Speaker: Shriram Srinivasan, Applied Mathematics & Plasma Physics Group, Los Alamos National Laboratory
Date: April 24, 2023
Time/location: 4pm @ GAB461
Title: A unified treatment of matrix calculus on structured spaces
Abstract: The modern viewpoint of calculus and linear algebra focusses on a study of derivatives of functions between normed linear spaces or inner-product spaces, a unified treatment that encompasses calculus of vector and matrix spaces.
However, the practice of matrix calculus as seen in most published work seems to follow an approach antithetical to this modern viewpoint.
As a result, there is no established technique to calculate the gradient for scalar functions defined on \emph{any} arbitrary matrix subspace.
Moreover, the gradient derived from extant methods is incorrect for functions defined on some special matrix subspaces (such as spaces of symmetric or skew-symmetric matrices), and these results populate several publications, as well as respected textbooks and handbooks on matrix calculus.
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One of our important contributions has been to wade through the published works carrying these wrong results and set up the calculation of the gradient defined on matrix subspaces in a rigorous and concrete mathematical setting of a finite-dimensional inner-product space.
Following the reasoning behind the spurious results in matrix calculus, we cast the derivative calculation in terms of a scalar function of a vector and in this process, uncover the reason behind the spurious results obtained for various matrix subspaces.
We also propose an alternative to correct the fundamental flaw.
Short Bio
Shriram Srinivasan earned his M.S. in Mathematics and Ph.D. in Mechanical Engineering, all from Texas A & M University, College Station. His research interests are in computational mechanics and reduced-order models of structured systems such as fracture networks and gas pipeline networks. He is a staff scientist in the Applied Mathematics and Plasma Physics Group of the Theoretical Division at Los Alamos National Laboratory, New Mexico.
Speaker: James Zhang, University of Washington
Date: March 6, 2023
Time/location: 4pm @ GAB461
Title: Automorphism problem of noncommutative algebras
Abstract:
Computing and understanding the automorphism group of an algebra is a notoriously difficult problem. We survey on recent results about the automorphism group of important classes of noncommutative algebras. New methods such as using rigidity, discriminant, and valuation will be discussed.
Speaker: Sascha Troscheit, University of Oulu (Finland)
Date: November 28, 2022
Time/location: 4pm @ GAB461
Title: Zoom in and enhance: Dimension theory and finer geometry of metric spaces
Abstract:
Dimension theory and fractal geometry study the local structures of geometric objects such as sets and measures. This is achieved through various 'dimensions' that encapsulate scaling behaviour. In recent years, much progress was made on the family of Assouad-type dimensions. These dimensions are linked to fine local information of geometry and have found their application in metric geometry and random geometry. I will give a tour of the last decade of research in the area and their connections to embeddability and structural theorems for random and deterministic structures.
Bio:
Sascha Troscheit is a research fellow at the University of Oulu (Finland). He completed his PhD at the University of St Andrews (Scotland) under the supervision of Kenneth J. Falconer and Mike Todd. Before coming to Oulu, he has held positions at the University of Waterloo (Canada) and the University of Vienna (Austria). In Vienna, Sascha was a Lise Meitner Fellow and is now a Marie Skłodowska-Curie Fellow funded by the Horizons Europe programme. His expertise lies in deterministic and random fractal geometry and their interactions with number theory. In his spare time, Sascha can be found sailing the waters off the British, Dutch, and German coasts.
Speaker: Mike P. Cohen, C.H. Robinson
Date: November 7, 2022
Time/location: 4pm @ GAB461
Title: Three math tracks: each alike in dignity?
Abstract: Many math Ph.D. seekers hope to find a career as either (1) a research mathematician at a doctoral granting institution; (2) a teaching professor at a liberal arts college; or (3) an industry mathematician at a Fortune 500 company. Nine years after completing my Ph.D. in math at UNT, I find myself in the weird, but perhaps not uncommon, situation of having experienced all three! I'll share some details of how I wandered through three very different environments (academic jobs at North Dakota State University and at Carleton College, and through to my current position as a data scientist at C.H. Robinson) and describe what I perceive as strengths, drawbacks, and major takeaways from these disparate career paths. The main audience for this talk is UNT math grad students, for whose benefit I will emphasize frankness, even around such shocking taboo topics as: research fatigue; two-body problems; money; what it might take to get hired (besides luck); data science bootcamps; undergraduate research mentorship; and whatever else you can think to ask about.
This special colloquium will follow a unique format: a short talk follows by a panel discussion. The PhD graduate panelist is Ms. Jill Kaiser and the moderator is Dr. Kiko Kawamura. Cookies and coffee/tea will be served in GAB 473 prior to his talk.
Speaker: Dominik Adolf, UNT
Date: October 17, 2022
Time/location: 4pm @ GAB461
Title: Mutual Stationarity at the least singular cardinal
Abstract: Let X be any set. We say a collection of subsets of X is stationary if it contains at least one set closed under any given algebra on X. This is, perhaps, the closest we get to a universal system of measure. In this talk we will discuss what sets on the least singular cardinal can be stationary using the notion of mutual stationarity introduced by Foreman and Magidor. We will see that a rich set of sequences can consistently be mutually stationarity if we have the right set of total, non-principal 0-1 measures of a fixed uncountable additivity. This is joint work with Omer Ben-Neria.
Speaker: Hung Tran, Texas Tech
Date: October 3, 2022
Time/location: 4pm @ GAB461
Title: Geometry and Topology: A Tale of Rigidity
Abstract: A mainstream in mathematics is to study the relation between the geometry and topology of a manifold. The geometry is about the distance, length, area, and volume and is determined by the curvature. The topology is about properties that are preserved under continuous deformations. It is generally observed that a suitable assumption on the geometry would lead to certain restrictions on the topology. In this talk, we'll discuss some recent progress in this direction, particularly from the perspective of the curvature of the second kind. There will be a resolution of a conjecture proposed by S. Nishikawa and rigidity results for Einstein metrics and gradient Ricci solitons. Much of the talk is based on joint work with X. Cao, X. M. Gursky, and E. Ribeiro Jr.
Speaker: Harrison Gaebler, UNT
Date: September 19, 2022
Time/location: 4pm @ GAB461
Title: Riemann Integration and Asymptotic Structure of Banach Spaces
Abstract: I will introduce the topic of Riemann integration in Banach spaces in this talk and show briefly the curious property that a Banach-valued Riemann-integrable function need not be Lebesgue a.e. continuous. A Banach space X such that every X-valued Riemann-integrable function is Lebesgue a.e. continuous is said to have the Property of Lebesgue (PL). It is an open problem to characterize the PL in terms of asymptotic structure. I will detail what is currently known with respect to this problem and where the research is now headed. In addition, I will discuss at the end some related questions such as non-linear stability of the PL and whether or not a Banach space with the PL has the Radon-Nikodym property.
Speaker: Kirill Lazebnik, UNT
Date: September 12, 2022
Time/location: 4pm @ GAB461
Title: A Geometric Approach to Runge's theorem
Abstract: We will recall Runge's theorem on approximation of analytic functions (in one complex variable) by polynomials, and remark how a lack of a global understanding of the polynomial approximant limits potential applications of this theorem in complex dynamics. We will discuss a different approach to this classical result which gives a fuller understanding of the polynomial approximants, and discuss potential applications. This talk is based on joint work with Chris Bishop.