We meet weekly on Wednesday at 3:00 PM (Central time), TBA
Organizers: Daniel Bernstein, Mahir Can, Tài Huy Hà, Kalina Mincheva.
Schedule: Fall 2022
August 31
Daniel Fretwell. Quaternary Lattices, Modular Forms and Elliptic Curves.
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A common theme in modern Number Theory is to find non-trivial links between objects coming from very different places, by relating their arithmetic.
In this talk we will (hopefully) see a surprising example of this, connecting the first and third objects in the title…using the second to bridge the gap.
Time permitting, we will sketch the proof, motivated by a hidden 1.5th object (Clifford algebras).
(Based on joint work with E. Assaf, C. Ingalls, A. Logan, S. Secord and J. Voight)
September 7
Victor Bankston, Tulane University. Nonlocality and the Pauli Group.
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The Pauli group is fundamental in quantum information and is related to certain extraspecial 2-groups.
In this talk, I will introduce the Pauli group and pose a novel problem about its structure in terms
of partial homomorphisms to products of 2-element groups. Finally, I will motivate the problem by describing
a relationship to a novel class of nonlocal games by using the eigenvalue expansion properties of a distance-regular graph.
The Wilf Conjecture is a longstanding conjecture regarding the complement finite submonoids of $\mathbb{N}$, the monoid of natural numbers.
There have been several attempts to generalize the conjecture for higher dimensions. In this talk, we will extend the conjecture
for complement finite submonoids of unipotent group with entries from $\mathbb{N}$. This extension gives a better bound compared to the previous
generalizations. Also, we will prove our conjecture for certain subfamilies (thick and thin) of the unipotent groups.
September 21
Karl Hofmann, Tulane University. Weakly Complete Universal Enveloping Algebras of Profnite-Dimensional Lie Algebras.
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Every associative algebra $A$ becomes a Lie algebra $A_{Lie}$ with the Lie bracket $[a, b] = ab-ba$.
The universal enveloping algebra $U(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$ is an associative algebra with identity such
that $U(\mathfrak{g})_{Lie}$ contains the Lie subalgebra $\mathfrak{g}$ which generates $U(\mathfrak{g})$ as associative algebra
such that every representation $\mathfrak{g} \rightarrow A_{Lie}$ for an associative algebra $A$ extends to an algebra homomorphism
$U(\mathfrak{g}) \rightarrow A$. The so called Poincare-Birkhoff-Witt Theorem secures its existence and structure.
In a seminar lecture on September 26 last year I introduced the class of weakly
complete topological vector spaces and weakly complete associative unital topological
algebras and used the latter to introduce the weakly complete group algebra $R[G]$
of a compact group $G$. In the seminar on March 16 this year I followed up with
introducing the weakly complete universal enveloping algebra $U(\mathfrak{g})$ of a weakly complete Lie algebra $\mathfrak{g}$.
In the current lecture I shall recall the background for now establishing the details
of the theory of a weakly complete universal enveloping algebras $U(\mathfrak{g})$ of a weakly
complete topological Lie algebra $\mathfrak{g}$ and the appropriate Poincare-Birkhoff-Witt
theory. [Joint work with Linus Kramer, University of Munster.]
In his seminal work on modular curves and the Eisenstein ideal, Mazur studied the existence of congruences between certain Eisenstein
series and newforms, proving that Eisenstein ideals associated to weight 2 cusp forms of prime level are locally principal. In this talk,
we'll explore generalizations of Mazur's result to squarefree level, focusing on recent work, joint with P. Wake and C. Wang-Erickson,
about a non-optimal level N that is the product of two distinct primes and where the Galois deformation ring is not expected to be Gorenstein.
First, we will outline a Galois-theoretic criterion for the deformation ring to be as small as possible, and when this criterion is satisfied,
deduce an R=T theorem. Then we'll discuss some of the techniques required to computationally verify the criterion.
October 19
Rose McCarty, Princeton University. Local structure for vertex-minors.
