Seminars and Colloquia Schedule

Reconstructing polytopes from projections

Series
Geometry Topology Seminar
Time
Monday, November 28, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sergii Myroshnychenko – Kent State University –
We are going to discuss one of the open problems of geometric tomography about projections. Along with partial previous results, the proof of the problem below will be investigated.Let $2\le k\le d-1$ and let $P$ and $Q$ be two convex polytopes in ${\mathbb E^d}$. Assume that their projections, $P|H$, $Q|H$, onto every $k$-dimensional subspace $H$, are congruent. We will show that $P$ and $Q$ or $P$ and $-Q$ are translates of each other. If the time permits, we also will discuss an analogous result for sections by showing that $P=Q$ or $P=-Q$, provided the polytopes contain the origin in their interior and their sections, $P \cap H$, $Q \cap H$, by every $k$-dimensional subspace $H$, are congruent.

Stochastic simulation and optimization under input uncertainty

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 28, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Enlu Zhou – Georgia Tech ISyE
Many real-life systems require simulation techniques to evaluate the system performance and facilitate decision making. Stochastic simulation is driven by input model — a collection of probability distributions that model the system stochasticity. The choice of the input model is crucial for successful modeling and analysis via simulation. When there are past observed data of the system stochasticity, we can utilize these data to construct an input model. However, there is only a finite amount of data in practice, so the input model based on data is always subject to uncertainty, which is the so-called input (model) uncertainty. Therefore, a typical stochastic simulation faces two types of uncertainties: one is the input (model) uncertainty, and the other is the intrinsic stochastic uncertainty. In this talk, I will discuss our recent work on how to assess the risk brought by the two types of uncertainties and how to make decisions under these uncertainties.

Scheme theoretic tropicalization

Series
Algebra Seminar
Time
Monday, November 28, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Oliver Lorscheid – IMPA
Recent work of Jeff and Noah Giansiracusa exhibits a scheme theoretic structure for tropicalizations of classical varieties in terms of so-called semiring schemes. This works well in the framework of closed subvarieties of toric varieties, and Maclagan and Rincon recover the structure of a weighted polyhedral complex from the scheme theoretic tropicalization of a variety embedded into a torus.In this talk, I will review these ideas and show how these results can be extended by using blue schemes. This leads to an intrinsic notion of a tropicalization, independent from an embedding into an ambient space, and generalizes the above mentioned results to the broader context of log-schemes.

Classification of Resonant Rossby Wave Triads

Series
PDE Seminar
Time
Tuesday, November 29, 2016 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gene Kopp – University of Michigan –
Linear wave solutions to the Charney-Hasegawa-Mima equation with periodic boundary conditions have two physical interpretations: Rossby (atmospheric) waves, and drift (plasma) waves in a tokamak. These waves display resonance in triads. In the case of infinite Rossby deformation radius, the set of resonant triads may be described as the set of integer solutions to a particular homogeneous Diophantine equation, or as the set of rational points on a projective surface. We give a rationalparametrization of the smooth points on this surface, answering the question: What are all resonant triads, and how may they be enumerated quickly? We also give a fiberwise description, yielding an algorithmic procedure to answer the question: For fixed $r \in \Q$, what are all wavevectors $(x,y)$ that resonate with a wavevector $(a,b)$ with $a/b = r$?

The nonlinear Schroedinger equation and the evolution of wave packets in nonlinear dispersive equations

Series
CDSNS Colloquium
Time
Wednesday, November 30, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Eugene Wayne – Boston University
The nonlinear Schroedinger equation (NLS) can be derived as a formal approximating equation for the evolution of wave packets in a wide array of nonlinear dispersive PDE’s including the propagation of waves on the surface of an inviscid fluid. In this talk I will describe recent work that justifies this approximation by exploiting analogies with the theory of normal forms for ordinary differential equations.

Probability Theory in SOM, a Personal Perspective

Series
Research Horizons Seminar
Time
Wednesday, November 30, 2016 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christian Houdré – Georgia Institute of Technology
I will start with a brief presentation of the Probability activities in SOM. I will continue by presenting results obtained in SOM, over the past ten years, answering long standing questions insequences comparison.

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.)

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).

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.

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.

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

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

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

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)