Abstract: In this talk, I will discuss some results on the transport properties of the class of limit-periodic continuum
Schr\"odinger operators whose potentials are approximated exponentially quickly by a sequence of periodic functions.
For such an operator $H$, and $X_H(t)$ the Heisenberg evolution of the position operator, we show the limit of $\frac{1}{t}X_H(t)\psi$ as $t\to\infty$ exists and is
nonzero for $\psi\ne 0$ belonging to a dense subspace of initial states which are sufficiently regular and of suitably rapid decay.
This is viewed as a particularly strong form of ballistic transport, and this is the first time it has been proven in a continuum almost periodic
non-periodic setting. In particular, this statement implies that for the initial states considered, the second moment grows quadratically in time.

The 1-loop conjecture proposed by Dimofte and Garoufalidis suggests a simple and explicit formula to compute the adjoint twisted Reidemeister torsion of hyperbolic 3-manifolds with toroidal boundary in terms of the shape parameters of any ideal triangulation of the manifolds. In this talk, I will give a brief overview of the conjecture and present our recent result on the 1-loop conjecture for fundamental shadow link complements. This is a joint work with Tushar Pandey.

We give a complete solution to the local classification program of higher rank partially hyperbolic algebraic actions. We show $C^\infty$ local rigidity of abelian ergodic algebraic actions for symmetric space examples, twisted symmetric space examples and automorphisms on nilmanifolds. The method is a combination of representation theory, harmonic analysis and a KAM iteration. A striking feature of the method is no specific information from representation theory is needed. It is the first time local rigidity for non-accessible partially hyperbolic actions has ever been obtained other than torus examples. Even for Anosov actions, our results are new: it is the first time twisted spaces with non-abelian nilradical have been treated in the literature.

In this joint paper with P. Vellaisamy, we first derive some explicit formulas for the computation
of the n-th order divergence operator in Malliavin calculus in the one-dimensional case. We then extend these results to the case of isonormal Gaussian space. Our results generalize some of the known results for the divergence operator. Our approach in deriving the formulas is new and simple.

Conway and Lagarias used combinatorial group theory to show that certain
roughly triangular regions in the hexagonal grid cannot be tiled by the
shapes Thurston later dubbed tribones. The ideas of Conway, Lagarias, and
Thurston have found many applications in the study of tilings in the plane.
Today I'll discuss a two-parameter family of roughly hexagonal regions in
the hexagonal grid I call benzels. A variant of Gauss’ shoelace formula
allows one to compute the signed area (aka algebraic area) enclosed by a
closed polygonal path, and by “twisting” the formula one can compute the
values of the Conway-Lagarias invariant for all benzels. It emerges that the
(a,b)-benzel can be tiled by tribones if and only if a and b are the paired
pentagonal numbers k(3k+1)/2, k(3k-1)/2. This is joint work with Jesse Kim.

Abstract: There comes a time in every mathematician's life when they are detained by an evil dictator or warden with a soft spot for riddles. Fortunately for us, these riddles are often rooted in simple mathematics. This talk will prepare you for that inevitable occurrence by going over the mathematics involved in several prominent riddles. Along the way, we'll pick up some tricks of the trade that you can use when facing a logic puzzle to make an escape of your own.

When modeling data, we would like to know that our models are extracting facts about the data itself, and not about something arbitrary, like the order of the factors used in the model-ing. Formally speaking, this means we want the model to be invariant with respect to certain transformations. Here we look at different models and the nature of their invariants. We find that regression, MLE and Bayesian estimation all are invariant with respect to linear transformations, whereas regularized regressions have a far more limited set of invariants. As a result, regularized regressions produce results that are less about the data itself and more about how it is parameterized. To correct this, we propose an alternative expression of regularization which we call functional regularization. Ridge regression and lasso can be recast in terms of functional regularization, as can Bayesian estimation. But functional regularization preserves model invariance, whereas ridge and lasso do not. It is also more flexible, easier to understand, and can even be applied to non-parametric models.

The pioneering work of Langlands has established the theory of reductive algebraic groups and their representations as a key part of modern number theory. I will survey classical and modern results in the representation theory of reductive groups over local fields (the fields of real, complex, or p-adic numbers, or of Laurent series over finite fields) and discuss how they relate to Langlands' ideas, as well as to the various reflections of the basic mathematical idea of symmetry in arithmetic and geometry.

Innovations and changes in financial markets do not come without
risks. We will discuss some recent innovations and changes and
discuss their implications for risk management, such as the risks
associated with cryptocurrency, and the impact on finance of
machine learning.

