Seminars and Colloquia by Series

Onsager's Conjecture

Series
PDE Seminar
Time
Tuesday, March 10, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tristan Buckmaster – Courant Institute, NYU
In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which dissipate energy. The first part of this conjecture has since been confirmed (cf. Eyink 1994, Constantin, E and Titi 1994). During this talk we will discuss recent work by Camillo De Lellis, László Székelyhidi Jr., Philip Isett and myself related to resolving the second component of Onsager's conjecture. In particular, we will discuss the construction of weak non-conservative solutions to the Euler equations whose Hölder $1/3-\epsilon$ norm is Lebesgue integrable in time.

The structure of space curve arrangements with many incidences

Series
Combinatorics Seminar
Time
Tuesday, March 10, 2015 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joshua Zahl – MIT –
In 2010, Guth and Katz proved that if a collection of N lines in R^3 contained more than N^{3/2} 2-rich points, then many of these lines must lie on planes or reguli. I will discuss some generalizations of this result to space curves in three dimensional vector spaces. This is joint work with Larry Guth.

Inclusion of Spectrahedra, the Matrix Cube Problem and Beta Distributions.

Series
Algebra Seminar
Time
Monday, March 9, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Igor Klep – University of Auckland
Given a tuple A=(A_1,...,A_g) of symmetric matrices of the same size, the affine linear matrix polynomial L(x):=I-\sum A_j x_j is a monic linear pencil. The solution set S_L of the corresponding linear matrix inequality, consisting of those x in R^g for which L(x) is positive semidefinite (PsD), is called a spectrahedron. It is a convex basic closed semialgebraic subset of R^g. Given a spectrahedron S_L, the matrix cube problem of Nemirovskii asks for the biggest cube [-r,r]^g included in S_L. We solve a relaxation of this problem based on``matricial’’ spectrahedra and estimate the error inherent in this relaxation. The talk is based on joint work with B. Helton, S. McCullough and M. Schweighofer.

Tight Surgeries on Knots in Overtwisted Contact Manifolds

Series
Geometry Topology Seminar
Time
Monday, March 9, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jamie Conway – Georgia Tech
Most work on surgeries in contact manifolds has focused upon determining the situations where tightness is preserved. We will discuss an approach to this problem from the reverse angle: when negative surgery on a fibred knot in an overtwisted contact manifold produces a tight one. We will examine the various phenomena that occur, and discuss an approach to characterising them via Heegaard Floer homology.

Transition path processes and coarse-graining of stochastic system

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 9, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Jianfeng Lu – Duke University –
Understanding rare events like transitions of chemical system from reactant to product states is a challenging problem due to the time scale separation. In this talk, we will discuss some recent progress in mathematical theory of transition paths. In particular, we identify and characterize the stochastic process corresponds to transition paths. The study of transition path process helps to understand the transition mechanism and provides a framework to design and analyze numerical approaches for rare event sampling and simulation.

Deviations of ergodic averages for systems coming from aperiodic tilings and self similar point sets.

Series
CDSNS Colloquium
Time
Monday, March 9, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rodrigo Trevino – Courant Inst. of Mathematical Sciences, NYU
A Penrose tiling is an example of an aperiodic tiling and its vertex set is an example of an aperiodic point set (sometimes known as a quasicrystal). There are higher rank dynamical systems associated with any aperiodic tiling or point set, and in many cases they define a uniquely ergodic action on a compact metric space. I will talk about the ergodic theory of these systems. In particular, I will state the results of an ongoing work with S. Schmieding on the deviations of ergodic averages of such actions for point sets, where cohomology plays a big role. I'll relate the results to the diffraction spectrum of the associated quasicrystals.

Introduction to regularity theory of second order Hamilton-Jacobi-Bellman equation

Series
PDE Working Seminar
Time
Friday, March 6, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Andrzej Sweich – GeorgiaTech –
I will give a series of elementary lectures presenting basic regularity theory of second order HJB equations. I will introduce the notion of viscosity solution and I will discuss basic techniques, including probabilistic techniques and representation formulas. Regularity results will be discussed in three cases: degenerate elliptic/parabolic, weakly nondegenerate, and uniformly elliptic/parabolic.

Dimension and matchings in comparability and incomparability graphs

Series
Graph Theory Seminar
Time
Thursday, March 5, 2015 - 00:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ruidong Wang – Math, GT
In the combinatorics of posets, many theorems are in pairs, one for chains and one for antichains. Typically, the statements are exactly the same when roles are reversed, but the proofs are quite different. The classic pair of theorems due to Dilworth and Mirsky were the starting point for this pattern, followed by the more general pair known respectively as the Greene-Kleitman and Greene theorems dealing with saturated partitions. More recently, a new pair has been discovered dealing with matchings in the comparability and incomparability graphs of a poset. We show that if the dimension of a poset P is d and d is at least 3, then there is a matching of size d in the comparability graph of P, and a matching of size d in the incomparability graph of P.

Fejer-Riesz type argument in non-linear dynamics

Series
Analysis Seminar
Time
Wednesday, March 4, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dmitriy Dmitrishin – Odessa National Polytechnic University
Some interesting applications of extremal trigonometric polynomials to the problem of stability of solutions to the nonlinear autonomous discrete dynamic systems will be considered. These are joint results with A.Khamitova, A.Korenovskyi, A.Solyanik and A.Stokolos

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