Colloquium Series
Asymptotic Cones Without Zorn's Lemma
Thursday, Feb 13, 2025 from 3:00-3:50 pm in Laws 304
Luke Serafin, Cornell University
Abstract: The asymptotic cone of a metric space is a form of scaling limit which can be thought of as the view of the space from infinitely far away. For example the asymptotic cone of the two-dimensional integer lattice is the Euclidean plane. Asymptotic cones capture the coarse geometry of spaces both in the sense of being scaling limits and in the sense that they are quasi-isometry invariants of metric spaces. In geometric group theory one considers groups as metric spaces via their Cayley graphs, and asymptotic cones of groups turn out to be of particular interest. Unfortunately asymptotic cones of the broad class of groups called hyperbolic groups are all isometric to the space called the universal continuum-valent R-tree, so they provide no useful information as quasi-isometry invariants for groups of this class. But the proof uses Zorn's lemma; could it be that a more constructive variant of asymptotic cones would be useful in the study of hyperbolic groups? We do not answer this question, but do indicate that there are models of set theory which fail to satisfy Zorn's lemma and in which there are hyperbolic groups which are not isometric to any continuum-valent R-tree. This work is joint with Justin Moore.
2024-2025 Colloquium Series
Speaker/Affiliation:
Michel Gaspar, University of São Paulo
Title: How many colors does Borel need?
Abstract: The chromatic number of a graph is the minimum number of colors necessary to color
each vertex so that no two vertices forming an edge have the same color. By restricting
to colorings with definable properties, we obtain variants of the chromatic number. In
this talk, we will explore the "Borel chromatic number" of a graph, which can always be
defined for graphs whose set of vertices has a natural topology. We will see that, for
various natural mathematical structures, the independence phenomenon cannot be
avoided: for some graphs, one cannot always uniquely determine the values of their
Borel chromatic numbers.
Date/Time/Location:
Thursday, September 12, 2024 from 3:00-3:50 pm in UPH 275
Speaker/Affiliation:
Michał Wojciechowski, Institute of Mathematics Polish Academy of Sciences
Title: Unexpected approximation of Gaussian density and Hermite functions by B-splines
Abstract: We prove, and provide a quantitative error estimate, that a B-spline based on
essentially arbitrarily distributed knots approximates the Gaussian density. This
construction provides an unexpected inequality for any polynomial with distinct real
roots. On the other hand, one can see this as a kind of generalization of CLT for the
uniform distribution, which is exactly what arises for equally spaced knots. Moreover,
we prove that the approximation could be made in the Sobolev norms which leads
to analogous approximations of Hermite functions
The talk is based on a joint work with M. Rzeszut.
Date/Time/Location: (Note the unusual day, time, and place)
Wednesday, September 18, 2024 from 1:20-2:10 pm in UPH 131
Speaker/Affiliation:
Martin J. Mohlenkamp, Ohio University
Title: The Hunt for the Swamp Monster
Abstract: There is a criminal on the loose in Tensorland. As innocent tensor approximation programs go about their work, sometimes they get ``swamped''. A swamped program has to work for many iterations without making appreciable progress before finally breaking free; sometimes they expire while trying. The criminal has been active since programs moved into Tensorland some 50 years ago, but has never been identified. Blurry images and indistinct tracks abound, leading to rumors and superstitions about the Swamp Monster. Join us as we construct and set a trap for the Swamp Monster. What will we find when the trap springs?
Date/Time/Location:
Thursday, October 3, 2024 from 3:00-3:50 pm in UPH 275
Speaker/Affiliation:
Borys Prydalnyi, Lutsk National Technical University, Lutsk, Ukraine
Title: Algorithms for the synthesis of complex technical systems with ill-defined parameters
Abstract: The work aims to improve the conditions for solving the problem of increasing the productivity and quality of the development of new technical systems in the context of the technological paradigm challenges of Industry 4.0 (and partly the prerequisites of the 5.0 paradigm). The research is focused on enhancing the conditions for automating the initial stages of technical systems creation, which encompass the formulation of its operational principles, structure, scheme, and design configuration. At present, these functions are carried out by a development engineer. In the studies, technical systems were conceptualized as an ordered set of structural elements and sustainable connections between them, which ensures the possibility of realizing their target functions. The main result of this research is the development of the preconditions of the theory for automated algorithmic structural-schematic synthesis of technical systems using a digital representation of their structural elements. To achieve this goal, the research project has established a set of principles for the digital representation of technical systems, a vector description of their structural elements, and the creation of databases of these elements. The use of vector representations of structural elements
allows the combination of structural elements and the generation of targeted structural schemes,
thereby enabling the achievement of the desired characteristics for the functioning of a given technical
system.
Date/Time/Location:
Wednesday, October 9, 2024 from 1:25-2:15 pm in UPH 235
Speaker/Affiliation:
Daniele Rosso, Indiana University Northwest
Title: Flag varieties and variations on the Robinson-Schensted-Knuth correspondence
Abstract: The Robinson-Schensted correspondence (which was generalized by Knuth) is a combinatorial algorithm that gives a bijection between permutations and pairs of standard Young tableaux of the same shape. Steinberg and Spaltenstein observed that it can be interpreted in terms of the geometry of flags (increasing sequences of subspaces in a vector space). I will discuss some variations on this construction, some of them from joint work with V. Nandkumar and N. Saunders.
Date/Time/Location:
Thursday, November 7, 2024 from 3:00-3:50 pm in UPH 275
Speaker/Affiliation:
Brett Mullins, UMass Amherst
Title: Efficient Marginal Reconstruction under Differential Privacy
Abstract: Differential privacy is the dominant standard for formal and quantifiable data privacy and has been used in major deployments that impact millions of people. Many differentially private mechanisms for query release and synthetic data contain steps that reconstruct answers to queries from answers to other queries privately measured. Reconstruction is an important subproblem for such mechanisms to economize the privacy budget, minimize error on reconstructed answers, and allow for scalability to high-dimensional datasets. In this talk, we introduce a principled and efficient postprocessing method ReM (Residuals-to-Marginals) for reconstructing answers to marginal queries under Gaussian noise. Our method builds on recent work on efficient mechanisms for marginal query release, based on making measurements using a residual query basis that admits efficient pseudoinversion, which is an important primitive used in reconstruction.
Date/Time/Location:
Thursday, November 14, 2024 from 3:00-3:50 pm in UPH 275
Speaker/Affiliation:
Pradyut Karmakar, Ohio University
Title: Fourier type convergence in groupoid C*-algebras
Abstract: If G is an étale groupoid and \(\Sigma\) is an associated twist, we have shown that if G has the Rapid Decay Property with respect to a conditional negative definite length function L, then we can recover an element of the groupoid C*-algebra from its support.
More precisely, if \(f \in C_r^{\ast}(\Sigma, G)\) with \(\mathrm{supp}(f) \subset U\) for some open subset of G, then it implies that \(f \in \overline{C_c(U)}^{r}\).
The construction is primarily based on the \(\ell^2\)-norm and is inspired by the findings of Bédos and Conti in a group context, as well as the work on amenable groupoids by Brown, Exel, Fuller, Pitts, and Reznikoff. Our results apply to certain non-amenable groupoids, including free groups and hyperbolic groups, which demonstrate the aforementioned phenomena—specifically, that elements can be recovered from the spectral subspace. This also highlights the fact that the Fourier series of elements can be recoverable for non-nuclear C*-algebras.
Date/Time/Location:
Tuesday December 3, 2024 from 3:00-3:50 pm in UPH 275