Totally positive matrices are matrices in which each minor is positive. Lusztig extended the notion to reductive Lie groups. He also proved that specialization of elements of the dual canonical basis in representation theory of quantum groups at q=1 are totally non-negative polynomials. Thus, it is important to investigate classes of functions on matrices that are positive on totally positive matrices. I will discuss two sourses of such functions. One has to do with multiplicative determinantal inequalities (joint work with M.Gekhtman). Another deals with majorizing monotonicity of symmetrized Fischer's products which are known for hermitian positive semidefinite case which brings additional motivation to verify if they hold for totally positive matrices as well (joint work with M.Skandera). The main tools we employed are network parametrization, Temperley-Lieb and monomial trace immanants.

Transcriptomic data, more specifically single cell RNA sequencing (scRNA-seq), is an emerging field in biology that is used to obtain molecular understanding of cells. Analyzing scRNA-seq gives insight to protein and gene regulatory networks, protein expression and diseases. In this talk, I will present ensemble clustering methods used to find clusters in scRNA-seq data, which can then be used for further analysis into differential gene expression, cell trajectory and cell-cell communication.
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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 .

This talk consists of two related but distinct parts, and should be accessible if you know some algebraic topology and/or differential geometry. The first part is about quantum invariants: I will sketch how to compute the colored Jones polynomials of a knot and discuss their origin in representation theory. The second part is about hyperbolic geometry: I will discuss the basics of hyperbolic knot theory and explain how to compute hyperbolic structures and their volumes using ideal triangulations. The goal is to motivate the volume conjecture discussed in my main talk, which relates the colored Jones polynomials to the hyperbolic volume.

Kuznetsov component A_X of an algebraic variety X is defined to be the right orthogonal of some exceptional collection in the bounded derived category of X. When X is a cubic fourfold or Gushel Mukai fourfold, A_X is a noncommutative K3 surface in the sense that its Serre functor is given by "shifting by 2". Whether or not A_X is equivalent to the bounded derived category of an actual K3 surface is believed to be related to the rationality of the variety X , therefore it has received extensive studies. Yet not many studies seem to answer the question of when the Kuznetsov component of a cubic fourfold is equivalent to that of a Gushel Mukai fourfold, we believe that the answer of this question should be interesting for it will give a part of "Torelli theorem for noncommutative K3 surfaces". In this talk, I will present some partial results which address the previous question.

In operator algebras, specifically free probability, free transport is a technique for producing state-preserving isomorphisms between C* and von Neumann algebras that was developed by Guionnet and Shlyakhtenko in their 2014 Inventiones paper. The inspiration for their work comes from the field of optimal transport, specifically work of Brenier from 1991 who showed that under very mild assumptions one can push forward a probability measure on $\mathbb{R}^n$ to the Gaussian measure. In the non-commutative case, Guionnet and Shlyakhtenko showed that if $x_1,\ldots, x_n$ are self-adjoint operators in a tracial von Neumann algebra $(M,\tau)$ whose distribution satisfies an "integration-by-parts" formula up to a small perturbation, then these operators generate a copy of the free group factor $L(\mathbb{F}_n)$. In this series of talks, I will give an overview of their proof, discuss some applications of their result, and survey the current state of free transport theory.

Quantum invariants of links like the colored Jones polynomial (which arise from the quantum Chern-Simons theory of Witten-Reshetikhin-Turaev) have a purely algebraic construction in terms of the representation theory of quantum groups. Despite this algebraic nature they appear to be connected to geometry: a class of related volume conjectures assert that their semi-classical asymptotics determine geometric invariants like the hyperbolic volume. To better understand these conjectures a number of authors have studied ways to twist quantum invariants by geometric data. In particular, Blanchet, Geer, Patureau-Mirand, and Reshetikhin recently defined quantum holonomy invariants depending on a link in S^3 and a flat 𝔰𝔩₂ connection on its complement. Their construction uses certain unusual cyclic modules of quantum 𝔰𝔩₂. For technical reasons the invariants are quite difficult to compute. In this talk (based on joint work with Nicolai Reshetikhin) I will explain how to effectively compute them using hyperbolic tensor networks constructed from quantum dilogarithms. Our construction reveals deep connections with hyperbolic geometry and suggests a way to break the Kashaev-Murakami-Murakami volume conjecture into two simpler pieces.

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 sequence 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. These results do not require the sequences of quantum operations that describe the quantum process to be trace non-increasing and hence can be applied to a larger class of compositions of random positive maps. 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.

