Week of November 10, 2019

Tue Nov 12, 2019
11:00am to 11:50am - RH 340N - Combinatorics and Probability
Elizaveta Rebrova - (UCLA)
Modewise Johnson-Lindenstrauss embeddings for tensors

The celebrated Johnson-Lindenstrauss lemma is a powerful tool for dimension reduction via simple (often random) projections that approximately preserve the geometry of the larger dimensional objects. I will discuss an extension of this result to low CP-rank tensors. I show how modewise tensor projections preserve tensor geometry in the analogous way, without doing any initial tensor matricization or vectorization. Time permitting, I will also talk about an application to the least squares fitting CP model for tensors. Based on our joint work with Mark Iwen, Deanna Needell, and Ali Zare.

1:00pm to 2:00pm - RH 440R - Dynamical Systems
Denis Gaidashev - ( Uppsala University)
Renormalization: from holomorphic dynamics to two dimensions

I will describe recent advances in promoting renormalization techniques from one dimensional holomirphic setting to two dimensions, together with the set of questions that these techniques are aiming to address, such as rigidity and geometry of attractors.

2:00pm to 3:00pm - RH 510R - Working Group in Information Theory
Liam Hardiman - (UCI)
Conditional Divergence

This week, we will continue to discuss Section 2.2 of the lecture notes of Wu and Polyanski: 
http://people.lids.mit.edu/yp/homepage/papers.html

Working Group in Information Theory is a self-educational project in the department. Techniques based on information theory have become essential in high-dimensional probability, theoretical computer science and statistical learning theory. On the other hand, information theory is not taught systematically. The goal of this group is to close this gap.

Wed Nov 13, 2019
1:00pm to 1:50pm - RH 310P - Algebra
Sergey Arkhipov - (Aarhus University)
Equivariant DG-modules over differential forms and coherent sheaves over derived Hamiltonian reduction

This is a joint work in progress with Sebastian Orsted.  Given an algebraic variety X acted by an affine algebraic group G, we make sense of  the derived category of DG-modules over the DG-algebra of differential forms on X  equivariant with respect to differential forms on G. The construction uses an explicit model for  a certain homotopy limit of a diagram of  DG-categories developed in our earlier work and generalizing a recent result of Block, Holstein and Wei. We compare the obtained category with a certain category of sheaves on the (shifted) cotangent bundle T^*X descending to the Hamiltonian reduction of the cotangent bundle. Two special cases are of interest. In the first, X is a point. Thus we compare comodules over Omega(G)  with G-equivariant coherent sheaves on Lie(G). In the second case, X is a simple algebraic group, with the action of the square of the upper triangular subgroup. We obtain a category closely related to the affine Hecke category.

Thu Nov 14, 2019
2:00pm to 3:00pm - RH 340P - Mathematical Physics
Wencai Liu - (Texas A&M)
The Christ-Kiselev's multi-linear operator technique and its applications to Schrodinger operators

We established an axiomatic version of Christ-Kiselev's multi-linear operator techniques.
As applications,  several spectral results of perturbed periodic Schrodinger operators are obtained, including WKB solutions, sharp transitions of preservation of absolutely continuous spectra, criteria of absence of singular spectra and sharp bounds of Hausdorff dimensions of singular spectra.

3:00pm to 4:00pm - RH 440R - Cryptography
Lynn Chua - (UC Berkeley)
On the concrete security of the unique Shortest Vector Problem

 

We study experimentally the Hermite factor of BKZ2.0 on uSVP lattices, with the motivation of understanding the concrete security of LWE in the setting of homomorphic encryption. We run experiments by generating instances of LWE in small dimensions, where we consider secrets sampled from binary, ternary or discrete Gaussian distributions. We convert each LWE instance into a uSVP instance and run the BKZ2.0 algorithm to find an approximation to the shortest vector. When the attack is successful, we can deduce a bound on the Hermite factor achieved for the given blocksize. This allows us to give concrete values for the Hermite factor of the lattice generated for the uSVP instance. We compare the values of the Hermite factors we find for these lattices with estimates from the literature and find that the Hermite factor may be smaller than expected for blocksizes 30, 35, 40, 45. Our work also demonstrates that the experimental and estimated values of the Hermite factor trend differently as we increase the dimension of the lattice, highlighting the importance of a better theoretical understanding of the performance of BKZ2.0 on uSVP lattices.
 

3:00pm to 4:00pm - RH 440R - Number Theory
Lynn Chua - (UC Berkeley)
On the concrete security of the unique Shortest Vector Problem

We study experimentally the Hermite factor of BKZ2.0 on uSVP lattices, with the motivation of understanding the concrete security of LWE in the setting of homomorphic encryption. We run experiments by generating instances of LWE in small dimensions, where we consider secrets sampled from binary, ternary or discrete Gaussian distributions. We convert each LWE instance into a uSVP instance and run the BKZ2.0 algorithm to find an approximation to the shortest vector. When the attack is successful, we can deduce a bound on the Hermite factor achieved for the given blocksize. This allows us to give concrete values for the Hermite factor of the lattice generated for the uSVP instance. We compare the values of the Hermite factors we find for these lattices with estimates from the literature and find that the Hermite factor may be smaller than expected for blocksizes 30, 35, 40, 45. Our work also demonstrates that the experimental and estimated values of the Hermite factor trend differently as we increase the dimension of the lattice, highlighting the importance of a better theoretical understanding of the performance of BKZ2.0 on uSVP lattices.

 

 

 

Fri Nov 15, 2019
1:00pm to 2:00pm - rh 340p - Mathematical Physics
Shiwen Zhang - (U Minnesota)
Landscape theory for tight-binding Hamiltonians 

In 2012, Filoche and Mayboroda introduced the concept of the landscape function u, for an elliptic operator L, which solves the inhomogeneous equation Lu=1. This landscape function has remarkable power to predict the shape and location of localized low energy eigenfunction. These ideas led to beautiful results in mathematics, as well as theoretical and experimental physics. In this talk, we first briefly review these results of landscape theory for differential operators on R^d. We will then discuss some recent progress of extending landscape theory to tight-binding Hamiltonians on discrete lattice Z^d. In particular, we show that the effective potential 1/u creates barrier for appropriate exponential decay eigenfunctions of Agmon type for some discrete Schrodinger operators. We also show that the minimum of 1/u leads to a new counting function, which gives non-asymptotic estimates on the integrated density of states of the Schrodinger operators. This talk contains joint work in progress with S. Mayboroda and some numerical experiments with W. Wang.