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Roughly, the vertex-minors of a graph G are the graphs that can be obtained from G by deleting vertices and by performing certain local
moves within the neighborhood of a vertex. We are interested in classes of graphs which are closed under vertex-minors and isomorphism
and which do not contain all graphs. Geelen conjectures that the graphs in any such class have a simple structural description. We discuss
progress towards proving this conjecture and a surprising connection with linear algebra. This is part of an ongoing project with Jim Geelen and Paul Wollan.
October 26
Katie Clinch, University of New South Wales. Abstract 3-Rigidity and Bivariate C12-Splines II.
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Combinatorially characterizing graph rigidity in three (and higher) dimensions is perhaps the most important and longstanding
open problem in rigidity theory. The graphs that are rigid in a given dimension are the spanning sets of a matroid called the
$d$-dimensional rigidity matroid. This talk will discuss recent progress on this problem. This is joint work with Bill Jackson
and Shin-Ichi Tanigawa.
Aida Maraj, University of Michigan. Colored Gaussian Graphical Models with Toric Vanishing Ideals.
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A Colored Gaussian graphical model is a collection of multivariate Gaussian distributions in which a colored graph encodes conditional
independence relations among the random variables. Its set of concentration matrices is a linear space of symmetric matrices intersected
with the cone of positive definite matrices. Its inverse space, the space of covariance matrices, on the contrary, most often is not a
friendly variety. We are interested in the equations that vanish on the covariance matrices of these models. The main result of the talk
is that colored Gaussian graphical models with block graph structure that satisfy certain permutation symmetries (RCOP) are toric in the
space of covariance matrices. We provide explicit Markov bases/generating sets for the vanishing ideal of this variety which can be read
from paths in the graph. The talk is based on the preprint arXiv:2111.14817.
November 16
Matthias Storzer. Nahm sums and their modularity.
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The classification of modular Nahm sums, special q-hypergeometric series, is known to be related to the vanishing of certain
associated elements in the Bloch group. Nevertheless, a first conjecture turned out to be false. In this talk, we will discuss
the motivation behind this conjecture and its failure. Moreover, we will describe how the associated elements in the Bloch group
give rise to the modularity of Nahm sums.
November 30
Louis Gaudet, Rutgers University. The least Euler prime via sieve.
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Euler primes are primes of the form $p = x^2+Dy^2$ with $D>0$. In analogy with Linnik’s theorem, we can ask if it is possible to show that $p(D)$, the least prime of this form, satisfies $p(D) \ll D^A$ for some constant $A>0$. Indeed Weiss showed this in 1983, but it wasn’t until 2016 that an explicit value for $A$ was determined by Thorner and Zaman, who showed one can take $A=694$. Their work follows the same outline as the traditional approach to proving Linnik’s theorem, relying on log-free zero-density estimates for Hecke L-functions and a quantitative Deuring-Heilbronn phenomenon. In an ongoing work (as part of my PhD thesis) we propose an alternative approach to the problem via sieve methods that (as far as results about zeros of the Hecke $L$-functions) only requires the classical zero-free region. We hope that such an approach may result in a better value for the exponent $A$.
December 7
Trung Chau, University of Utah. Barile-Macchia resolutions.
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Resolving a monomial ideal over a polynomial ring is easy, but resolving it minimally isn't. Batzies and Welker (2002), using discrete Morse theory, provided an algorithm to produce free resolutions for monomial ideals, called Lyubeznik resolutions, and showed that these are minimal for large classes of ideals (generic + shellable). We introduce an alternate algorithm and produce what we call "Barile-Macchia resolutions". These resolutions are also minimal for large classes of ideals, where we put a a focus on edge ideals of weighted oriented graphs. We also make a comparison between Barile-Macchia resolutions and Lyubeznik resolutions, and in in the same vein, compare them to Taylor, Lyubeznik and Scarf resolutions. This is joint work with Selvi Kara.