has a closed-form solution. We present new iteration complexity results for this method when applied to convex problems. We also propose a nonmonotone RBCG method for solving a class of nonconvex problems with the above structure, and establish their global convergence and iteration complexity. In addition, we present new complexity results for the accelerated RBCG method proposed by Nesterov for solving unconstrained convex optimization problems. Finally, we discuss the application of these methods for solving some support vector machine problems and report some computational results. (This is a joint work with Lin Xiao at Microsoft Research Redmond.) Wed02/12/20144:00pm ColloquiumFast algorithms for electronic structure analysisLin Lin, Lawrence Berkeley National LaboratoryContact: James Nagy, nagy@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 46.9 kB)Show abstractKohn-Sham density functional theory (KSDFT) is the most widely used electronic structure theory for molecules and condensed matter systems. For a system with N electrons, the standard method for solving KSDFT requires solving N eigenvectors for an $O(N) * O(N)$ Kohn-Sham Hamiltonian matrix. The computational cost for such procedure is expensive and scales as $O(N^3)$. We have developed pole expansion plus selected inversion (PEXSI) method, in which KSDFT is solved by evaluating the selected elements of the inverse of a series of sparse symmetric matrices, and the overall algorithm scales at most $O(N^2)$ for all materials including insulators, semiconductors and metals. The PEXSI method can be used with orthogonal or nonorthogonal basis set, and the physical quantities including electron density, energy, atomic force, density of states, and local density of states are calculated accurately without using the eigenvalues and eigenvectors. The recently developed massively parallel PEXSI method has been implemented in SIESTA, one of the most popular electronic structure software packages using atomic orbital basis sets. The resulting method can allow accurate treatment of electronic structure in a unprecedented scale. We demonstrate the application of the method for solving graphene-like structures with more than 30,000 atoms, and the method can be efficiently parallelized 10,000 - 100,000 processors on Department of Energy (DOE) high performance machines. Tue02/11/20144:00pm Seminar: AlgebraCanceled due to weatherMatt Ballard, University of South CarolinaContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W302Download printable flyer (PDF, 42.6 kB)Show abstractOne of the most basic invariants of an algebra is its global dimension, the maximal $n$ for which $Ext^n_A(M,N)$ does not vanish. For an algebra A of finite global dimension, what are the possible global dimensions of algebras Morita equivalent to A? Derived Morita equivalent to A? I will review these notions, discuss these questions, and then their extensions to differential graded algebras, which will naturally lead into Orlov spectra. Mon02/10/20144:00pm ColloquiumComputational Large-Scale Continuous Optimization, Uncertainty and RobustnessSomayeh Moazeni, Princeton UniversityContact: James Nagy, nagy@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 37.9 kB)Show abstractOptimal decisions often rely on assumptions about the models and their associated parameter values. Therefore, it is essential to assess the suitability of these assumptions and to understand the sensitivity of outcomes when they are altered. More importantly, appropriate approaches should be developed to achieve a robust solution. In this talk, we first present a sensitivity analysis on parameter values as well as model specification of a problem in portfolio management, namely the optimal portfolio execution problem. We then propose more robust solution techniques and models including regularized robust optimization for convex optimization programs and computational stochastic optimization. Extensions of these approaches for energy storage operational management and electricity price modeling are discussed. Fri02/07/20144:00pm ColloquiumIndependent Sets in HypergraphsDhruv Mubayi, The University of Illinois at ChicagoContact: Dwight Duffus, dwight@mathcs.emory.eduVenue: Mathematics and Science Center, Room W201Download printable flyer (PDF, 40.5 kB)Show abstractAbstract: The problem of determining the independence number of (hyper)graphs has tight connections to questions in discrete geometry, coding theory, number theory, theoretical computer science and combinatorics. One of the most famous early examples is the result of Komlos-Pintz-Szemeredi from 1982 on the independence number of 3-uniform hypergraphs which made important progress on the decades old Heilbronn problem. I will begin by explaining this result and some of these connections. I will then describe recent work in this area which shows that hypergraphs have a significantly different behavior than graphs when it comes to independent sets. This answers a question posed by Ajtai-Erdos-Komlos-Szemeredi (1981), and disproves conjectures of deCaen (1986), Frieze and the speaker (2007), and several others. Tue02/04/20144:00pm Seminar: AlgebraContinuous analogues of methods used to calculate component groups of JacobiansJoe Rabinoff, Georgia TechContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W302Download printable flyer (PDF, 45 kB)Show abstractLet K be complete, discretely-valued field and let X be a smooth projective K-curve equipped with a semistable model over the valuation ring. A series of classical theorems, mostly due to Raynaud, give two ways of calculating the component group of the Jacobian J of X: one using the intersection matrix on the special fiber of the model of X, and the other using cycles on its incidence graph G. These calculations can be interpreted in terms of divisors on G (in the sense of Baker-Norine) and the uniformization theory of G, respectively. If K is complete and non-Archimedean but not discretely valued, these theorems are no longer applicable, as Néron models do not exist in this situation. Replacing the component group with the skeleton of J (in the sense of Berkovich), a principally polarized real torus canonically associated to J, and the incidence graph with a skeleton Gamma of X, a metric graph, we will prove "continuous" analogues of these theorems. Specifically, we will show that the Jacobian of Gamma is canonically identified with the skeleton of J as principally polarized real tori, in a way that is compatible with the descriptions of the two Jacobians in terms of divisors and in terms of uniformizations. As a consequence, we will show that, when K is algebraically closed, essentially any piecewise-linear function on Gamma is the restriction to Gamma of $-\log |f|$, where f is a nonzero rational function on X. Fri01/31/20143:00pm ColloquiumNumerical Methods for Hyperelastic Image RegistrationLars Ruthotto, University of British ColumbiaContact: James Nagy, nagy@mathcs.emory.eduVenue: Mathematics and Science Center, Room W201Download printable flyer (PDF, 38 kB)Show abstractImage registration is an essential task in almost all areas involving imaging techniques. The goal of image registration is to find geometrical correspondences between two or more images. Image registration is commonly phrased as a variational problem that is known to be ill-posed and thus regularization is commonly used to ensure existence of solutions and/or introduce prior knowledge about the application in mind. This talk presents a nonlinear regularization functional based on the theory of hyperelastic materials, which overcomes limitations of the most commonly used linear elastic model. In particular, the hyperelastic regularization functional guarantees that solutions to the variational problem exist and are one-to-one correspondences between the images, which is a key concern in most applications. The focus of this talk is on accurate and fast numerical methods for solving hyperelastic image registration problems. Further, the potential of hyperelastic schemes is demonstrated for real-life medical imaging problems.