Antoine Marmat and Nikolay Moshchevitin (Univ. Graz and Moscow State Univ.)
X
We will discuss the problem of optimal inequalities between the ordinary and the uniform Diophantine exponents, for simultaneous approximation as well as for approximation with one linear form. The first result, dealing with approximation to two numbers, was obtained by V. Jarnik in 1940. Quite recently W,M. Schmidt and L. Summerer introduced a new method of Parametric Geometry of Numbers which gave new understanding of related problems. In particular they formulated a conjecture about the optimal bounds between exponents.We will speak about a proof of this conjecture and related topics
Uniform and ordinary Diophantine exponents
CANCELED Jan. 29, 2019
Kempner
Catherine Sulem (University of Toronto)
X
The Derivative Nonlinear Schrodinger (DNLS) equation is a long-wave, weakly non-linear model that arises in the context of the Hall magnetohydrodynamic equations for conducting fluids.
It was known since the work of Hayashi (1992) that solutions exist globally in time for small initial data but the case of large data remained open. Exploiting the complete integrability of the DNLS equation discovered by Kaup and Newell (1978), we prove global wellposedness for general initial conditions in weighted Sobolev spaces. Furthermore, for initial conditions that support bright solitons (but exclude spectral singularities), we give a full description of the long-time behavior of the solutions in the form of a finite sum of localized solitons and a dispersive component. This property is usually referred to in the literature as the soliton resolution conjecture.
This is a joint work with Robert Jenkins (Colorado State University), Jiaqi Liu (University of Toronto) and Peter Perry (University of Kentucky).
Global wellposedness and soliton resolution for the derivative nonlinear Schrodinger equation
Jan. 29, 2019 1pm (MATH 220)
Grad Algebra/Logic
Peter Mayr (CU Boulder)
X
We show that solvable quotients of subdirect products of perfect groups are nilpotent of arbitrary degree. This answers a question by Derek Holt. Joint work with Keith Kearnes and Nik Ruskuc.
Subdirect products of perfect groups
Jan. 29, 2019 2pm (MATH 220)
Michael Wheeler (CU Boulder)
X
Until recently, the question of whether p < t is consistent with ZFC had been the longest open problem regarding cardinal invariants of the continuum. Malliaris and Shelah proved that p < t is inconsistent and so p = t for all models of ZFC. In these talks we will present a simplified version of Malliaris and Shelah's argument due to Douglas Ulrich. In this first week, we define omega-nonstandard models of ZFC, give some sense of what these models look like, and give a brief outline of the proof that p = t. We will present the basic definitions needed for the argument, and, time permitting, will present the result that "small" types of omega-nonstandard models that are consistently realized in a given pseudofinite set are always realized.
p = t via Nonstandard Models of Set Theory, Part 1
Jan. 29, 2019 4pm (MATH 350)
Topology
Agnes Beaudry (CU Boulder)
X
I will give some background on topological field theories with a view towards the Freed-Hopkins-Teleman results on invertible field theories. This will partly be an organizational meeting and talks will be assigned for the rest of the semester.
Uniform and ordinary Diophantine exponents