The Favard length of the planar $1/4$-corner Cantor set is $0$. Estimates exists about the rate with which the Favard length of the previous steps goes to $0$, but the exact rate of decay is unknown. However, if one considers a random construction of the $1/4$-corner Cantor set, things might seem better. In fact, Peres and Solomyak showed that the rate of decay for the average Favard length for the random $1/4$-corner Cantor set is of order exactly $1/n$. We show that the rate of decay for a random disk-like analogue has again order $1/n$. This suggests that any ``reasonable'' random Cantor set of positive and finite length might decay at the same rate.

One of the most well-known examples of a cluster structure comes from Penner's lambda-length coordinates on the decorated Teichmuller space of a surface. In 2019, Penner and Zeitlin defined a super-manifold generalizing the decorated Teichmuller space, which involves new anti-commuting variables. I wall talk about some recent work with Gregg Musiker and Sylvester Zhang, where we showed that the coordinates on the decorated super Teichmuller space have many of the nice properties associated to a cluster structure, such as a kind of Laurent phenomenon, positivity, and some interesting combinatorial interpretations of the Laurent expressions, involving double dimer covers of certain graphs.

In this talk, we give an overview of some basic properties of wavelets. We then introduce the Windowed Scattering Transform and go over stability and invariance properties that make it desirable as a feature extractor. Finally, we provide a generalization of the Windowed Scattering Transform that is translation invariant and discuss other stability and invariance properties of our generalized Scattering Transform.
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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 .

Mathematics education research is overwhelmingly assimilationist in its desire
to change people, teachers, students, researchers, instead of changing the mathematics. In this teaching-research collaboration, one teacher (Tina Haselius), two graduate research assistants (Sofía Abreu and Melvin Peralta), and one mathematics education researcher (Higinio Dominguez) will share their emerging experiences learning how to animate mathematical concepts. While one key goal in our collaboration has been to resist the violence of trying to change, assimilate, and colonize learners, the process of animating mathematical concepts has, in beautiful and nonviolent ways, allowed us to experience change in and among ourselves as we learn to (co)respond to the animacy and agency of the mathematical concepts that we set out to animate in our teaching-research group.

The birational automorphism group is a natural birational invariant associated to an algebraic variety. In this talk, we study the specialization homomorphism for the birational automorphism group. As an application, building on work of Kollár and of Chen–Stapleton, we show that a very general n-dimensional complex hypersurface X of degree ≥ 5⌈(n+3)/6⌉ has no finite order birational automorphisms. This work is joint with Nathan Chen and David Stapleton.

In operator algebras, specifically free probability, free transport is a technique for producing state-preserving isomorphisms between C* and von Neumann algebras that was developed by Guionnet and Shlyakhtenko in their 2014 Inventiones paper. The inspiration for their work comes from the field of optimal transport, specifically work of Brenier from 1991 who showed that under very mild assumptions one can push forward a probability measure on $\mathbb{R}^n$ to the Gaussian measure. In the non-commutative case, Guionnet and Shlyakhtenko showed that if $x_1,\ldots, x_n$ are self-adjoint operators in a tracial von Neumann algebra $(M,\tau)$ whose distribution satisfies an "integration-by-parts" formula up to a small perturbation, then these operators generate a copy of the free group factor $L(\mathbb{F}_n)$. In this series of talks, I will give an overview of their proof, discuss some applications of their result, and survey the current state of free transport theory.

In this talk I will detail a construction of symmetric link
homology. In particular, this provides a non-trivial categorification of
1 and a finite dimensional categorification of the colored Jones
polynomial and a new categorification of the Alexander polynomial. I
will also explain how this relates to the triply graded homology and
knot Floer homology.

In joint work with Russ Woodroofe, we showed that the order complex of the poset of all cosets of all proper subgroups of a finite group, ordered by inclusion, has noncontractible order complex using Smith Theory. A key part of our proof involves invariable generation of finite groups: two subsets $S,T$ of a group $G$ generate $G$ invariably if, for every $g,h \in G$, $g^{-1}Sg$ and $h^{-1}Th$ together generate $G$. It remains open whether the alternating group $A_n$ can be generated invariably by $\{s\}$ and $\{t\}$ with both $s,t$ having prime power order. This question is closely related to a (still open) question about prime divisors of binomial coefficients. I will discuss all of this, along with current work joint with Bob Guralnick and Russ Woodroofe about invariable generation of arbitrary simple groups by two elements of prime or prime power order.

The damped wave equation models the behavior of vibrating systems exposed to some damping force, which causes the total energy to decay. In this talk, I will discuss classical results that give upper and lower bounds on decay, based on the dynamics of the geodesic flow and the support of the damping. I will discuss recent generalizations of these results to time dependent, unbounded, or anisotropic damping.
(Note location: this talk will be held in C517 due to hiring meeting in C304.)

