Seminars and Colloquia by Series

1-d parabolic Anderson model with rough spatial noise

Series
Stochastics Seminar
Time
Thursday, March 7, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Samy Tindel – Purdue University
In this talk I will first recall some general facts about the parabolic Anderson model (PAM), which can be briefly described as a simple heat equation in a random environment. The key phenomenon which has to be observed in this context is called localization. I will review some ways to express this phenomenon, and then single out the so called eigenvectors localization for the Anderson operator. This particular instance of localization motivates our study of large time asymptotics for the stochastic heat equation. In the second part of the talk I will describe the Gaussian environment we consider, which is rougher than white noise, then I will give an account on the asymptotic exponents we obtain as time goes to infinity. If time allows it, I will also give some elements of proof.

Packing and covering triangles

Series
Graph Theory Working Seminar
Time
Wednesday, March 6, 2019 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Youngho Yoo – Georgia Tech

Let $\nu$ denote the maximum size of a packing of edge-disjoint triangles in a graph $G$. We can clearly make $G$ triangle-free by deleting $3\nu$ edges. Tuza conjectured in 1981 that $2\nu$ edges suffice, and proved it for planar graphs. The best known general bound is $(3-\frac{3}{23})\nu$ proven by Haxell in 1997. We will discuss this proof and some related results.

Global Convergence of Neuron Birth-Death Dynamics

Series
Applied and Computational Mathematics Seminar
Time
Wednesday, March 6, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joan Bruna Estrach – New York University –
Neural networks with a large number of parameters admit a mean-field description, which has recently served as a theoretical explanation for the favorable training properties of "overparameterized" models. In this regime, gradient descent obeys a deterministic partial differential equation (PDE) that converges to a globally optimal solution for networks with a single hidden layer under appropriate assumptions. In this talk, we propose a non-local mass transport dynamics that leads to a modified PDE with the same minimizer. We implement this non-local dynamics as a stochastic neuronal birth-death process and we prove that it accelerates the rate of convergence in the mean-field limit. We subsequently realize this PDE with two classes of numerical schemes that converge to the mean-field equation, each of which can easily be implemented for neural networks with finite numbers of parameters. We illustrate our algorithms with two models to provide intuition for the mechanism through which convergence is accelerated. Joint work with G. Rotskoff (NYU), S. Jelassi (Princeton) and E. Vanden-Eijnden (NYU).

A restriction estimate in $\mathbb{R}^3$

Series
Analysis Seminar
Time
Wednesday, March 6, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hong Wang – MIT

If $f$ is a function supported on a truncated paraboloid, what can we say about $Ef$, the Fourier transform of f? Stein conjectured in the 1960s that for any $p>3$, $\|Ef\|_{L^p(R^3)} \lesssim \|f\|_{L^{\infty}}$.

We make a small progress toward this conjecture and show that it holds for $p> 3+3/13\approx 3.23$. In the proof, we combine polynomial partitioning techniques introduced by Guth and the two ends argument introduced by Wolff and Tao.

On the reconstruction error of PCA

Series
Stochastics Seminar
Time
Tuesday, March 5, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 168
Speaker
Martin Wahl – Humboldt University, Berlin.

We identify principal component analysis (PCA) as an empirical risk minimization problem with respect to the reconstruction error and prove non-asymptotic upper bounds for the corresponding excess risk. These bounds unify and improve existing upper bounds from the literature. In particular, they give oracle inequalities under mild eigenvalue conditions. We also discuss how our results can be transferred to the subspace distance and, for instance, how our approach leads to a sharp $\sin \Theta$ theorem for empirical covariance operators. The proof is based on a novel contraction property, contrasting previous spectral perturbation approaches. This talk is based on joint works with Markus Reiß and Moritz Jirak.

