Harrison GaeblerDepartment of Mathematics
University of Kansas
1460 Jayhawk Blvd
Lawrence, Kansas 66045-7567
Office: 560 Snow Hall
Office hours: M 1-2pm and by appointment
I am entering my 6th year as a PhD student in the KU math department. See my CV for more details.
I am an analyst and my background is in functional analysis. My advisor, Milena Stanislavova, and I work on infinite-dimensional dynamical systems and in particular our focus has been to derive uniform and sub-exponential bounds for certain Hamiltonian linearizations of NLS and KdV (with periodic boundary conditions), as well as the Dirac Equation.
I also work independently in the area of Banach space theory (see item 1 of the ``Non-peer reviewed/Expository" section below). A time-honored analysis exercise is to prove, for real-valued functions on [0,1] that boundedness and Lebesgue-almost everwhere continuity is equivalent to both Darboux and Riemann integrability. These familiar notions of integrability can be quite easily adapted for general Banach-valued functions on [0,1] and boundedness and Lebesgue-almost everywhere continuity is equivalent in this context to Darboux integrability and implies Riemann integrability. There do, however, exist Banach-valued Riemann-integrable functions on [0,1] that are everywhere discontinuous. If, for example, r1, r2,... is a listing of the rationals in [0,1], then the function f:[0,1]-->c0 defined by f(s)=0 for irrational s and f(rj)=ej where e1, e2,... is the canonical basis for c0 is Riemann-integrable but clearly discontinuous everywhere on [0,1]. A Banach space, X, is therefore defined to have the ``Property of Lebesgue" (and consequently to be a PL-space) if every Riemann-integrable X-valued function on [0,1] is Lebesgue-almost everywhere continuous.
Whether or not a Banach space is a PL-space is closely related to its l1 structure. Indeed, it is known that l1 is a PL-space while lp for 1 < p < ∞ is not. The classical Figiel-Johnson-Tsirelson space is a reflexive Banach space containing no isomorphic copy of c0 or of lp for any 1 <= p < ∞ and it is also a PL-space. Tsirelson's space and several of its variants are so-called asymptotic-l1 with respect to their (countable) bases, as is l1 itself. In 2007, K.M. Naralenkov proved that every asymptotic-l1 Banach space is a PL-space and I re-proved this result independently (I was not aware of the Naralenkov paper at the time) in the first item of the ``Non-peer reviewed/Expository" section below. I further proved that an asymptotic-lp Banach space for 1 < p <= ∞ is not a PL-space if this asymptotic condition is met with respect to a democratic basis (this is only a slight generalization of the lp case for 1 < p < ∞). It is, however, worth noting that the asymptotic-lp condition is a global estimate for all 1 <= p <= ∞ and control over the behavior of particular Riemann sums is certainly a local problem. In view of this realization, I lastly proved in my paper that a spreading model of a PL-space is equivalent to the canonical l1 basis if it is non-trivial, unconditional, and generated by a democratic basic sequence. This can be regarded as a special case of the unpublished result credited by K.M. Naralenkov jointly to A. Pelczynski and G.C. da Rocha Filho that every spreading model of a PL-space that is generated by a normalized basic sequence is equivalent to the canonical l1 basis.
A natural question to ask is if the converse of A. Pelczynski and G.C. da Rocha Filho's result is true. Namely, is X a PL-space if every spreading model generated by a normalized basic sequence is equivalent to the canonical l1 basis? It seems likely that the Property of Lebesgue can be characterized in general Banach spaces in terms of their local asymptotic structure. However, such a characterization remains elusive. One line of thought that has not yet been explored is the relationship between the Property of Lebesgue and the block finite representability of the canonical l1 basis in other basic sequences. Let X be a Banach space that has a normalized countable basis, (ei)i. A well-known result due to Brunel and Shuston implies that there is a subsequence of (ei)i that generates a spreading model and the unpublished result of A. Pelczynski and G.C. da Rocha Filho then implies that this spreading model is equivalent to the canonical l1 basis if X is a PL-space. It is not difficult to prove in this case that the canonical l1 basis is within an epsilon of being nearly (in this sense) crudely block finitely represented in (ei)i. Conversely, one wonders whether or not some variant of crude block finite representation of the canonical l1 basis in the given basis for X is sufficient to ensure that X is a PL-space. A full characterization of the Property of Lebesgue at least among those Banach spaces with countable bases is perhaps possible if so. It is lastly worth noting that certain subsets of Riemann-integrable X-valued functions on [0,1], in particular regulated functions (i.e. uniform limits of step functions) and BV functions (being dense in the regulated functions with respect to the supremum norm), are necessarily Lebesgue-almost everywhere continuous. This is significant in that if one is trying to show that an arbitrary bounded function whose set of discontinuities has positive Lebesgue measure is not Riemann-integrable, then one is assured that such a function can be neither BV nor regulated. There exist real-valued Riemann-integrable functions on [0,1] that are not regulated (for example, f(s)=1 if s=2-k for k=0,1,2,... and f(s)=0 else) and it is safe to assume that Banach-valued versions of such a function are likely to be similarly ill-behaved.
Fall 2020 - Math 115 (by means of Zoom)