Seminar Calendar
for events the week of Wednesday, December 17, 2014.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Monday, December 15, 2014

Mathematics Colloquium - Special Lecture 2014-15
4:30 pm   in 245 Altgeld Hall,  Monday, December 15, 2014
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Melody Chan (Harvard)
Combinatorics and degenerations in algebraic geometry
Abstract: I will give two examples from my own research showing how one may prove theorems about algebraic curves and their moduli using degenerations and combinatorics, and conversely, how surprising combinatorial theorems can arise from the study of algebraic curves. Joint work, in part, with Lopez, Pflueger, and Teixidor; and with Galatius and Payne.

Tuesday, December 16, 2014

Mathematics Colloquium - Special Lecture 2014-15
4:00 pm   in 245 Altgeld Hall,  Tuesday, December 16, 2014
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Tudor Dimofte (Institute for Advanced Study)
Chern-Simons theory with complex gauge group
Abstract: I will discuss aspects of Chern-Simons theory with complex gauge group, and the rich interaction between mathematics and physics that has fueled its development in recent years. Like its close cousin with compact gauge group, complex Chern-Simons is a topological field theory in three dimensions. Its partition functions on a large class of three-manifolds can be defined by convergent, finite-dimensional integrals, providing a new class of topological invariants with remarkable properties. Their perturbative expansions are arithmetic and (conjecturally) display modular properties. The invariants are closely related to cluster algebras and to Teichmuller theory. The physics of complex Chern-Simons theory also suggests that the invariants admit a categorification (analogous to Khovanov homology), which has been found in some simple cases.

Wednesday, December 17, 2014

Mathematics Colloquium - Special Lecture 2014-15
4:00 pm   in 245 Altgeld Hall,  Wednesday, December 17, 2014
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Qin Li (CalTech)
Intrinsic Sparse Mode Decomposition of High Dimensional Random Fields with Application to Stochastic Elliptic PDEs
Abstract: Inspired by the recent developments in data sciences, we introduce an intrinsic sparse mode decomposition method for high dimensional random fields. This sparse representation of the random field allows us to break a high dimensional stochastic field into many spatially localized modes with low stochastic dimension locally. Such decomposition enables us to break the curse of dimensionality in our local solvers. To obtain such representation, we first decompose the covariance function into low part plus sparse parts. We then extract the spatially localized modes from the sparse part by solving an $L^0$ minimization. We further relax this $L^0$ minimization problem into an $L^1$ minimization and prove rigorously the equivalence of the two formulations. Moreover, we provide an efficient algorithm to solve it. As an application, we apply our method to solve elliptic PDEs with random media having high stochastic dimension. Using this localized representation, we propose various combinations of local and global solver that achieve different level of accuracy and efficiency. At the end of the talk, I will also discuss other applications of the intrinsic sparse mode extraction.This work is in collaboration with Thomas Y. Hou and Pengchuan Zhang.

Thursday, December 18, 2014

Mathematics Colloquium - Special Lecture 2014-15
4:00 pm   in 245 Altgeld Hall,  Thursday, December 18, 2014
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Choongbum Lee (MIT)
Some advances in Sidorenko's conjecture
Abstract: An important conjecture of Erdos-Simonovits and Sidorenko states that if $H$ is a fixed bipartite graph, then the random $n$-vertex graph ($n$ is large) has asymptotically the minimum number of copies of $H$ over all graphs of the same order and edge density. This conjecture also has an equivalent analytic form and has connections to a broad range of topics such as matrix theory, Markov chains, graph limits, and quasirandomness. In this talk, I will provide an overview on this beautiful conjecture and discuss some recent results. Joint w/ Jeong Han Kim (KIAS) and Joonkyung Lee (Oxford).