I will give an overview of some relations between finite group theory, G-equivariant topological quantum field theory, and the computational complexity of invariants of 3-manifolds, both classical and quantum. We will start with one of the simplest kinds of invariants in knot theory: the coloring invariants, introduced by Fox when giving a talk to undergraduates in the 1950s. We will then build up to the idea of G-equivariant TQFT (aka homotopy QFT with target K(G,1)), which mathematically describes the topological order determined by a symmetry-enriched topological phase of matter. Physicists have studied these in part motivated by the search for new universal topological quantum computing architectures.
Our goal will be to convey two complexity-theoretic lessons. First, when G is sufficiently complicated (nonabelian simple), the simple-to-define coloring invariants associated to G are, in fact, very difficult to compute, even on a quantum computer. Second, no matter what finite group G one uses, a 3-dimensional G-equivariant TQFT can not be used for universal topological quantum computation if the underlying non-equivariant theory is not already universal. This talk is based on joint works with Greg Kuperberg and Colleen Delaney.
https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

The first part of the talk will review background material on the differential geometry of 7-dimensional manifolds with the exceptional holonomy group $G_{2}$. There are now many thousands of examples of deformation classes of such manifolds and there are good reasons for thinking that many of these have fibrations with general fibre diffeomorphic to a K3 surface and some singular fibres: higher dimensional analogues of Lefschetz fibrations in algebraic geometry. In the second part of the talk we will discuss some questions which arise in the analysis of these fibrations and their "adiabatic limits". The key difficulties involve the singular fibres. This brings up a PDE problem, analogous to a free boundary problem, and similar problems have arisen in a number of areas of differential geometry over the past few years, such as in Taubes' work on gauge theory. We will outline some techniques for handling these questions.

A grid associahedron is a simple polytope whose faces correspond to nonkissing collections of routes inside a grid graph. The usual associahedron arises as a special case when the graph is a 2 by n rectangle. Other examples of grid associahedra have been considered in connection with combinatorial properties of Grassmannians and with the representation theory of gentle algebras. The 1-skeleton of the grid associahedron has a natural orientation that induces the grid-Tamari order, a poset with many remarkable properties. I will present a new construction of the grid associahedron as a Minkowski sum of order polytopes of fence posets. Using this construction, I will show that the grid-Tamari order has the non-revisiting chain property. This is based on joint work with Alexander Garver.

First, I will introduce Hilbert spaces, reproducing kernel Hilbert spaces, and von Neumann algebras. Then, I will show how one may construct a sequence of RKHS's associated to a finite-dimensional von Neumann algebra, which serves as a complete algebraic invariant. Finally, I will describe an RKHS structure which recovers whether or not an arbitrary von Neumann algebra has property Gamma.

What's the difference between a continuous function and a Hermitian matrix? From the perspective of operator algebras, not much! Operator algebras is a branch of mathematics that is equal parts analysis and linear algebra, and operator algebraists spend a lot of time thinking about mathematical objects called C*-algebras. If you've taken a course in calculus, then you are already familiar with one example of a C*-algebra: the continuous functions on a closed interval [a,b]. If you've taken linear algebra, then you're familiar with another example: the nxn square matrices. In this talk I will introduce the definition of a C*-algebra by way of these examples, and show how each example can provide insights into the other.

The Peterson-Thom conjecture asserts that any diffuse, amenable subalgebra of a free group factor is contained in a unique maximal amenable subalgebra. This conjecture is motivated by related results in Popa's deformation/rigidity theory and Peterson-Thom's results on $L^{2}$-Betti numbers. We present an approach to this conjecture in terms of so-called strong convergence of random matrices by formulating a conjecture which is a natural generalization of the Haagerup-Thorbjornsen theorem whose validity would imply the Peterson-Thom conjecture. This random matrix conjecture is related to recent work of Collins-Guionnet-Parraud.

