In this talk, we introduce a new framework called Variance-Reduced Sketching (VRS), specifically designed to estimate density functions in higher dimensions with a reduced curse of dimensionality. Our framework conceptualizes multivariable functions as infinite-size matrices\/tensors, facilitating a new matrix\/tensor-based bias-variance tradeoff in various nonparametric contexts.<\/p>\n\n\n\n
We demonstrate the robust numerical performance of VRS through a series of simulated experiments and real-world data applications. Notably, VRS shows remarkable improvement over existing neural network estimators and classical kernel methods in numerous density models. Additionally, we will discuss theoretical guarantees for VRS to support its ability to deliver density estimation with a reduced curse of dimensionality.<\/p>\n\n\n\n
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09\/12\/2024 – Chase Bender: Julia Set on the Projective Line<\/h2>\n\n\n\n
Abstract:<\/strong> The field of complex dynamics was born from the work of Gaston Julia and Pierre Fatou centered around studying the limiting behavior of the sequence of iterates of a rational function of one complex variable. In this talk I will give a rough outline of the classical theory of Julia sets on one variable, and then transition to a more modern perspective which yields sharper results and is more applicable to the several variable setting.<\/p>\n\n\n\n Abstract:<\/strong> The talk is based on \u201cOn Pseudo-Finite Dimensions\u201d by Hrushovski and \u201cApproximate Subgroups of Linear Groups\u201d by Breuillard, Green and Tao. For an ultraproduct of first order structures, given a convex subset of the nonstandard real numbers, a pseudo-finite dimension can be defined for each of the pseudo-finite subset of the ultraproduct. In the first part of the talk, I will discuss the definition and properties of pseudo-finite dimensions. The second part of the talk will be focused on application of this concept to the study of approximate subgroups of simple algebraic groups.<\/p>\n\n\n\n Abstract:<\/strong> We will attempt to explain what operads are, where they came from, and more importantly, what they are used for. Roughly, an operad can be thought of as a “gadget” that parameterizes a class of algebraic structures, such as associative unital algebras or Lie algebras..<\/p>\n\n\n\n Abstract:<\/strong> Manifolds with positive or nonnegative sectional curvature are important and interesting mathematical objects. Some fundamental results, like Gromov\u2019s Betti number Theorem, Bonnet\u2013Myers or Synge\u2019s Theorem have helped understand the topology of these manifolds. However, we are far from being able to classify all of them. A classification of these manifolds in lower dimensions has been proven successful in the case when they have a large group of isometries. In this talk we will give a sketch of the proof of the classification of all closed, simply-connected, nonnegatively curved 4-manifolds with circle symmetry. This will lead us to a crash course in Alexandrov geometry, a pivotal tool essential for the proof. We will also see that the classification eventually boils down to showing that some curve in the 3-sphere is the unknotted.<\/p>\n\n\n\n Abstract:<\/strong> In this presentation, we will show the difference between one and several complex variables by introducing the Hartogs’s extension theorem. In one variable, the isolated singularities are classified into removable, pole and essential; however, Hartogs said that in several complex variables, every isolated singularity is removable.<\/p>\n\n\n\n Abstract:<\/strong> I will prove the Peter-Weyl theorem on compact groups, and, if time permits, I will discuss some applications of it in model theory and in additive combinatorics.<\/p>\n\n\n\n Abstract:<\/strong> The theory of total positivity was developed in the 1930’s to study matrices whose minors were all nonnegative. This notion was generalized in the 1990s when Lustzig introduced the notion of a totally nonnegative variety $G_{\\geq 0}$ in a reductive group $G$. Fomin and Zelevinsky further studied $G_{\\geq 0}$ for $G$ a simply connected complex Lie group with Weyl Group $W$. In this setting, we can describe $G$ as a disjoint union of varieties $G^{u, v}$ indexed by $(u, v) \\in W \\times W$.<\/p>\n\n\n\n In this presentation, we will cover three results of Fomin and Zelevinsky concerning special properties of $G^{u, v}$. We will first show that $G^{u, v}$ is a smooth variety, and describe it as an open subset of affine space. Then, we will give natural coordinates on $G^{u, v}$ coming from Lie Theory. Finally, we will use these coordinates to describe $G_{\\geq 0}$ in $G$. <\/p>\n\n\n\n Abstract:<\/strong> This seminar will be a panel discussion featuring some former Notre Dame students in mathematics who are currently working in a job outside of academia, or in academia but not as a math professor.