Massimo Lanza de Cristoforis (University of Padova)

Title: A necessary and sufficient condition on a singular kernel for the continuity of an integral operator in Hölder spaces and applications to the double layer potential

Abstract: Volume and layer potentials are integrals on a subset Y of the Euclidean space R^{n} that depend on a variable in a subset X of R^{n}. Here we follow a unified approach by assuming that X and Y are subsets of a metric space M and that Y is equipped with a measure ν that satisfies upper Ahlfors growth conditions that include non-doubling measures as done by J. García-Cuerva and A.E. Gatto in a series of papers in case X=Y and for standard kernels, and we prove a necessary and sufficient condition on the kernel for an integral operator to be bounded in Hölder spaces.

Then we present an application to the case of the double layer potential that is associated to the fundamental solution of an arbitrary constant coefficient second order elliptic operator with real principal coefficients.

Will Brian (University of North Carolina, Charlotte)

Title: Partitioning the real line into Borel sets

Abstract: For which cardinals κ is there a partition of the real line into precisely κ Borel sets? If the Continuum Hypothesis holds, then the answer to this question is fairly simple: all κ ≤ |R|. But in other models of set theory, where the Continuum Hypothesis fails, the answer to this question is surprisingly subtle, involving forcing constructions, singular cardinal combinatorics, and large cardinal axioms. In this talk, I will survey some of the history of the question, along with some recent developments.

January 26

3:30 PM - SDRICH 207

Jennifer Loe (Sandia National Laboratory)

Title: A Mathematician's Work at Sandia National Laboratories

Abstract: What is a National Lab? What kind of work do scientists do there? Can I work there with my graduate degree from Baylor? Dr. Jennifer Loe will talke about her path from the Baylor math department to Sandia National Laboratories. She will share how she got an internship there, how that internship turned into a postdoc position, and finally how she became a staff member. She will discuss various research paths in mathematics and computer science at Sandia. Finally, she will conclude wit a briefe presentation of her research in graph partitioning algorithms and in mixed precision linear solver.

January 19

3:30 PM - SDRICH 207

Edmund Y. M. Chiang (Hong Kong University of Science and Technology)

Title: A D-module approach to generating functions and polynomials of some special functions

Abstract: We demonstrate how a Weyl-algebraic treatment to Truesdell’s F-equation theory published in 1948 to derive generating functions of classical special functions would allow a unified treatment of both some classical special functions and their (new) difference analogues. Our approach also illustrates that D-modules is a nature language to describe many classical special functions. In this talk we shall illustrate how the methodology can be applied to Bessel functions and Bessel polynomials, etc. Some realistic applications will be discussed.

November 18

3:30 PM - SDRICH 207

Michail Savvas (The University of Texas at Austin)

Title: Enumerative geometry through sheaves

Abstract: Enumerative geometry is concerned with counts of geometric objects satisfying given constraints. In algebraic geometry, these can be related to geometric spaces, such as curves, and objects of more algebraic flavor, like sheaves, which encode equations and are a generalization of vector bundles. Under certain assumptions, one hopes to consider a moduli space that parametrizes the objects in question and then obtain numerical invariants by integrating cohomology classes capturing the constraints against an appropriate fundamental class in its homology. In this talk, we will review sheaf counting and then discuss how to extend this approach to cases of interest where the usual assumptions do not hold. An important application motivated by physics is the construction of new generalized Donaldson-Thomas invariants enumerating sheaves on Calabi-Yau threefolds. Based on joint works with Young-Hoon Kiem and Jun Li.

November 16

3:30 PM - SDRICH 225

Marcos Mazari-Armida (The University of Colorado Boulder)

Title: Abstract elementary classes of modules

Abstract: Abstract elementary classes constitute a model theoretic framework to study classes of structures. They were introduced by Saharon Shelah in the seventies. The framework is general enough to encompass many interesting classes of structures, but it still allows the development of a rich theory. In this talk, we will introduce some key model theoretic notions, uncover their ring theoretic content, and show how these logical notions can be used to answer an algebraic question. More specifically, we will show that the classical model theoretic notion of superstability is a natural algebraic property by characterizing some classical classes of rings, such as noetherian rings and perfect rings, via superstability of certain classes of modules. Moreover, we will show how the model theoretic notion of stability can be used to solve a problem in abelian group theory due to László Fuchs.

November 10

3:30 PM - SDRICH 207

Ben Hayes (University of Virginia)

Title: An Operator theoretic approach to the convergence of rearranged Fourier series

Abstract: This is joint with Keaton Hamm and Armenak Petrosyan. We study the rearrangement problem for Fourier series introduced by P.L. Ulyanov, who conjectured that every continuous function on the torus admits a rearrangement of its Fourier coefficients such that the rearranged partial sums of the Fourier series converge uniformly to the function. The main theorem here gives several new equivalences to this conjecture in terms of the convergence of the rearranged Fourier series in the strong (equivalently in this case, weak) operator topologies on B(L2(T)). We also provide characterizations of unconditional convergence of the Fourier series in the SOT and WOT. These considerations also give rise to some interesting questions regarding weaker versions of the rearrangement problem. Towards the end of this talk, I might indicate how these problems can be generalized from Z^{d} to arbitrary general groups.

October 27

3:30 PM - SDRICH 207

Heather Wilber (Oden Institute at the University of Texas)

Title: Zolotarev rational functions in computational mathematics

Abstract: In the late 1800s, Y. Zolotarev (a student of Chebyshev) posed and solved two important rational approximation problems. These problems (and variations of them) arise in many modern applications in numerical linear algebra, signal processing, and computational mathematics. This talk highlights the role of Zolotarev's problems in modern computing and illustrates how classical ideas in approximation theory, such as conformal mapping, can be put to use in numerical contexts. We focus primarily on Zolotarev's so-called 4th problem, the best approximation to the sign function, and use it to inspire and develop new, spectrally accurate methods for solving the spectral fractional Poisson equation.

October 11

3:30 PM - MMSci 101

Annalisa Crannell (Franklin and Marshall College)

Title: Drawing conclusions from drawing a square

Abstract: The Renaissance famously brought us amazingly realistic perspective art. Creating that art was the spark from which projective geometry caught fire and grew. This talk looks directly at projective geometry as a tool to illuminate the way we see the world around us, whether we look with our eyes, with our cameras, or with the computer (via our favorite animated movies). One of the surprising results of projective geometry is that it implies that every quadrangle (whether convex or not) is the perspective image of a square. We will describe implications of this result for computer vision, for photogrammetry, for applications of piecewise planar cones, and of course for perspective art and projective geometry.

October 6

3:30 PM - SDRICH 207

Eric Ricard (University of Caen)

Title: The non commutative Khintchine inequalities

Abstract: The Khintchine inequalities are basic and fundamental inequalities at the interface of probabilties and functional analysis. We will first review some classical facts about them. Then, we will introduce what is meant by non commutative analysis by illustrating it with matrices. It took almost 30 years to get a full version of the non commutative Khintchine inequalities. Their formulation is at the heart a truly non commutative difficulty. We will explain the history that led to a satisfactory but not totally full understanding. The talk only requires very basic notions.

Click here to see a list of past colloquium talks.

DEPARTMENT OF MATHEMATICS

Sid Richardson Science Building 1410 S.4th Street Waco, TX 76706