The talks will typically take place on Tuesdays at 4:00-5:00pm in Adel Room 164. Please contact Nandor Sieben if you would like to give a talk or have a question about the colloquium.
Short organizational meeting
Speaker: Jin Wang Title: Generalized Depth-Based Trimmed Means and Trimmed Scatter Matrices (Sabbatical report)
Abstract: Multivariate descriptive measures for location and scatter are the foundation of multivariate statistics and underpin almost all methods in the field. In this paper, we propose and study new general depth-based trimmed means and scatter matrices, along with their sample versions (estimators). In addition to their basic properties, we establish the strong consistency and asymptotic distributions of these estimators. Using the asymptotic distributions, we compute the asymptotic relative efficiencies of the sample trimmed means and sample trimmed scatter matrices based on the halfspace depth, with respect to the sample mean vector and the sample covariance matrix, respectively. Robustness is explored through influence function and finite-sample breakdown point. The results show that the sample trimmed means and scatter matrices are not only highly efficient but also exceptionally robust, making them highly competitive estimators for multivariate location and scatter.
Speaker: Annie Boyd, Ben Jefferies, Gina Nabours Title: LMC Course Data Update
Abstract: Courses in the LMC have undergone numerous curricular changes since the LMC was established in 2012. This summer, the LMC Administrative team analyzed data to see if there was a statistically significant impact on student pass rates due to these changes. We will discuss the curricular changes in the 4 LMC math courses and share results from our analysis on student success and the impact of the curricular changes in subsequent math courses.
Speaker: Mike Falk Title: Oriented matroids and Orlik-Solomon algebras
Abstract: A theorem of Orlik and Solomon from 1982 shows that the cohomology ring of the complement X of a union of complex hyperplanes is determined by certain combinatorial data associated with the collection of hyperplanes, encoded in a matroid M. One can define this ring directly from the matroid, resulting in the so-called Orlik-Solomon (OS) algebra of M. For arrangements of complexified real hyperplanes the matroid has additional structure, known as an orientation. For oriented matroids there is an abstract simplicial complex, the Salvetti complex, that models the homotopy type of X. It is a theorem of Gelfand and Rybnikov from 1989 that the cohomology of this simplicial complex is isomorphic to the OS algebra of the underlying matroid, for any oriented matroid, whether or not it arises from a hyperplane arrangement. No proof of this more general theorem has ever appeared.
With Emanuele Delucchi we have written a proof of this result using an alternative to the Salvetti complex called the tope-pair complex, the order complex of the tope-pair poset, introduced in our earlier work from 2017. The argument is an analogue of the classical inductive topological argument in the context of poset topology. We’ll explain the classical argument and the poset analogues of the main steps, and discuss implications with regard to newly-discovered examples of complex-realizable matroids with non-realizable orientations.
Speaker: Mikhail Baltushkin Title: Isomorphism theorems for gamegraphs
Abstract: Combinatorial Game Theory typically focuses on two-player games that involve no elements of chance. We model these games using specialized directed graphs, called rulegraphs, and develop a theory of rulegraphs analogous to universal algebra, where homomorphisms are replaced by a special class of digraph maps known as option-preserving maps. By introducing congruence relations, we define quotient rulegraphs and establish results that parallel the four isomorphism theorems in universal algebra within the framework of rulegraphs.
Speaker: Nandor Sieben Title: Impartial Loopy Games
Abstract: We introduce impartial combinatorial games where a draw is possible as a result of infinite play. We define the remoteness function to measure the length of optimal play. The remoteness function can be used to compute extended nim-values. The relationship between extended nim-values and game sums is also explored.
Cancelled:
Speaker: Giorgio Cipolloni (UA) Title: A story of non-Hermitian random matrices
Abstract: We will discuss recent progresses in the study of the fluctuations in the spectrum of non-Hermitian random matrices. In particular, we will present a new connection between non-Hermitian matrices and two- and three-dimensional logarithmically correlated fields.
Speaker: Anne Carter Title: Two-variable polynomials with dynamical Mahler measure zero
Abstract: Introduced by Lehmer in 1933, the classical Mahler measure of a complex rational function $P$ in one or more variables is given by integrating $\log |P(x_1, \ldots, x_n)|$ over the unit torus. Lehmer asked whether the Mahler measures of integer polynomials, when nonzero, must be bounded away from zero, a question that remains open to this day. In this talk we generalize Mahler measure by associating it with a discrete dynamical system $f: \mathbb{C} \to \mathbb{C}$, replacing the unit torus by the $n$-fold Cartesian product of the Julia set of $f$ and integrating with respect to the equilibrium measure on the Julia set. We then characterize those two-variable integer polynomials with dynamical Mahler measure zero, conditional on a dynamical version of Lehmer’s conjecture.
Speaker: Andrew Schultz (Wellesley College) Title: Galois module structure of psth power classes of a field
Abstract: When a field $K$ contains a primitive $p$ th root of unity, Kummer theory tells us that the $\mathbb{F}_p$-space $K^{\times p}/K^\times$ is a parameterizing space for elementary $p$-abelian extensions of $K$. In previous work, the authors computed the Galois module structure of this set when the Galois group came from an extension $K/F$ whose Galois group is isomorphic to $\mathbb{Z}/p^n\mathbb{Z}$. In this talk we consider the more refined group $K^{\times p^s}/K^\times$ as a Galois module, and we determine its structure. Although the modular representation theory in this case is unwieldy, it turns out that there is only one summand in the decomposition of $K^{\times p^s}/K^\times$ which is not free (either under the full ring or one of its natural quotients). Furthermore, this “exceptional” summand’s structure is connected to the cyclotomic character and a certain family of embedding problems along the tower $K/F$. This work is joint with J'{a}n Min'{a}\v{c} and John Swallow.
Speaker: Jim Swift Title:
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Speaker: Joe Polman (CU Boulder) CSTL STEM Education speaker series Title:
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Speaker: Jeff Hovermill Title:
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Speaker: UGRADS Robert Title:
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Speaker: Robert ? Title:
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Speaker: Prabath Silva Title:
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