In the past decade, exemplar-based texture synthesis algorithms have seen strong gains in performance by matching statistics of deep convolutional neural networks. However, these algorithms require regularization terms or user-added spatial tags to capture long range constraints in images. Thus, we propose a new set of statistics for exemplar based texture synthesis based on Sliced Wasserstein Loss and create a multi-scale algorithm to synthesize textures without any regularization terms or user-added spatial tags. Lastly, we study the ability of our proposed algorithm to capture long range constraints in images and compare our results to other exemplar-based neural texture synthesis algorithms.
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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’ll talk about C*-modules and representations on them, developing a loose parallel with the Hilbert space case. For specialized C*-modules, much can be said, and a classification of these modules suggests a vast generalization of the Stone-von Neumann Theorem which accommodates all of the data of generalized C*-dynamical systems.

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 .

The eigencurve is a rigid analytic curve that $p$-adically interpolates eigenforms of finite slope. The global geometry of the eigencurve is somewhat mysterious. However, over the boundary, it is predicted to behave rather nicely (by the so-called Halo conjecture). This conjecture has been studied by Liu--Wan--Xiao for definite quaternion algebras. In this talk, we will report on some work in progress on this conjecture in the case of $\rm{GL}(2)$. If time permits, we will discuss some generalizations towards groups beyond $\rm{GL}(2)$. This is partially joint with H. Diao.

Based on the work of Grothendieck, in the 1960's Atiyah and Hirzebruch developed K-theory as a tool for algebraic geometry. Adapted to the topological setting K-theory can be regarded as the study of a ring generated by vector bundles. In the 1970's it was introduced as a tool in C*-algebras. C*-algebras are often considered to be "noncommutative topology", additionally they are an algebra over the complex numbers. In this setting the algebraic and topological definitions of K-theory overlap giving us a powerful tool. Essential for the Elliott classification program, for certain classes of C*-algebras, K-theory is a complete invariant. K-theory is also a natural setting for higher index theory.
We will begin by looking at different types of equivalence for projections. Then we will build a monoid where these types of equivalences are equivalent. We then use the Grothendieck construction to turn our monoid into an abelian group. This group is called the $K_0$ group of our algebra and can be thought of as the "connected components" of projections in our C*-algebra.
Next, in a similar manner, we construct the $K_1$ group using unitaries from our C*-algebra.
Once we have the $K_0$ and $K_1$ groups we will discuss Bott periodicity and the six-term exact sequence, a tool used to calculate K-theory.

I’ll start by introducing infinite-type surfaces—those with infinite genus or infinitely many punctures—and the emerging study of their mapping class groups. One difference from the finite-type setting is that these mapping class groups come with natural non-discrete topologies. I’ll discuss joint work with Nick Vlamis where we fully characterize which surfaces have mapping class groups with dense conjugacy classes, so that there exists an element that well approximates every mapping class, up to conjugacy.

I will report on joint work with D. Jordan, I. Le and A. Shapiro in which we construct categorical invariants of decorated surfaces using the stratified factorization homology of Ayala, Francis and Tanaka, together with the representation theory of quantum groups. The categories we obtain can be regarded as `quantizations' of the categories of quasicoherent sheaves on the stacks of decorated local systems on surfaces, and satisfy strong functoriality and locality properties reminiscent of those of a TQFT. I will give an overview of their construction, and explain how to recover Fock-Goncharov-Shen's cluster quantizations of related moduli spaces within this framework.

Schubert polynomials are a very interesting family of polynomials in algebraic geometry due to their relation with the cohomology of the flag variety. Moreover, they are also very interesting from a combinatorial point of view because they can be considered generalizations of Schur functions. In this talk, we will talk about how to multiply a Schubert polynomial by a Schur function indexed by a hook and how we can extend this multiplication to the quantum world. This is a current work with C. Benedetti, N. Bergeron, F. Saliola, and F. Sottile.

Why do some high school mathematics lessons captivate high school students and others not? This study explores this question by comparing how the content unfolds in the lessons that students rated highest with respect to their aesthetic affordances (e.g., using terms like “intriguing”, “surprising”) with those the same students rated lowest with respect to their aesthetic affordances (e.g., “just ok”, “dull”). Using a framework that interprets the unfolding content across a lesson as a mathematical story, we identified characteristics of lessons that provoked curiosity or enabled surprise. This talk will explain the methodological approach to studying this question, as well as share the lesson characteristics that related strongly to student experience. These findings point to the characteristics of future lesson designs that could enable more students to experience curiosity and wonder in secondary mathematics classrooms. Also on Zoom: https://msu.zoom.us/j/98171049554 Passcode: GOGREEN