We introduce directed graphs and demonstrate how to generate a C*-algebra which reflects certain features of the graph. Time permitting, we will introduce two uniqueness theorems for their representations and explore a few of their consequences.

We construct the first positive examples to the Connes' Rigidity Conjecture, i.e., we construct groups $G$ with Kazhdan's property (T) such that if $H$ is a group with the same von Neumann algebra as $G$, then $H\cong G$. In this talk, I will focus on the group-theoretic side of this result and talk about how we applied geometric group theory to solve problems from von Neumann algebra. This is joint work with Ionut Chifan, Adrian Ioana, and Denis Osin.

In previous work of Akhmechet, Krushkal, and the speaker, a unification of lattice cohomology and the $\widehat{Z}$-invariant was established. Both theories are combinatorially defined invariants of plumbed 3-manifolds, but with quite different origins. Lattice cohomology, due to Némethi, is motivated by the study of normal surface singularities and is isomorphic to Heegaard Floer homology for plumbing trees. On the other hand, $\widehat{Z}$, due to Gukov-Pei-Putrov-Vafa, is a power series coming from a physical theory and is conjectured to recover quantum invariants of 3-manifolds at roots of unity. In this talk, I will discuss work in progress relating knot lattice homology and the Gukov-Manolescu 2-variable series, the knot theoretic counterparts to lattice homology and $\widehat{Z}$. This is joint work with Ross Akhmechet and Sunghyuk Park.

I will discuss a discrete non-deterministic flow-firing process for topological cell complexes. The process is a form of discrete diffusion; a flow is repeatedly diverted according to a discrete Laplacian. The process is also an instance of higher-dimensional chip-firing. I will motivate and introduce the system and then focus on two important features – whether or not the system is terminating and whether or not the system is confluent.

Polynuclear growth is one of the basic models in the Kardar-Parisi-Zhang universality class, which describes a one-dimensional crystal growth. For a particular initial state, it describes the length of the longest increasing subsequence for uniformly random permutations (the problem first studied by S. Ulam). In my joint work with J. Quastel and D. Remenik we expressed the distribution functions of the polynuclear growth in terms of the solutions of the Toda lattice, one of the classical integrable systems. A suitable rescaling of the model yields a non-trivial continuous limit of the polynuclear growth (the KPZ fixed point) and the respective equations (Kadomtsev-Petviashvili).

What Should a Good Deep Neural Network Look Like? Insights from a Layer-Peeled Model and the Law of Equi-Separation
See https://sites.google.com/view/minds-seminar/home

In favorable circumstances, topological 4-manifolds and surfaces in them can be classified. In contrast, little is known about smooth 4-manifolds and smooth surfaces. Several of the hardest problems in 4-dimensional topology (eg. the Poincare conjecture) simply ask whether the topological classification fails in the smooth setting; such failures are called exotica. In this talk, I will discuss some historic and recent progress towards detecting exotic phenomena, and outline some promising approaches.

Recovery of sparse vectors and low-rank matrices from a small number of linear measurements is well-known to be possible under various model assumptions on the measurements. The key requirement on the measurement matrices is typically the restricted isometry property, that is, approximate orthonormality when acting on the subspace to be recovered. Among the most widely used random matrix measurement models are (a) independent subgaussian models and (b) randomized Fourier-based models, allowing for the efficient computation of the measurements.
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For the now ubiquitous tensor data, direct application of the known recovery algorithms to the vectorized or matricized tensor is memory-heavy because of the huge measurement matrices to be constructed and stored. In this talk, we will discuss two different modewise measurement schemes and related recovery algorithms. These modewise operators act on the pairs or other small subsets of the tensor modes separately. They require significantly less memory than the measurements working on the vectorized tensor, and experimentally can recover the tensor data from fewer measurements and do not require impractical storage.
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This will be a hybrid seminar and take place in C117 Wells Hall and via Zoom at https://msu.zoom.us/j/99426648081?pwd=ZEljM3BPUXg2MjVUMVM5TnlzK2NQZz09 .

