Seminars and Colloquia by Series

Discrete geometry and representation theory

Series
Combinatorics Seminar
Time
Friday, December 2, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ben Steinberg – CUNY
One can associate regular cell complexes to various objects from discrete and combinatorial geometry such as real and complex hyperplane arrangements, oriented matroids and CAT(0) cube complexes. The faces of these cell complexes have a natural algebraic structure. In a seminal paper from 1998, Bidigare, Hanlon and Rockmore exploited this algebraic structure to model a number of interesting Markov chains including the riffle shuffle and the top-to-random shuffle, as well as the Tsetlin library. Using the representation theory of the associated algebras, they gave a complete description of the spectrum of the transition matrix of the Markov chain. Diaconis and Brown proved further results on mixing times and diagonalizability for these Markov chains. Bidigare also noticed in his thesis a natural connection between Solomon's descent algebra for a finite Coxeter group and the algebra associated to its Coxeter arrangement. Given, the nice interplay between the geometry, the combinatorics and the algebra that appeared in these two contexts, it is natural to study the representation theory of these algebras from the point of view of the representation theory of finite dimensional algebras. Building on earlier work of Brown's student, Saliola, for the case of real central hyperplane arrangements, we provide a quiver presentation for the algebras associated to hyperplane arrangements, oriented matroids and CAT(0) cube complexes and prove that these algebras are Koszul duals of incidence algebras of associated posets. Key to obtaining these results is a description of the minimal projective resolutions of the simple modules in terms of the cellular chain complexes of the corresponding cell complexes.This is joint work with Stuart Margolis (Bar-Ilan) and Franco Saliola (University of Quebec at Montreal)

Multiscale Crystal Plasticity Modeling for Metals

Series
GT-MAP Seminar
Time
Friday, December 2, 2016 - 15:00 for 2 hours
Location
Skiles 006
Speaker
Prof. David McDowell and Shouzhi Xu – GT ME and MSE

Please Note: Talk by Shuozhi Xu, Title: Algorithms and Implementation for the Concurrent Atomistic-Continuum Method. Abstract: Unlikemany other multiscale methods, the concurrent atomistic-continuum (CAC) method admits the migration of dislocations and intrinsic stacking faults through a lattice while employing an underlying interatomic potential as the only constitutive relation. Here, we build algorithms and develop a new CAC code which runs in parallel using MPI with a domain decomposition algorithm. New features of the code include, but are not limited to: (i) both dynamic and quasistatic CAC simulations are available, (ii) mesh refinement schemes for both dynamic fracture and curved dislocation migration are implemented, and (iii) integration points in individual finite elements are shared among multiple processors to minimize the amount of data communication. The CAC program is then employed to study a series of metal plasticity problems in which both dislocation core effects at the nanoscale and the long range stress field of dislocations at the submicron scales are preserved. Applications using the new code include dislocation multiplication from Frank-Read sources, dislocation/void interactions, and dislocation/grain boundary interactions.

Crystal plasticity modeling is useful for considering the influence of anisotropy of elastic and plastic deformation on local and global responses in crystals and polycrystals. Modern crystal plasticity has numerous manifestations, including bottom-up models based on adaptive quasi-continuum and concurrent atomistic-continuum methods in addition to discrete dislocation dynamics and continuum crystal plasticity. Some key gaps in mesoscale crystal plasticity models will be discussed, including interface slip transfer, grain subdivision in large deformation, shock wave propagation in heterogeneous polycrystals, and dislocation dynamics with explicit treatment of waves. Given the mesoscopic character of these phenomena, contrasts are drawn between bottom-up (e.g., atomistic and discrete dislocation simulations and in situ experimental observations) and top-down (e.g., experimental) information in assembling mesoscale constitutive relations and informing their parameters.