The alternating genus of a knot is the minimum genus of a surface onto which the knot has an alternating diagram satisfying certain conditions. Very little is currently known about this knot invariant. We study spanning surfaces for knots, and define an alternating distance from the extremal spanning surfaces. This gives a lower bound on the alternating genus and can be calculated exactly for torus knots. We prove that the alternating genus can be arbitrarily large, find the first examples of knots where the alternating genus is equal to n for each n>2, and classify all toroidally alternating torus knots.
Zoom: https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

Starting from a short review of Riemann's sketch of a geometry on manifolds (1854ff.) this talk discusses how Hermann Weyl proposed to take up Riemann's ideas and how to modify them in the light of his views of the new developments in the foundations of analysis and in physics (the general theory of relativity, GTR). Although in his early work (1913) Weyl contributed to the later axiomatisation of the concept of manifolds, his own preferences lay clearly in a constructive approach (for a time even an intuitionistic one). Moreover, he saw a basic problem for accepting Riemannian geometry as an adequate geometric framework for a field theoretic foundation of physics. As an alternative he proposed a ``purely infinitesimal'' approach to the metric -- later called Weyl geometry. It contained the first instance of an (explicit) gauge structure which --in transformed form -- became the grandfather of the later gauge theories in physics.

We consider posets of (projective) sign vectors ordered by reverse inclusion of zero entries with restrictions on sign variation. We show many of these posets are partitionable and give an interpretation of the h-vector in terms of descents in signed permutations. This is joint work with Nantel Bergeron and Aram Dermenjian. Time permitting we will discuss conjectural (strong) Spernicity.

I’ll discuss some explicit examples of Calabi-Yau threefolds (CY3s) with torsion in various cohomology groups and consequences of such torsion; in particular, I’ll present computations using p-adic Hodge theory which answer some outstanding questions about CY3s over finite fields. I will not assume familiarity with Calabi-Yau manifolds or p-adic Hodge theory.

I will discuss recent joint work with Lewis Bowen and Frank Lin. In it, we consider a natural metric satisfying the CAT(0) condition (a certain natural negative curvature condition) on a space of operators affiliated to a tracial von Neumann algebra (a version of this space appeared in previous work of Andruchow-Larotonda). We also investigate how the geometric properties of this CAT(0) space reflect algebraic/analytic properties of the underlying von Neumann algebra.

A central problem of machine learning is the following. Given data of the form (y_i, f(y_i) + ϵ_i)_{i = 1}^M, where y_i’s are drawn randomly from an unknown (marginal) distribution μ* and ϵ_i are random noise variables from another unknown distribution, find an approximation to the unknown function f, and estimate the error in terms of M. The approximation is accomplished typically by neural/rbf/kernel networks, where the number of nonlinear units is determined on the basis of an estimate on the degree of approximation, but the actual approximation is computed using an optimization algorithm. Although this paradigm is obviously extremely successful, we point out a number of perceived theoretical shortcomings of this paradigm, the perception reinforced by some recent observations about deep learning. We describe our efforts to overcome these shortcomings and develop a more direct and elegant approach based on the principles of approximation theory and harmonic analysis.

The talk starts an introduction to weights and some history in the one and two weight theory and Calderon-Zygmund operators. There is also an introduction to T1 and Tb Theorems and I present a recent result of mine together with Grigoriadis, Sawyer, Shen and Uriarte-Tuero about the two-weight local Tb theorem for fractional singular integral operators.

In this talk we will share how our work as researchers in math education and, respectively, special education has led to an ongoing collaboration to support pre-service teachers to facilitate meaningful mathematical conversations and conceptual learning for all students. Anna will describe how she has applied techniques of discourse analysis to document different aspects of students’ talk, including how students negotiate authority and co-construct mathematical ideas. Casey will summarize his research on mathematics interventions for students with mild disabilities, including the use of gestures and visual representations. We will discuss how our research trajectories have led to our collaborative work, as well as the affordances and challenges of bridging different perspectives. Finally, we will share an example from our current research to illustrate how we integrate our analysis discourse and gesturing between students and pre-service teachers.
Register in advance for the colloquium:
https://msu.zoom.us/meeting/register/tJEsce2pqjwoGtRHK0KTvrsyYiH6GvjuNHSU
After registering, you will receive a confirmation email containing information about joining the meeting.