Skein algebras are spanned by webs or links in a thickened surface subject to skein relations. When the skein relations are the Kauffman bracket relations associated to SL(2), they provide a diagrammatic way to encode cluster algebras, as shown by Muller, and also quantum groups, as shown by Costantino and Le.
In this talk, we will explore a construction of a basis for the stated skein algebra for Sp(4) which is built from Kuperberg's web relations along with extra skein relations along the boundary of the surface. We will use the basis to obtain results about the structure of the skein algebra, relating it to the quantum group associated to Sp(4). We will also recover Kuperberg's result about the Sp(4) web category.

Let $K/\bf{Q}_p$ be a local number field of absolute ramification index $e$, and let $X$ be a proper smooth $O_K$-scheme. I will discuss how one can obtain bounds on ramification of the mod $p$ Galois representations arising as the étale cohomology of (the geometric generic fiber of) $X$ in terms of $e$, the given prime $p>2$ and the cohomological degree $i$. The key tools for achieving this are the Breuil-Kisin and $A_{\rm inf}$-cohomology theories of Bhatt, Morrow and Scholze, and a series of conditions based on a criterion of Gee and Liu regarding crystallinity of the representation attached to a free Breuil-Kisin-Fargues $G_K$-module.

We illustrate various aspects of graph algebras introduced last week, including usage of the universal property, aspects of their representation theory, and ideals, pointing to relationships with the structure of the underlying graph.

A quantum particle restricted to a lattice of points has been well studied in many different contexts. In the absence of disorder or environmental interaction, the particle simply undergoes ballistic transport for many suitable Hamiltonian operators. Recently, progress has been made on introducing a Lindbladian interaction term to the model, which drastically changes the dynamics in the large time limit. We prove that indeed diffusion is present in this context for an arbitrary periodic Hamiltonian. Additionally, we show that the diffusion constant is inversely proportional to the particles' coupling strength with its environment.

For a planar bipartite graph G equipped with a SLn-local system, we show that the determinant of the associated Kasteleyn matrix counts “n-multiwebs” (generalizations of n-webs) in G, weighted by their web-traces. We use this fact to study random n-multiwebs in graphs on some simple surfaces. Time permitting, we will discuss some relations to Fock-Goncharov theory. This is joint work with Rick Kenyon and Haolin Shi.

Intersections between varieties and subschemes can result in structures with many varying dimensions, which naturally leads us to consider something like a homology structure. The non-Hausdorff nature of the Zariski topology limits our ability to utilize algebraic topology, so we must first define cycles, the formal sum of subvarieties, to get something we can work with.

When Björner and Wachs introduced one of the main forms of lexicographic shellability, namely CL-shellability, they also introduced the notion of recursive atom ordering, and they proved that a finite bounded poset is CL-shellable if and only if it admits a recursive atom ordering. We generalize the notion of recursive atom ordering, and we prove that any such generalized recursive atom ordering may be transformed via a reordering process into a recursive atom ordering. We also prove that a finite bounded poset admits a generalized recursive atom ordering if and only if it is ``CC-shellable'' by way of a CC-labeling which is self-consistent in a certain sense. This allows us to conclude that CL-shellability is equivalent to self-consistent CC-shellability. As an application, we prove that the uncrossing orders, namely the face posets for stratified spaces of planar electrical networks, are dual CL-shellable.
During this talk, we will review plenty of background on poset topology and specifically regarding the technique of lexicographic shellability. This is joint work with Grace Stadnyk

Eigenalgebra is a construction attaching to a family of commuting operators acting on some space or module that parameterizes the system of eigenvalue for some operators.

The generalization error of a classifier is related to the complexity of the set of functions among which the classifier is chosen. We study a family of low-complexity classifiers consisting of thresholding a random one-dimensional feature. The feature is obtained by projecting the data on a random line after embedding it into a higher-dimensional space parametrized by monomials of order up to k. More specifically, the extended data is projected n-times and the best classifier among those n, based on its performance on training data, is chosen. We show that this type of classifier is extremely flexible, as it is likely to approximate, to an arbitrary precision, any continuous function on a compact set as well as any Boolean function on a compact set that splits the support into measurable subsets. In particular, given full knowledge of the class conditional densities, the error of these low-complexity classifiers would converge to the optimal (Bayes) error as k and n go to infinity. On the other hand, if only a training dataset is given, we show that the classifiers will perfectly classify all the training points as k and n go to infinity. We also bound the generalization error of our random classifiers. In general, our bounds are better than those for any classifier with VC dimension greater than O (ln n) . In particular, our bounds imply that, unless the number of projections n is extremely large, there is a significant advantageous gap between the generalization error of the random projection approach and that of a linear classifier in the extended space. Asymptotically, as the number of samples approaches infinity, the gap persists for any such n. Thus, there is a potentially large gain in generalization properties by selecting parameters at random, rather than optimization.
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A preprint of this work can be found here: https://arxiv.org/abs/2108.06339
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This will be a hybrid seminar and take place in C117 Wells Hall and via Zoom at https://msu.zoom.us/j/99426648081?pwd=ZEljM3BPUXg2MjVUMVM5TnlzK2NQZz09 .

