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.