The universal quantum invariant and colored ideal triangulations

Series
Geometry Topology Seminar
Time
Friday, December 2, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
S. Suzuki – RIMS, Kyoto University
The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal R matrix, and we obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant is an invariant of framed links, and is constructed diagrammatically using a ribbon Hopf algebra. In that construction, a copy of the universal R matrix is attached to each positive crossing, and invariance under the Reidemeister III move is shown by the quantum Yang-Baxter equation of the universal R matrix. On the other hand, R. Kashaev showed that the Heisenberg double has the canonical element (the universal S matrix) satisfying the pentagon relation. In this talk we reconstruct the universal quantum invariant using Heisenberg double, and extend it to an invariant of colored ideal triangulations of the complement. In this construction, a copy of the universal S matrix is attached to each tetrahedron and the invariance under the colored Pachner (2,3) move is shown by the pentagon equation of the universal S matrix

Lazifying Conditional Gradient Algorithms

Series
ACO Student Seminar
Time
Friday, December 2, 2016 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Daniel Zink – Georgia Tech
Conditional gradient algorithms (also often called Frank-Wolfe algorithms) are popular due to their simplicity of only requiring a linear optimization oracle and more recently they also gained significant traction for online learning. While simple in principle, in many cases the actual implementation of the linear optimization oracle is costly. We show a general method to lazify various conditional gradient algorithms, which in actual computations leads to several orders of magnitude of speedup in wall-clock time. This is achieved by using a faster separation oracle instead of a linear optimization oracle, relying only on few linear optimization oracle calls.

Asymptotic equivalence between density estimation and Gaussian white noise revisited

Series
Job Candidate Talk
Time
Thursday, December 1, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kolyan Ray – Leiden Univ. –
Asymptotic equivalence between two statistical models means that they have the same asymptotic (large sample) properties with respect to all decision problems with bounded loss. In nonparametric (infinite-dimensional) statistical models, asymptotic equivalence has been found to be useful since it can allow one to derive certain results by studying simpler models. One of the key results in this area is Nussbaum’s theorem, which states that nonparametric density estimation is asymptotically equivalent to a Gaussian shift model, provided that the densities are smooth enough and uniformly bounded away from zero.We will review the notion of asymptotic equivalence and existing results, before presenting recent work on the extent to which one can relax the assumption of being bounded away from zero. We further derive the optimal (Le Cam) distance between these models, which quantifies how close they are for finite-samples. As an application, we also consider Poisson intensity estimation with low count data. This is joint work with Johannes Schmidt-Hieber.

From stochastic calculus to geometric inequalities

Series
School of Mathematics Colloquium
Time
Thursday, December 1, 2016 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ronen Eldan – Weizmann Institute of Science –
The probabilistic method, pioneered by P. Erdös, has been key in many proofs from asymptotic geometric analysis. This method allows one to take advantage of numerous tools and concepts from probability theory to prove theorems which are not necessarily a-priori related to probability. The objective of this talk is to demonstrate several recent results which take advantage of stochastic calculus to prove results of a geometric nature. We will mainly focus on a specific construction of a moment-generating process, which can be thought of as a stochastic version of the logarithmic Laplace transform. The method we introduce allows us to attain a different viewpoint on the method of semigroup proofs, namely a path-wise point of view. We will first discuss an application of this method to concentration inequalities on high dimensional convex sets. Then, we will briefly discuss an application to two new functional inequalities on Gaussian space; an L1 version of hypercontractivity of the convolution operator related to a conjecture of Talagrand (joint with J. Lee) and a robustness estimate for the Gaussian noise-stability inequality of C.Borell (improving a result of Mossel and Neeman).

The Homfly skein and elliptic Hall algebras

Series
Geometry Topology Seminar
Time
Wednesday, November 30, 2016 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Peter Samuelson – Edinburgh –
The Homfly skein algebra of a surface is defined using links in thickened surfaces modulo local "skein" relations. It was shown by Turaev that this quantizes the Goldman symplectic structure on the character varieties of the surface. In this talk we give a complete description of this algebra for the torus. We also show it is isomorphic to the elliptic Hall algebra of Burban and Schiffmann, which is an algebra whose elements are (formal sums of) sheaves on an elliptic curve, with multiplication defined by counting extensions of such sheaves. (Joint work with H. Morton.)

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