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# Departmental Colloquia

Our department is proud to host weekly colloquium talks featuring research by leading mathematicians from around the world. Most colloquia are held on Fridays at 4pm in Parker Hall, Room 250 (unless otherwise advertised) with refreshments preceding at 3:30pm in Parker Hall, Room 244.

DMS Colloquium: Honglang Wang

Apr 20, 2018 04:00 PM

Speaker: Honglang Wang, IUPUI (Indiana University--Purdue University Indianapolis)

Title: Unified empirical likelihood ratio tests for functional concurrent linear models and the phase transition from sparse to dense functional data

Abstract: We consider the problem of testing functional constraints in a class of functional concurrent linear models where both the predictors and the response are functional data measured at discrete time points. We propose test procedures based on the empirical likelihood with bias‐corrected estimating equations to conduct both pointwise and simultaneous inferences. The asymptotic distributions of the test statistics are derived under the null and local alternative hypotheses, where sparse and dense functional data are considered in a unified framework. We find a phase transition in the asymptotic null distributions and the orders of detectable alternatives from sparse to dense functional data. Specifically, the tests proposed can detect alternatives of √n‐order when the number of repeated measurements per curve is of an order larger than with n being the number of curves. The transition points  for pointwise and simultaneous tests are different and both are smaller than the transition point in the estimation problem. Simulation studies and real data analyses are conducted to demonstrate the methods proposed.

Faculty host: Guanqun Cao

Apr 10, 2018 04:00 PM

Title: Special Quadrature Processes for Summation of Slowly Convergent Series

Abstract: Slowly convergent series  appear in many problems in mathematics, physics  and other  sciences. There are several numerical methods based on linear and nonlinear transformations (e.g., Cesàro-transformation,  Aitken $$\Delta^2$$-transformation, Wynn-Shanks $$\varepsilon$$-algorithm, $$E$$-algorithm, Levin's transformation,  $$\rho$$-algorithm, etc.). In this lecture  we present the so-called summation/integration methods, which are very efficient. Methods are based on certain transformations of sums to weighted integrals over the real line or the half-line and  on an application of special Gaussian quadrature formulas with respect to some non-classical weight functions (Bose-Einstein, Fermi-Dirac, hyperbolic weights, $$\ldots$$). For constructing such quadrature rules we use a recent progress in symbolic computation and variable-precision arithmetic, implemented through our Mathematica package “OrthogonalPolynomials.” Several interesting applications will also be presented.

Faculty host: Narendra K. Govil

DMS Colloquium: Claudiu Raicu

Apr 06, 2018 04:00 PM

Speaker: Claudiu Raicu (Notre Dame)

Title: Koszul modules

Abstract: The Cayley-Chow form of a projective variety $$X$$ is an equation that detects when a linear space intersects $$X$$ non-trivially. I will explain how it can be described when $$X$$ is the Grassmannian of lines in its Plücker embedding, by relating it to a fascinating class of modules called Koszul modules. Despite their elementary definition, Koszul modules have close ties to the study of syzygies of generic canonical curves, and provide important applications to the structure of certain invariants of finitely presented groups.

Joint work with M. Aprodu, G. Farkas, S. Papadima, and J. Weyman.

DMS Colloquium: Emily King

Mar 30, 2018 04:00 PM

Speaker: Emily King (U. Bremen)

Title: (Hilbert Space) Frames, Algebraic Combinatorics, and Geometry

Abstract: Frames are generalizations of orthonormal bases which yield "nice" decompositions of data. Such systems are the foundation of applied harmonic analysis and are also closely related to quantum measurements and linear codes.  When one wants an optimally robust representation of data, one often looks for frames that have some sort of spread, be it geometric (as non-parallel as possible) or algebraic (no nontrivial linear dependencies).  Over the last few years, it has been discovered that the relationship between these two types of spread is more complicated than had previously been believed. Furthermore, methods from algebra, geometry, and combinatorics have recently proven themselves to be very useful in the study of frames.

For example, algebraic graph theory and combinatorial design theory have led to new characterizations and novel constructions of optimal line (or, more generally, subspace) configurations which are also frames. Also, almost all desirable classes of frames form real algebraic varieties, and certain known results in frame theory have also been found to be equivalent to concepts in matroid theory and arrangements of hyperplanes.

In this talk, the currently known connections between these objects from harmonic analysis / quantum information theory and combinatorics / algebraic & discrete geometry will be presented.

(photo courtesy Uni Bremen/Kai Uwe Bohn)

DMS Colloquium: Jan Rosinski

Mar 23, 2018 04:00 PM

Speaker: Jan Rosinski, University of Tennessee

Title: Series expansions of time-continuous random walks and solutions to random differential equations

Abstract: In 1923 N. Wiener constructed two random trigonometric series which converge uniformly to a limit satisfying conditions for a Brownian motion. The conditions were  earlier  postulated by A. Einstein in terms of partial differential equations.  K. Ito, the founder of Ito stochastic calculus, unified results on the convergence of these and other similar series expansions in the so-called  Ito-Nisio theorem (1968). O. Kallenberg (1974) gave the first generalization of the Ito-Nisio theorem to the uniform convergence of processes with jumps.

Brownian motion describes continuous in space random walk. In this talk we will concentrate on random walks with jumps, their series expansions, and strongest possible modes of their convergence. To this aim we will establish further generalization of the Ito-Nisio theorem.  We will discuss the Ito map, which is just an ODE with a rough input. Using these tools we obtain strong pathwise convergence in numerical solutions of stochastic differential equations driven by Levy processes.

This talk is based on a joint work with Andreas Basse-O’Connor and Jorgen Hoffmann-Jorgensen.

Faculty host: Erkan Nane

DMS AU-AUM Joint Colloquium: Robert Underwood

Mar 09, 2018 04:00 PM

AU-AUM Joint Math Colloquium

Speaker: Robert Underwood, AUM

Title: A Class of Automatic Sequences