All Seminars
| Title: Extremal problems on optimizing the number of nonnegative subsets |
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| Colloquium: N/A |
| Speaker: Hao Huang of The Institute for Advanced Study and DIMACS |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2014-02-20 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Extremal combinatorics studies the maximum or minimum possible size of a combinatorial structure satisfying certain properties. It is one of the central themes of modern discrete mathematics, and has numerous natural connections to other areas including probability, number theory and theoretical computer science. As an example, in this talk I will discuss some recent progress on a fifty-year-old conjecture of Erdos on hypergraph matching, and describe its relation with several other extremal problems on optimizing the number of nonnegative. Our work settles conjectures of Manickam, Miklos and Singhi, and of Tsukerman. |
| Title: Applications of Flag Algebras in Hypercubes and Permutations |
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| Colloquium: N/A |
| Speaker: Bernard Lidicky of The University of Illinois at Urbana-Champaign |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2014-02-19 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Flag 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. |
| Title: Lifting Tropical Curves and Linear Systems on Graphs |
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| Seminar: Algebra |
| Speaker: Eric Katz of University of Waterloo |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2014-02-18 at 4:00PM |
| Venue: W302 |
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| Abstract: Tropicalization 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. |
| Title: Randomized Block Coordinate Gradient Methods for a Class of Structured Nonlinear Programming |
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| Colloquium: N/A |
| Speaker: Zhaosong Lu of Simon Fraser University |
| Contact: James Nagy, nagy@mathcs.emory.edu |
| Date: 2014-02-17 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Nowadays 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.) |
| Title: Fast algorithms for electronic structure analysis |
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| Colloquium: N/A |
| Speaker: Lin Lin of Lawrence Berkeley National Laboratory |
| Contact: James Nagy, nagy@mathcs.emory.edu |
| Date: 2014-02-12 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Kohn-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. |
| Title: Canceled due to weather |
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| Seminar: Algebra |
| Speaker: Matt Ballard of University of South Carolina |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2014-02-11 at 4:00PM |
| Venue: W302 |
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| Abstract: One 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. |
| Title: Computational Large-Scale Continuous Optimization, Uncertainty and Robustness |
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| Colloquium: N/A |
| Speaker: Somayeh Moazeni of Princeton University |
| Contact: James Nagy, nagy@mathcs.emory.edu |
| Date: 2014-02-10 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Optimal 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. |
| Title: Independent Sets in Hypergraphs |
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| Colloquium: N/A |
| Speaker: Dhruv Mubayi of The University of Illinois at Chicago |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2014-02-07 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: Abstract: 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. |
| Title: Continuous analogues of methods used to calculate component groups of Jacobians |
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| Seminar: Algebra |
| Speaker: Joe Rabinoff of Georgia Tech |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2014-02-04 at 4:00PM |
| Venue: W302 |
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| Abstract: Let 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. |
| Title: Numerical Methods for Hyperelastic Image Registration |
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| Colloquium: N/A |
| Speaker: Lars Ruthotto of University of British Columbia |
| Contact: James Nagy, nagy@mathcs.emory.edu |
| Date: 2014-01-31 at 3:00PM |
| Venue: MSC W201 |
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| Abstract: Image 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. |