I will talk about Ogg's conjecture and its generalization. This is joint work with Ken Ribet (and should serve as an introduction to Ken's talk on Friday).

A discrete parameter quantum process is represented by a sequence of quantum operations, which are completely positive maps that are trace non-increasing. Given a stationary and ergodic sequences of such maps, an ergodic theorem describing convergence to equilibrium for a general class of such processes was recently obtained by Movassagh and Schenker. Under irreducibility conditions we obtain a law of large numbers that describes the asymptotic behavior of the processes involving the Lyapunov exponent. Furthermore, a central limit type theorem is obtained under mixing conditions. In the continuous time parameter, a quantum process is represented by a double-indexed family of positive map valued random variables. For a stationary and ergodic family of such maps, we extend the results by Movassagh and Schenker to the continuous case.

Nakajima's graded quiver varieties are complex algebraic varieties associated with quivers. They are introduced by Nakajima in the study of representations of universal enveloping algebras of Kac-Moody Lie algebras, and can be used to study cluster algebras. In the talk, I will explain how to precisely locate the supports of the triangular basis of skew-symmetric rank 2 quantum cluster algebras by applying the decomposition theorem to various morphisms related to quiver varieties, thus prove a conjecture proposed by Lee-Li-Rupel-Zelevinsky in 2014.

The contact invariant, defined by Kronheimer and Mrowka, is
an element in the monopole Floer homology of a 3-manifold canonically
attached to a contact structure. I will discuss how the contact
invariant places constraints on the topology of families of contact
structures, and how it can be used to detect non-trivial
contactomorphisms given by "Dehn twists" on spheres. The main new tool
is a generalisation of the contact invariant to an invariant of
families of contact structures.

The KPZ universality class contains one-dimensional random growth models, which under quite general assumptions exhibit similar (non-Gaussian) scaling behavior. For special initial states, the limiting distributions surprisingly coincide with those from the random matrix theory. The physical explanation is that in the space of Markov processes, these models are all being rescaled to a universal fixed point. This scaling invariant fixed point was first characterized in joint work with Jeremy Quastel and Daniel Remenik. In our work, we found a surprising relation between random growing interfaces and the solutions of the classical integrable systems.

We present results on the stability of equilibria (time-independent solutions) of the Vlasov-Maxwell equation. In particular, linear stability criteria for certain classes of equilibria are discussed. We also give a result on the nonlinear stability of an initial-boundary value problem for the Vlasov-Poisson equation.
**Note: speaker will present Virtually. Participants can join in person to view the presentation in C304, or through the Zoom link.**

If a and b are integers that satisfy a simple nonvanishing
condition, the cubic equation y^2 = x^3 + ax + b defines an elliptic
curve over the field of rational numbers. Elliptic curves have been
studied for millennia and seem to occur all over the place in
mathematics, physics and other sciences. In my talk, I'll explain how a
specific elliptic curve provides the solution to a surprisingly hard "brain
teaser" that had a big run on social media a few years ago.

Over 40 years ago, I proved the finiteness of the group of cyclotomic torsion points on an abelian variety over a number field. (A torsion point is cyclotomic if its coordinates lie in the field obtained by
adjoining all roots of unity to the base field.) If the abelian variety is one that we know well, and if the number field is the field of rational numbers, we can hope to determine explicitly the group of
its cyclotomic torsion points. I will illustrate this theme in the situation studied by Barry Mazur in his landmark "Eisenstein ideal" article, i.e., that where the abelian variety is the Jacobian of the
modular curve $X_0(p)$.

