Events
Department of Mathematics and Statistics
Texas Tech University
We develop a theory of self-similar solutions to the critical surface quasi-geostrophic equations. We construct self-similar solutions for arbitrarily large data in various regularity classes and demonstrate, in the small data regime, uniqueness and global asymptotic stability. These solutions are non-decaying in space which leads to ambiguity in the drift velocity. This ambiguity is corrected by imposing m-fold rotational symmetry. The self-similar solutions exhibited here lie just beyond the known well-posedness theory and are expected to shed light on potential non-uniqueness, due to symmetry-breaking bifurcations, in analogy with work of Jia and Sverak on the Navier-Stokes equations. This is joint work with Dallas Albritton of Courant Institute, NYU.
This week's Biomath seminar details available at this pdf
In their 1987 paper, Eliahou and Kervaire constructed a minimal
resolution of a class of monomial ideals of a polynomial ring, and
they showed that this resolution has the structure of a differential
graded algebra. I will discuss how the minimal resolution they
constructed can be extended to a skew polynomial ring, giving the
minimal resolution the structure of a color commutative differential
graded algebra.
Join Zoom Meeting https://zoom.us/j/96217128540?pwd=NjU5dzE2RjZvV0prejhOOWVjVENadz09
Meeting ID: 962 1712 8540
Passcode: 474170
The talk is accessible to beginning graduate students. Consider a geometric functional such as the area or the volume. In order to find local extrema, we calculate the second variation which is normally a symmetric bilinear form in a function space. The Morse index, which counts the maximal dimension of a subspace on which the form is negative definite, is the generalization of the 2nd derivative test in elementary calculus. Now consider the natural extension of finding extrema subject to certain constraints. How would we examine the index with constraints? In this talk, we’ll answer that question in a general abstract framework and then apply it to study capillary surfaces. This is joint work with Detang Zhou.
Rational homotopy theory is an extremely rich source of algebraic models for geometry and topology. For example, minimal Sullivan models are in 1-1 correspondence with rational homotopy types. In this expository talk we explore this correspondence, along with the rational equivalence between simply connected spaces and connected differential graded Lie algebras.