An operator algebra is an algebra of bounded linear operators acting on a Hilbert space that is closed in a certain norm topology. When that algebra is closed with respect to the adjoint operation (an abstract conjugate transpose), we call it a C*-algebra. The prototypical examples of C*-algebras include the ring of n x n matrices over the complex numbers and the ring of complex-valued continuous functions on a compact Hausdorff space. The latter example gives an algebraic perspective for studying topological dynamics. In particular, one can build an operator algebra called a crossed product that encodes the dynamical information of a group of homeomorphisms acting on a topological space.
In the 1960s, W. Arveson determined that the action of a homeomorphism on a topological space is better encoded in a crossed product via the action of a semigroup on that space, rather than a group, which led to many important results in operator algebra theory.
I will discuss how and why operator algebraists have been returning in recent years to crossed products in the context of groups acting on non-adjoint closed operator algebras, and I will discuss a recent partial solution to when dynamics are encoded fully in this crossed product context.

Not all the augmentations of a Legendrian knot come from embedded exact Lagrangian fillings. In this talk, we show that all augmentations come from possibly immersed exact Lagrangian fillings. We first associate an immersed cobordism with a DGA map from the top Legendrian knot to the DGA of the cobordism. This gives a functor from a Legendrian category whose morphisms involve immersed Lagrangian cobordisms to a DGA category. With this functorality, an immersed filling L together with an augmentation of L induce an augmentation of the top Legendrian knot. This is a joint work with Dan Rutherford.
Zoom: https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

Charles Hermite’s name has been attributed to several objects and results in mathematics, from Hermitian matrices to Hermite polynomials to Hermite’s identity or Hermite-Minkowski theorem. Despite his achievements and central role in the mathematical life of the 19th century, he often appears as an anti-hero, opposed to anything modern, be it ideals, non-Euclidean geometry or set theory. I will try to explain his point of view which is linked to a vision of mathematics as a natural, observational science, and show how this perspective shaped his mathematical work and his requirements on what good mathematics should be.

A Schwarz function on an open domain $\Omega$ is a holomorphic function satisfying $S(\zeta)=\overline{\zeta}$ on the boundary $\Gamma$ of $\Omega$. Sakai in 1991 managed to give a complete characterisation of the boundary of a domain admitting a Schwarz function. In fact, if $\Omega$ is simply connected $\Gamma$ has to be regular real analytic. Here we try to describe $\Gamma$ when the boundary condition is slightly relaxed. In particular, we are interested in three different conditions over a simply connected domain $\Omega$: When $f_1(\zeta)=\overline{\zeta}f_2(\zeta)$ with $f_1,f_2$ holomorphic, when $\mathcal{U}/\mathcal{V}$ equals some real analytic function on $\Gamma$ with $\mathcal{U},\mathcal{V}$ harmonic and when $S(\zeta)=\Phi(\zeta,\overline{\zeta})$ with $\Phi$ a holomorphic function of two variables. It turns out the boundary can be from analytic to just $C^1$, regular except finitely many points, or regular except for a measure zero set, respectively.

