Fibered knots show up all over low-dimensional topology, as they provide a robust way to investigate interactions between phenomena of different dimensions. In this talk, I'll survey what they are, why you should care, and how to identify them. Then, as time permits, I'll also sketch a proof that positive braid knots are fibered. I will assume very little background for this talk -- all are welcome!

Dimensionality reduction is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally close to its intrinsic dimension.
A classical embedding result is the well-know “Johnson–Lindenstrauss”. The JL lemma shows how a $n$-set of points in $\mathbb{R}^N$ can be embedded into a smaller dimensional space. In this talk we present a result similar to the JL-embedding in the interesting case where instead of a discrete set we embed a compact $d$-dimensional submanifold $\mathcal{M}$ of $\mathbb{R}^N$ into $\mathbb{R}^m $ where $m$ depends on the volume, reach and dimension of $\mathcal{M}.$
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This will be a hybrid seminar and take place in C329 Wells Hall and via Zoom at https://msu.zoom.us/j/99426648081?pwd=ZEljM3BPUXg2MjVUMVM5TnlzK2NQZz09 .

A vector bundle on projective space is called "Steiner" if it can be recognized simply as the cokernel of a map given by a matrix of linear forms. Such maps arise from various geometric setups and one can ask: from the Steiner bundle, can we recover the geometric data used to construct it? In this talk, we will mention an interesting Torelli-type result of Dolgachev and Kapranov from 1993 that serves as an origin of this story, as well as other work that this inspired. We'll then indicate our contribution which amounts to analogous Torelli-type statements for certain tautological bundles on the very ample linear series of a polarized smooth projective variety. This is joint work with R. Lazarsfeld.

In classical probability theory, Fisher information is one of the important concepts. Voiculescu introduced the free probability analogue of this quantity, called free fisher information. In this talk, we will discuss how Free Fisher information helps us to understand a von Neumann algebra.

Cloning systems are a method for generalizing Thompson's groups, for example $V_d$, that result in a family of groups, $\mathcal{T}_d(G_*)$, whose group von Neumann algebras have been intensely studied by Bashwinger and Zarmesky in recent years. We consider the group actions of a large class of $\mathcal{T}_d(G_*)$ and show they are stable, that is, $G \sim_{OE} G \times \mathbb{Z}.$ As a corollary, we answer Bashwinger and Zaremsky question about when $\mathcal{T}_d(G_*)$ is a McDuff Group in the sense of Deprez and Vaes. As a contrasting result, we show $L(V_d)$ is a prime II$_1$ factor. This is joint work with Rolando de Santiago and Krishnendu Khan.

In this talk, I’ll describe a braid word theoretic property, called “twist positivity”, which often puts strong restrictions on quantitative and geometric properties of a braid. I’ll describe some old and new results about twist positivity, as well as some new applications towards knot concordance. In particular, I’ll describe how using a suite of numerical knot invariants (including the braid index) in tandem allows one to prove that there is an infinite family of L-space knots (containing all positive torus knots and also an infinite family of hyperbolic knots) where every knot represents a distinct smooth concordance class. This confirms a prediction of the slice-ribbon conjecture. Everything I’ll discuss is joint work with Hugh Morton. I will assume little background about knot invariants for this talk – all are welcome!

In this talk, we will explore and make comparisons between various models that exist for spherical tensor categories associated to the category of representations of the quantum group $U_q(sl_n).$ In particular, we will discuss the combinatorial model of Murakami-Ohtsuki-Yamada (MOY), the n-valent ribbon model of Sikora and the trivalent spider category of Cautis-Kamnitzer-Morrison (CKM). We conclude by showing that the full subcategory of the spider category from CKM, whose objects are monoidally generated by the standard representation and its dual, is equivalent as a spherical braided category to Sikora's quotient category. This proves a conjecture of Le and Sikora and also answers a question from Morrison's Ph.D. thesis.

Given a finite poset that is not completely ordered, is it always possible to find two elements x and y, such that the probability that x is less than y in the random linear extension of the poset, is bounded away from 0 and 1? Kahn-Saks gave an affirmative answer and showed that this probability falls between 3/11 (0.273) and 8/11 (0.727). The currently best known bound is 0.276 and 0.724 by Brightwell-Felsner-Trotter, and it is believed that the optimal bound should be 1/3 and 2/3, also known as the 1/3-2/3 Conjecture. Most notably, log-concave and cross product inequalities played the central role in deriving both bounds. In this talk we will discuss various generalizations of these results together with related open problems. This talk is joint work with Igor Pak and Greta Panova, and is intended for the general audience.

A rotating cosmic string spacetime has a singularity along a timelike curve corresponding to a one-dimensional source of angular momentum. Such spacetimes are not globally hyperbolic: they admit closed timelike curves near the so-called "string". This presents challenges to studying the existence of solutions to the wave equation via conventional energy methods. In this work, we show that forward solutions to the wave equation (in an appropriate microlocal sense) do exist. Our techniques involve proving a statement on propagation of singularities and using the resulting estimates to show existence of solutions. This is joint work with Jared Wunsch.

The two-scale relation in wavelet analysis dictates that a square-integrable function can be written as a linear combination of scaled and shifted copies of itself. This fact is equivalent to the existence of square-integrable functions whose time-scale shifts are linearly dependent. By contrast, by replacing the scaling operator with a modulation operator one would think that the linear dependency of the resulting time-frequency shifts of a square-integrable function might be easily inferred. However, more than two decades after C.~Heil, J.~Ramanatha, and P.~Topiwala conjectured that any such finite collection of time-frequency shifts of a non-zero square-integrable function on the real line is linearly independent, this problem (the HRT Conjecture) remains unresolved.
The talk will give an overview of the HRT conjecture and introduce an inductive approach to investigate it. I will highlight a few methods that have been effective in solving the conjecture in certain special cases. However, despite the origin of the HRT conjecture in Applied and Computational Harmonic Analysis, there is a lack of experimental or numerical methods to resolve it. I will present an attempt to investigate the conjecture numerically.

We define a natural, purely geometrical bijection between the set solutions of Bethe ansatz equations for the Gaudin magnet chain and the set of arc diagrams of Frenkel-Kirillov-Varchenko. The former set is in natural bijection with monodromy-free sl_2-opers (aka projective structures) on the projective line with the prescribed type of regular singularities at prescribed real marked points (according to Feigin and Frenkel), while the latter indexes the canonical base in a tensor product of U_q(sl_2)-modules (via the Schechtman-Varchenko isomorphism). Both sets carry a natural action of the cactus group, i.e., the fundamental group of the real Deligne-Mumford space of stable rational curves with marked points (by monodromy of solutions to Bethe ansatz equations on the former and by crystal commuters on the latter). We prove that our bijection is compatible with this cactus group action. This is joint work with Nikita Markarian.

Deep learning has had transformative impacts in many fields including computer vision, computational biology, and dynamics by allowing us to learn functions directly from data. However, there remain many domains in which learning is difficult due to poor model generalization or limited training data. We'll explore two applications of representation theory to neural networks which help address these issues. Firstly, consider the case in which the data represent an $G$-equivariant function. In this case, we can consider spaces of equivariant neural networks which may more easily be fit to the data using gradient descent. Secondly, we can consider symmetries of the parameter space as well. Exploiting these symmetries can lead to models with fewer free parameters, faster convergence, and more stable optimization.