Field Theoretical Interpretation of QM Wave Functions and Quantum Mechanism of High Tc Superconductivity

Series
PDE Seminar
Time
Tuesday, March 5, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Professor Shouhong Wang – Indiana University –

First, we introduce a new field theoretical interpretation of quantum mechanical wave functions, by postulating that the wave function is the common wave function for all particles in the same class determined by the external potential V, of the modulus of the wave function represents the distribution density of the particles, and the gradient of phase of the wave function provides the velocity field of the particles. Second, we show that the key for condensation of bosonic particles is that their interaction is sufficiently weak to ensure that a large collection of boson particles are in a state governed by the same condensation wave function field under the same bounding potential V. For superconductivity, the formation of superconductivity comes down to conditions for the formation of electron-pairs, and for the electron-pairs to share a common wave function. Thanks to the recently developed PID interaction potential of electrons and the average-energy level formula of temperature, these conditions for superconductivity are explicitly derived. Furthermore, we obtain both microscopic and macroscopic formulas for the critical temperature. Third, we derive the field and topological phase transition equations for condensates, and make connections to the quantum phase transition, as a topological phase transition. This is joint work with Tian Ma.

The nu+ equivalence class of genus one knots

Series
Geometry Topology Seminar
Time
Monday, March 4, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 154
Speaker
Kouki Sato – University of Tokyo
The nu+ equivalence is an equivalence relation on the knot concordance group. It is known that the equivalence can be seen as a certain stable equivalence on knot Floer complexes, and many concordance invariants derived from Heegaard Floer theory are invariant under the equivalence. In this talk, we show that any genus one knot is nu+ equivalent to one of the unknot, the trefoil and its mirror.

Chow rings of matroids, ring of matroid quotients, and beyond

Series
Algebra Seminar
Time
Monday, March 4, 2019 - 12:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chris Eur – University of California, Berkeley –
We introduce a certain nef generating set for the Chow ring of the wonderful compactification of a hyperplane arrangement complement. This presentation yields a monomial basis of the Chow ring that admits a geometric and combinatorial interpretation with several applications. Geometrically, one can recover Poincare duality, compute the volume polynomial, and identify a portion of a polyhedral boundary of the nef cone. Combinatorially, one can generalize Postnikov's result on volumes of generalized permutohedra, prove Mason's conjecture on the log-concavity of independent sets for certain matroids, and define a new valuative invariant of a matroid that measures its closeness to uniform matroids. This is an on-going joint work with Connor Simpson and Spencer Backman.

A partial order on nu+ equivalence classes

Series
Geometry Topology Seminar Pre-talk
Time
Monday, March 4, 2019 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 257
Speaker
Kouki Sato – University of Tokyo
I will review the definition of nu+ equivalence, which is an equivalence relation on the knot concordance group, and introduce a partial order on the equivalence classes. This partial order is preserved by all satellite maps and some concordance invariants. We also consider full-twist operations and its relationship to the partial order.

Local Guarantees in Graph Cuts and Clustering

Series
ACO Student Seminar
Time
Friday, March 1, 2019 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Roy Schwartz – CS, Technion –

Correlation Clustering is an elegant model that captures fundamental graph cut problems such as Minimum s-t Cut, Multiway Cut, and Multicut, extensively studied in combinatorial optimization.

Here, we are given a graph with edges labeled + or - and the goal is to produce a clustering that agrees with the labels as much as possible: + edges within clusters and - edges across clusters.

The classical approach towards Correlation Clustering (and other graph cut problems) is to optimize a global objective, e.g., minimizing the total number of disagreements or maximizing the total number of agreements.

We depart from this and study local objectives: minimizing the maximum number of disagreements for edges incident on a single node, and the analogous max min agreements objective.

This naturally gives rise to a family of basic min-max graph cut problems.

A prototypical representative is Min-Max s-t Cut: find an s-t cut minimizing the largest number of cut edges incident on any node.

In this talk we will give a short introduction of Correlation Clustering and discuss the following results:

  1. an O(\sqrt{n})-approximation for the problem of minimizing the maximum total weight of disagreement edges incident on any node (thus providing the first known approximation for the above family of min-max graph cut problems)
  2. a remarkably simple 7-approximation for minimizing local disagreements in complete graphs (improving upon the previous best known approximation of 48)
  3. a (1/(2+epsilon))-approximation for maximizing the minimum total weight of agreement edges incident on any node, hence improving upon the (1/(4+epsilon))-approximation that follows from the study of approximate pure Nash equilibria in cut and party affiliation games.

Joint work with Moses Charikar and Neha Gupta.

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