<\/p>\n\n\n\n The panelists include: Abstract:<\/strong> In this talk I will introduce a one-player game (a sequence of moves governed by some rules) and challenge you, the audience, to determine the game’s outcome: Does it always terminate? Does the starting position matter? Are there different strategies, and do those affect whether the game terminates? Are these strategies optimal? After the introduction you will place a bet on the answer to those questions; winners get candy. Are you ready to play? Abstract:<\/strong> Work by David Reutter and Chris Schommer\u2013Pries has shown that TQFTs can distinguish stable diffeomorphism classes of closed, connected, even-dimensional manifolds subject to certain finiteness conditions. In particular, simply connected closed 6-manifolds with finite Abstract:<\/strong> The interpolation problem has occupied a central position in mathematics for several millennia, with roots tracing back to Ancient Greece. Roughly speaking, it asks: When can a curve of a given type pass through a collection of points? The problem has a long and rich history, with many chapters, and a complete solution has been obtained only very recently by Eric Larson and Isabel Vogt in 2022. This concrete problem motivates fundamental concepts such as moduli spaces and their properties, deformation theory, normal bundles, and more. In this talk I will give a historical introduction to the interpolation problem, and I will proceed to discuss the interpolation problem from\u00a0a more modern perspective, namely Brill\u2013Noether theory. Time permitting, I will talk about the architecture of the complete solution to the interpolation problem.<\/p>\n\n\n\n 09\/05\/2024 – Prof. Daren Wang: Nonparametric Density Estimation via Variance-Reduced Sketching Abstract: Nonparametric models are of great interest in various scientific and engineering disciplines. Classical kernel methods, while numerically robust and statistically sound in low-dimensional settings, become inadequate in higher-dimensional settings due to the curse of dimensionality. In this talk, we introduce a new framework […]<\/p>\n","protected":false},"author":4879,"featured_media":0,"parent":55,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-692","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/pages\/692","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/users\/4879"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/comments?post=692"}],"version-history":[{"count":20,"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/pages\/692\/revisions"}],"predecessor-version":[{"id":805,"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/pages\/692\/revisions\/805"}],"up":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/pages\/55"}],"wp:attachment":[{"href":"https:\/\/sites.nd.edu\/mgsa\/wp-json\/wp\/v2\/media?parent=692"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\n09\/19\/2024 – Yuyan He: Pseudo-finite Dimension and Simple Algebraic Groups<\/h2>\n\n\n\n
\n\n\n\n09\/26\/2024 – Cory Gillette: Introduction to Operads<\/h2>\n\n\n\n
\n\n\n\n10\/03\/2024 – Luis Atzin: Classifying non-negatively curved 4-manifolds<\/h2>\n\n\n\n
\n\n\n\n10\/10\/2024 – Chen-Kuan Lee: Hartogs’s extension theorem<\/h2>\n\n\n\n
\n\n\n\n10\/17\/2024 and 10\/31\/2024 – Atticus Stonestrom: The Peter\u2013Weyl Theorem<\/h2>\n\n\n\n
\n\n\n\n11\/07\/2024 – Samuel Heard: Double Bruhat Cells and Total Positivity<\/h2>\n\n\n\n
\n\n\n\n11\/14\/2024 – Prof. Sam Evens: Panel on Non-academic Jobs<\/h2>\n\n\n\n
Jeremy Mann<\/strong> (Notre Dame PhD, 2019, Data Scientist)
John Siratt<\/strong> (Notre Dame PhD, 2024, researcher in the Formal Methods Program at NASA)
James Schmidt<\/strong> (UIUC math PhD and Notre Dame undergrad math student, research engineer)<\/p>\n\n\n\n
\n\n\n\n11\/21\/2024 – Lorenzo Riva: [Redacted]<\/mark> and the [Redacted]<\/mark> game<\/h2>\n\n\n\n
Note: Since this is a pretty famous game that can be easily found on the internet I decided to omit its name and the machinery used to study it from the title of the talk, otherwise the betting round would be pretty trivial.<\/p>\n\n\n\n
\n\n\n\n12\/05\/2024 – Katherine Novey: Ordinary TQFTs Cannot Distinguish Simply Connected 6-Manifolds<\/h2>\n\n\n\n
![\"\\pi_2\"](\"https:\/\/ci3.googleusercontent.com\/meips\/ADKq_NaLQ8gXuekpOeM3P5Y3iRzGb__PIAdJm62s1XRZ8RKV-shvdcxN3fyGhL5fvLzzK8QG_N4kr4F1b3PgF85b0xAvGerBQGEXhL_fLMnyAtqJfzdjyAhL3vFsmkLmn71Eug=s0-d-e1-ft#https:\/\/s0.wp.com\/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=%5Cpi%5F2\")
\n\n\n\n12\/12\/2024 – Emma Dooley: The Interpolation Problem<\/h2>\n\n\n\n
\n","protected":false},"excerpt":{"rendered":"