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1:00pm to 2:00pm - RH 440R - Dynamical Systems Victor Kleptsyn - (CNRS, University of Rennes 1, France) Dynamics of the Sturmian Trace Skew Product Schrödinger operators with Sturmian potential are highly analogous to the Almost Matthieu ones: the potential is also quasi-periodic with some frequency $\alpha$. In particular, the corresponding spectrum (plotted as a function of $\alpha$) forms so-called Kohmoto butterfly, that can be seen as a sibling to the Hofstadter butterfly, associated to the Almost Matthieu operator. These operators and their spectra have been already extensively studied by many authors; in particular, many results have been obtained previously for the frequencies $\alpha$ that are quadratic irrationalities or of bounded type. A key element of many of these works is the study of corresponding renormalisation operators, acting on the associated Markov surface. My talk will be devoted to our joint result with Anton Gorodetski and Seung uk Jang: we study the dynamics of the skew product that joins the renormalisation of the (traces of the) transition matrices and of the frequency $\alpha$. For this skew product, we construct explicitly a ``stable cone field’’ (over all irrational $\alpha$ and for an arbitrary coupling constant $\lambda$). This is a first step in our strategy of obtaining a dynamical/renormalization point of view on the self-similarity of the Kohmoto butterfly. |
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1:00pm to 1:50pm - RH 340N - Algebraic Geometry Maksym Zubkov - (University of British Columbia ) The Geometry of Rational Neural Networks Rational neural networks are feedforward neural networks with a rational activation function. These networks found their applications in approximating the solutions of PDE, as they are able to learn the poles of meromorphic functions. In this talk, we are going to consider the simplest rational activation function, sigma = 1 / x, and study the geometry of family such architectures. We will show that the closure of all possible shallow (one hidden layer) networks is an algebraic variety, which called a neurovariety. |
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4:00pm to 5:00pm - RH 306 - Differential Geometry Ronan Conlon - (University of Texas, Dallas) A family of Kahler flying wing steady Ricci solitons Steady Kahler-Ricci solitons are eternal solutions of the Kahler-Ricci flow. I will present new examples of such solitons with strictly positive sectional curvature that live on C^n and provide an answer to an open question of H.-D. Cao in complex dimension 2. This is joint work with Pak-Yeung Chan and Yi Lai. |
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3:00pm to 4:00pm - Rowland Hall 510 R - Combinatorics and Probability Deanna Needell - (UCLA) Fairness and Foundations in Machine Learning In this talk, we will address areas of recent work centered around the themes of fairness and foundations in machine learning as well as highlight the challenges in this area. We will discuss recent results involving linear algebraic tools for learning, such as methods in non-negative matrix factorization that include tailored approaches for fairness. We will showcase our derived theoretical guarantees as well as practical applications of those approaches. Then, we will discuss new foundational results that theoretically justify phenomena like benign overfitting in neural networks. Throughout the talk, we will include example applications from collaborations with community partners, using machine learning to help organizations with fairness and justice goals. |
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1:00pm - 306 Rowland Hall - Harmonic Analysis Beatrice-Helen Vritsiou - (University of Alberta) Illuminating certain high-dimensional 1-unconditional convex bodies Let us think of a convex body in R^n (convex, compact set, with non-empty interior) as an opaque object, and let us place point light sources around it, wherever we want, to illuminate its entire surface. What is the minimum number of light sources that we need? The Hadwiger-Boltyanski illumination conjecture from 1960 states that we need at most as many light sources as for the n-dimensional hypercube, and more generally, as for n-dimensional parallelotopes. For the latter their illumination number is exactly 2^n, and they are conjectured to be the only equality cases. The conjecture is still open in dimension 3 and above, and has only been fully settled for certain classes of convex bodies (e.g. zonoids, bodies of constant width, etc.). Moreover, there are some rare examples for which a basic, folklore argument could quickly lead to the upper bound 2^n, while at the same time understanding the equality cases has remained elusive for decades. One such example would be convex bodies very close to the cube, which was settled by Livshyts and Tikhomirov in 2017. In this talk I will discuss another such instance, which comes from the class of 1-unconditional convex bodies, and which also 'forces' us to settle the conjecture for a few more cases of 1-unconditional bodies. This is based on joint work with Wen Rui Sun, and our arguments are primarily combinatorial. |
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4:00pm to 5:00pm - RH 306 - Colloquium Omer Ben-Neria - (Einstein Institute of Mathematics, the Hebrew University, Jerusalem, Israel.) In Search of Stronger Axioms The search for an ultimate axiomatization of mathematics is inevitably incomplete. However, this does not preclude the possibility of strong and natural extensions to the standard axioms of set theory (ZFC) which shed light on many unresolved mathematical questions. Gödel suggested searching for strong axioms of infinity (known as large cardinals) as potential candidates for such extensions -- a pursuit that has continued for over half a century. This talk will survey several central developments in this search, including results concerning finite graphs, questions about Lebesgue measurability of projective sets of reals, the inner model program, and current efforts toward identifying an "ultimate" model of set theory via canonical inner models. |