The classical question of determining which varieties are rational has led to a huge amount of interest and activity. On the other hand, one can take on a complementary perspective - given a smooth projective variety whose nonrationality is known, how far is it from being rational? I will survey what is currently known, with an emphasis on hypersurfaces and complete intersections.

We performed a systematic study of permutation statistics and
bijective maps on permutations, looking for the homomesy phenomenon.
Homomesy occurs when the average value of a statistic is the same on
each orbit of a given map. The maps that exhibit homomesy include the
Lehmer code rotation, the reverse, the complement, and the
Kreweras complement, all of which have some geometric interpretations.
The statistics studied relate to familiar notions such as inversions,
descents, and permutation patterns, among others. Beside the many new
homomesy results, I’ll discuss our research method, in which we used
SageMath to search the FindStat combinatorial statistics database to
identify potential instances of homomesy, and what this experiment
taught us about the maps themselves and the homomesy phenomenon at large.
This is joint work with Jennifer Elder, Erin McNicholas, Jessica Striker
and Amanda Welch.

We prove the existence of the Minkowski content of the intersection of an SLE$_\kappa(\rho)$ curve with a real interval using the standard approach, which is to estimate the convergence rate of one-point and two-point boundary Green's functions of SLE$_\kappa(\rho)$. Then we show the existence of a conformally covariant measure called Minkowski content measure on the intersection of an SLE$_\kappa(\rho)$ curve with a half real line, which is closely related to the Minkowski content. Using the Minkowski content measure, we construct rooted and unrooted SLE$_\kappa(\rho)$ bubble measures, which are supported on loops and satisfy SLE$_\kappa(\rho)$-type domain Markov property.

Continuation of previous talk. Following the eigenalgebra book, after considering eigenalgebra over field, if time allow, we will talks about eigenalgebra over dvr

We will give a gentle introduction to a class of finite semigroups
bearing the cryptic name "left-regular bands" (LRBs). These LRBs show up, for example, in the combinatorics of reflection groups and hyperplane arrangements, in the analysis of mixing times for certain card-shuffling Markov chains, as well as in the space of phylogenetic trees.
We focus on examples with large groups of symmetries that act on the semigroup algebra of the LRB. Here the well-understood LRB representation theory, together with some combinatorics, allow one to answer two invariant-theory questions: What is the structure of the invariant subalgebra, and how does it act on the whole semigroup algebra?
This is based on joint work with Sarah Brauner and Patty Commins (arXiv:2206.11406).

Let G be a complex simple group. Let $\beta$ be a positive braid whose Demazure product is the longest Weyl group element. The braid variety X($\beta$) generalizes many well known varieties, including positroid cells, open Richardson varieties, and double Bott-Samelson cells. We provide a concrete construction of the cluster structure on X($\beta$), using the weaves of Casals and Zaslow. We show that the coordinate ring of X($\beta$) is a cluster algebra, which confirms a conjecture of Leclerc as special cases. As an application, we show that X($\beta$) admits a natural Poisson structure and can be further quantized. If
time permits, I will explain several of its applications on representation theory and knot theory,
including its connections with the Kazhdan-Lusztig R-polynomials and a geometric interpretation of the
Khovanov-Rozansky homology (following the work of Lam-Speyer and Galashin-Lam). This talk is based on joint work with Roger Casals, Eugene Gorsky, Mikhail Gorsky, Ian Le, and Jose Simental (arXiv:2207.11607).

In this talk, we will give an overview of exemplar-based texture synthesis. For the first part of the talk, we will discuss a classical approach via matching statistics of wavelet coefficients and its shortcomings. In the second part of the talk, we will discuss more recent work using statistics of deep convolutional neural networks for more realistic texture synthesis.
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This will be a hybrid seminar and take place in C117 Wells Hall and via Zoom at https://msu.zoom.us/j/99426648081?pwd=ZEljM3BPUXg2MjVUMVM5TnlzK2NQZz09 .

Given an elliptic curve over a number field with its Weierstrass model, we can study the integral points on the curve. Taking an infinite family of elliptic curves and imposing some ordering, we may ask how often a curve has an integral point and how many integral points there are on average. We expect that elliptic curves with any non-trivial integral points are generally very sparse. In certain quadratic and cubic twist families, we prove that almost all curves contain no nontrivial integral points.