New Mexico State University

Monday, October 6

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Tuesday, October 7

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The Graduate Student Seminar is designed to give graduate students a low-pressure environment to gain experience presenting their work to their peers, and to practice public speaking skills more generally. Interested in speaking? Fill out this form: https://forms.gle/w2WNNJ2XgaYApYgv8. 

Questions? Contact Trevor Jess at tkjess@nmsu.edu or Susan Harding at susaneh@nmsu.edu.

Wednesday, October 8

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Thursday, October 9

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The modal algebra of a graph is its powerset algebra endowed with a unary operation induced by the edges. We consider relations in metric spaces, in particular those consisting of pairs of points that are more than 1 apart, less than 1 apart, or exactly 1 apart. The corresponding operations are expressive enough to capture various aspects of metric spaces such as colorability and packing problems. I will give an overview of some results in this area with a focus on recent joint work with Gabriel Agnew, Uzias Gutierrez-Hougardy, John Harding, and Jackson West.

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Friday, October 10

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Beyond its conceptual impact on noncommutative Choquet theory, the counterexample to Arveson's hyperrigidity we discovered with Bilich informs us on how to prove functoriality properties for the C*-envelope in non-commutative dynamical contexts, which then leads to insight on structural and classification problems through the use of the C*-envelope. For instance, for the seemingly unrelated, long-standing Hao--Ng isomorphism problem, which asks whether the Cuntz--Pimsner C*-algebra construction commutes with taking the full or reduced crossed product by a locally compact Hausdorff group action, beyond the amenable case of Hao and Ng.

Building on ideas that emerge from the discovery of the counterexample to Arveson's hyperrigidity conjecture with Bilich, together with ideas arising from a recent amendment to Arveson's hyperrigidity conjecture by Clouatre and Thompson, through the use of the C*-envelope we resolve the reduced version of the Hao-Ng isomorphism problem in full generality.  More precisely, for a non-degenerate \mathrm{C}^*-correspondence $X$ and a generalized gauge action $G \curvearrowright X$ by a locally compact Hausdorff group $G$, we prove the commutation ${\mathcal{O}}_{X\rtimes_rG}\cong {\mathcal{O}}_X\rtimes_rG$ of the reduced crossed product with the Cuntz-Pimsner C*-algebra construction.

Based on joint work with Ian Thompson.

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The Hilbert consistency program of the 1920s aimed to prove the consistency of mathematical theories by trusted means. This program was effectively cancelled in the 1930s when Gödel's Second Incompleteness Theorem, G2, was interpreted as “there exists no consistency proof of a system that can be formalized in the system itself” (Encyclopædia Britannica). We demonstrate that this Unprovability of Consistency Thesis, UCT, has been based on an oversight: the unprovable consistency formula used in G2 was stronger than the original consistency property. Therefore, G2 did not actually yield UCT.  Building on Hilbert's efforts, we found a consistency proof of the formal Peano Arithmetic PA, which can be formalized in PA itself. We extended this result to other mathematical theories, including Set Theory. This yields an unexpected refutation of UCT and removes the principal roadblock of the consistency studies.

References

Serial properties, selector proofs and the provability of consistency, Journal of Logic and Computation, Volume 35, Issue 3, April 2025.
Consistency formula is strictly stronger in PA than PA-consistency. arXiv:2508.20346

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A question going back to Halmos asks when two approximately commuting matrices of a certain kind are close to genuinely commuting matrices of the same kind. It was quickly realized that dimension independent results were far more difficult to obtain, and in work of Lin the result was settled for self-adjoint matrices. However, the precise control in terms of the commutator was unknown until effective bounds were achieved in the seminal work of Kachkovskiy and Safarov. On the other hand, it has long been known that there are obstruction for dimension independent approximately commuting unitary matrices to be close to commuting unitary matrices. In work of Gong and Lin, and work of Eilers, Loring and Pedersen, it was finally shown that when an index obstruction vanishes, approximately commuting unitary matrices are close to commuting unitary matrices. However, effective bounds in terms of the commutator of the unitary matrices are still unknown.

In this talk I will report on joint work with Hall and Kachkovskiy where we show that under the vanishing of the same index obstruction, we can find effective bounds for the distance to commuting unitary matrices in terms of the commutator of the original pair of unitary matrices.

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Knot Floer homology (HFK) is a powerful suite of invariant of knots in three manifolds. A knot K in S^3 is strongly invertible if it is invariant under an orientation preserving involution of S^3 which has fixed set equal to an unknot that intersects K in two points. The knot K is freely 2-periodic if it is invariant under a free order two symmetry of S^3. These symmetric knots admit so called "quotient knots." For each of these symmetries, we construct spectral sequences in knot Floer homology with E^1 page given by a localization of HFK(K) and E-infinity page isomorphic to a localization of HFK of the quotient. We extract rank inequalities in knot Floer homology from these spectral sequences.

Saturday, October 11

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