Work in progress joint with Samuele Anni and Alexandru Ghitza. For N
prime to p, we count the number of classical modular forms of level Np
and weight k with fixed residual Galois representation and
Atkin-Lehner-at-p sign, generalizing both recent results of Martin (no
residual representation constraint) and rhobar-dimension-counting
formulas of Jochnowitz and Bergdall-Pollack. One challenge is the
tension between working modulo p and the need to invert p when working
with the Atkin-Lehner involution. To address this, we use the trace
formula to establish up-to-semisimplifcation isomorphisms between
certain mod-p Hecke modules (namely, refinements of weight-graded
pieces of spaces of mod-p forms) by exhibiting ever-deeper congruences
between traces of prime-power Hecke operators acting on
characteristic-zero Hecke modules. This last technique is new and
combinatorial in nature; it relies on a theorem discovered by the
authors and beautifully proved by Gessel, and may be of independent
interest.

The famous Shannon-Nyquist theorem has become a landmark in the development of digital signal and image processing. However, in many modern applications, the signal bandwidths have increased tremendously, while the acquisition capabilities have not scaled sufficiently fast. Consequently, conversion to digital has become a serious bottleneck. Furthermore, the resulting digital data requires storage, communication and processing at very high rates which is computationally expensive and requires large amounts of power. In the context of medical imaging sampling at high rates often translates to high radiation dosages, increased scanning times, bulky medical devices, and limited resolution.
In this talk, we present a framework for sampling and processing a large class of wideband analog signals at rates far below Nyquist in space, time and frequency, which allows to dramatically reduce the number of antennas, sampling rates and band occupancy. Our framework relies on exploiting signal structure and the processing task. We consider applications of these concepts to a variety of problems in communications, radar and ultrasound imaging and show several demos of real-time sub-Nyquist prototypes including a wireless ultrasound probe, sub-Nyquist MIMO radar, super-resolution in microscopy and ultrasound, cognitive radio, and joint radar and communication systems. We then discuss how the ideas of exploiting the task, structure and model can be used to develop interpretable model-based deep learning methods that can adapt to existing structure and are trained from small amounts of data. These networks achieve a more favorable trade-off between increase in parameters and data and improvement in performance, while remaining interpretable.

I have often been vexed by textbooks or professors offering one word proofs to quickly justify concepts that do not come so lazily to me. (That's a pangram!) The purpose of this talk will be to shed some light on some of these one word proofs, by giving an introduction to categories. This talk will be accessible to anyone who has ever been similarly frustrated (so all of us), as my examples include things such as the integers, partially ordered sets, and matrices. On the other hand, as a warning, if you can define a natural transformation without looking it up, then this talk will probably be boring for you.

This is the first talk in a series on "ergodic quantum processes." A quantum process is a sequence of quantum channels, which in turn are completely positive, trace preserving maps. In this talk, I will discuss the definition of a quantum channel and describe a general ergodic theorem for a quantum process formed from an ergodic sequence of stochastic channels. The proof of the result and applications to Matrix Product States will be discussed if time permits (but likely in a future talk).
(Zoom password: CPTP)

The stable Gauss map of a Lagrangian $L$ in a cotangent $T^*M$ is a map $g
\colon L \to U/O$ obtained by stabilization of the usual Gauss map from $L$ to
the Lagrangian Grassmannian of $T^*M$. Arnold's conjecture on nearby
Lagrangians implies in particular that $g$ is homotopic to a constant map. We
will see the weaker result that the map induced by $g$ on the homotopy groups is
trivial.
By a theorem of Giroux and Latour $g$ is homotopic to a constant map if and only
if $L$ admits a generating function. We introduce ``twisted'' generating
functions as a tool to the study of $L$ and make the link with difficult results
of pseudo-isotopy theory.
This is a joint work with Mohammed Abouzaid, Sylvain Courte and Thomas Kragh.
https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

This talk is about the coherent-constructible correspondence (CCC). CCC is a version of homological mirror symmetry for toric varieties. It equates the derived category of coherent sheaves on a toric variety and the category of constructible sheaves on a torus that satisfy some condition on singular support. Recently, Harder-Katzarkov conjectured that there should be a version of CCC for toric fiber bundles and they proved their conjecture for $\mathbb{P}_1$-bundles. I will explain how we can prove (half of) their conjecture for
$\mathbb{P}_n$-bundles. If time permits, I will give a more precise version of the conjecture for arbitrary toric fiber bundles.

