With every Coxeter system one can associate a family of algebras considered as deformation of its group algebra. These are so-called Hecke algebras, which are classical objects of study in combinatorics and representation theory. Complex Hecke algebras admit a natural *-structure and a *-representation on Hilbert space. Taking the norm- and SOT-closure in such representation, one obtains Hecke operator algebras, which have recently seen increased attention.
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In this talk, I will introduce Hecke operator algebras from scratch, focusing on the case of right-angled Coxeter groups. This case is particularly interesting from an operator algebraic perspective, thanks to its description by iterated amalgamated free products. I will survey known results on the structure of Hecke operator algebras, before I describe recent work that clarified the factor decomposition of Hecke von Neumann algebras. Two applications to representation theory will be presented. I will finish with some results on the scope and limits of K-theoretic classification of right-angled Hecke C*-algebras.
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This is joint work with Adam Skalski.

I discuss an optimization framework for solving problems with group sparsity inducing regularization. Such regularizers include Lasso (L1), group Lasso, and latent group Lasso. The framework computes iterates by optimizing over small dimensional subspaces, thus keeping the cost per iteration relatively low. Theoretical convergence results and numerical tests on various learning problems will be presented.

In this talk we consider a canonical clustering problem where one receives unlabeled samples drawn from a balanced mixture of two elliptical distributions and aims for a classifier to estimate the labels. Many popular methods including PCA and k-means require individual components of the mixture to be somewhat spherical, and perform poorly when they are stretched. To overcome this issue, we propose a non-convex program seeking for an affine transform to turn the data into a one-dimensional point cloud concentrating around -1 and 1, after which clustering becomes easy. Our theoretical contributions are two-fold: (1) we show that the non-convex loss function exhibits desirable geometric properties when the sample size exceeds some constant multiple of the dimension, and (2) we leverage this to prove that an efficient first-order algorithm achieves near-optimal statistical precision without good initialization. We also propose a general methodology for clustering with flexible choices of feature transforms and loss objectives.

In this talk I will describe pathways to invite and retain underrepresented minorities and women local to San Diego and Tijuana in mathematics and sciences at UCSD. I will describe Math Jams, which I pioneered at San Diego City College, and how I can lead Math Jams at Living Learning Communities (LLCs) such as the African Black Diaspora LLC at Sixth College and Raza LLC at Eleanor Roosevelt College. I will discuss the Hesabu Circle, a safe space for black students of all ages pre-K to post-doc, and how math circles can be created for underrepresented minorities and women at UCSD supporting undergraduate and graduate students. I will discuss student success statistics with my Umoja and Puente students at San Diego City College and Upward Bound students.

Deep Learning is much publicized and has had great empirical success on challenging problems in learning. Yet there is no quantifiable proof of performance and certified guarantees for these methods. This talk will give an overview of Deep Learning from the viewpoint of mathematics and numerical computation.

The classical problem of counting elliptic curves with a
rational N-isogeny can be phrased in terms of counting rational points
on certain moduli stacks of elliptic curves. Counting points on stacks
poses various challenges, and I will discuss these along with a few ways
to overcome them. I will also talk about the theory of heights on stacks
developed in recent work of Ellenberg, Satriano and Zureick-Brown and
use it to count elliptic curves with an $N$-isogeny for certain $N$. The
talk assumes no prior knowledge of stacks and is based on joint work
with Brandon Boggess.

Kodaira classified the singular elliptic fibers
occurring on relatively minimal elliptic surfaces (over C). I will
explain a birational Kodaira's classifications for higher dimensional
elliptic fibrations. (Based on work in collaboration with T. Weigand)

The FFT problem, which was inspired by work of Guldemond, can be stated as follows: how can you fill a 3x3 grid with F's and T's such that it contains as many copies of the word "FFT" as possible? For example, the following two grids each contain 5 copies of the word FFT (we allow the word to be written forwards or backwards, and to appear in rows, columns, or diagonals):
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\[\begin{matrix} F F T\\ F F T\\ F F T\end{matrix}\hspace{30pt} \begin{matrix} F T F\\ T F F\\ F F T\end{matrix}\]
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Grubb claimed that there exists a grid containing 6 copies of FFT. Eight minutes later he claimed that actually, the best you could do is 5. He offered no proof of either claim. In this talk we consider a generalization of the FFT problem. Namely, given a word $w$ of length $n$ and a grid $G$ of letters, let $f(w,G)$ be the number of times $w$ appears in $G$, and let $f(w)=\max_G f(w,G)$. We determine $f(w)$ for a number of words, and in particular we determine $f(FFT)$, solving the FFT problem. I, Sam Spiro, will be the only person talking for the entire hour that the talk is given. Absolutely nothing out of the ordinary will happen during the talk.