Emmy Noether is famous as the “mother of modern algebra,” but her influence extended far beyond algebra alone. This talk, based on my recent book with the title above, will focus on Noether’s broader influence as an international figure in the 1920s. Beyond her immediate circle of students, Noether’s courses drew talented mathematicians from all over the world. Four of the most important were B.L. van der Waerden, Pavel Alexandrov, Helmut Hasse, and Olga Taussky. Noether’s classic papers on ideal theory inspired van der Waerden to recast his research in algebraic geometry. Her lectures on group theory motivated Alexandrov to develop links between point set topology and combinatorial methods. Noether’s vision for a new approach to algebraic number theory gave Hasse the impetus to pursue a line of research that led to the Brauer-Hasse-Noether Theorem, whereas her abstract style clashed with Taussky’s approach to classical class field theory during a difficult time when both were trying to find their footing in a foreign country. Hermann Weyl, her colleague before both fled to the United States in 1933, fully recognized that Noether’s dynamic school was the very heart and soul of the famous Göttingen community.
Two recent books on Emmy Noether:
Emmy Noether – Mathematician Extraordinaire
https://www.springer.com/gp/book/9783030638092
Proving It Her Way: Emmy Noether, a Life in Mathematics
https://www.springer.com/gp/book/9783030628109

In complex algebraic geometry, positivity of direct images of relative canonical bundles are important for the study of geometry of algebraic morphisms. In this talk, I would like to discuss a notion of metric positivity for coherent sheaves and prove that a large class of sheaves from Hodge theory, including the direct images of relative canonical bundles, always satisfy the metric positivity. This result unifies and strengthens several results of positivity on the algebraic side (i.e. weak positivity). Based on joint work with Christian Schnell.

(Note the unusual time: 4:30pm Shanghai, 10:30am Paris.)
The group synchronization problem calls for the estimation of a ground-truth vector from the noisy relative transforms of its elements, where the elements come from a group and the relative transforms are computed using the binary operation of the group. Such a problem provides an abstraction of a wide range of inverse problems that arise in practice. However, in many instances, one needs to tackle a non-convex optimization formulation. It turns out that for synchronization problems over certain subgroups of the orthogonal group, a simple projected gradient-type algorithm, often referred to as the generalized power method (GPM), is quite effective in finding the ground-truth when applied to their non-convex formulations. In this talk, we survey the related recent results in the literature and focus in particular on the techniques for analyzing the statistical and optimization performance of the GPM.
This talk covers joint works with Huikang Liu, Peng Wang, Man-Chung Yue, and Zirui Zhou.

In this talk, we formulate and discuss the proof of a generalized Stone-von Neumann Theorem. The generalization extends to certain C*-dynamical systems represented on Hilbert C*-modules. Along the way, we will discuss imprimitivity bimodules and unbounded operators on Hilbert C*-modules. The novelty of our generalization stems from our representation of the Weyl Commutation Relation on Hilbert C*-modules instead of just Hilbert spaces, and our introduction of two additional commutation relations, which are necessary to obtain a uniqueness theorem. This is joint work with Leonard Huang (University of Nevada, Reno).

Permutations can be described in many ways, including as words and as graphs. The graphical perspective lets us think about "heights" and "depths" of a permutation. Peak sets of permutations have a long history in the literature. Inspired by work of Billey, Burdzy, and Sagan on those sets, we introduced the pinnacle sets of permutations. Despite natural symmetries, pinnacle sets and peak sets have notably different properties. We will explore some of those differences in this talk, giving characterization and enumerative results about pinnacle sets. Very recently, several papers have added to the literature on pinnacle sets, and we will describe those updates here.
This talk includes joint work with Robert Davis, Sarah Nelson, Kyle Petersen, and Irena Rusu.