BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Henry Adams (Colorado State University) DTSTART:20210825T133000Z DTEND:20210825T143000Z DTSTAMP:20260404T094310Z UID:Danger2021/1 DESCRIPTION:Title: Applied topology: from global to local\nby Henry Adams (Colo rado State University) as part of DANGER: Data\, Numbers\, and Geometry\n\ n\nAbstract\nThrough the use of examples\, I will explain one way in which applied topology has evolved since the birth of persistent homology in th e early 2000s. The first applications of topology to data emphasized the g lobal shape of a dataset\, such as the three-circle model for 3 x 3 pixel patches from natural images\, or the configuration space of the cyclo-octa ne molecule\, which is a sphere with a Klein bottle attached via two circl es of singularity. More recently\, persistent homology is being used to me asure the local geometry of data. How do you vectorize geometry for use in machine learning problems? Persistent homology\, and its vectorization te chniques including persistence landscapes and persistence images\, provide popular techniques for incorporating geometry in machine learning. I will survey applications arising from machine learning tasks in agent-based mo deling\, shape recognition\, materials science\, and biology.\n LOCATION:https://stable.researchseminars.org/talk/Danger2021/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Sonja Petrovic (Illinois Institute of Technology) DTSTART:20210825T143000Z DTEND:20210825T153000Z DTSTAMP:20260404T094310Z UID:Danger2021/2 DESCRIPTION:Title: Learning in commutative algebra & models for random algebraic st ructures\nby Sonja Petrovic (Illinois Institute of Technology) as part of DANGER: Data\, Numbers\, and Geometry\n\n\nAbstract\nA commutative alg ebraist's interest in randomness has many facets\, of which this talk high lights two. Namely\, we will discuss 1) how to use basic statistics and le arning for improving Buchberger's algorithm and 2) how to generate samples of ideals in a `controlled' way. The two topics\, based on joint work wit h various collaborators and students\, form a two-step process in learning on algebraic structures\, designed with the aim of avoiding the 'danger z one' of blind machine learning over uninteresting distributions. \nFor lea rning\, we show that a multiple linear regression model built from a set o f easy-to-compute ideal generator statistics can predict the number of pol ynomial additions somewhat well\, better than an uninformed model\, and be tter than regression models built on some intuitive commutative algebra in variants that are more difficult to compute. We also train a simple recurs ive neural network that outperforms these linear models. Our work serves a s a proof of concept\, demonstrating that predicting the number of polynom ial additions in Buchberger's algorithm is a feasible problem from the poi nt of view of machine learning.\nAs a first example of sampling\, we prese nt random monomial ideals\, using which we prove theorems about the probab ility distributions\, expectations and thresholds for events involving mon omial ideals with given Hilbert function\, Krull dimension\, first graded Betti numbers\, and present several experimentally-backed conjectures abou t regularity\, projective dimension\, strong genericity\, and Cohen-Macaul ayness of random monomial ideals. The models for monomial ideals can be us ed as a basis for generating other types of algebraic objects\, and provin g existence of desired properties.\n LOCATION:https://stable.researchseminars.org/talk/Danger2021/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Siu-Cheong Lau (Boston University) DTSTART:20210825T160000Z DTEND:20210825T170000Z DTSTAMP:20260404T094310Z UID:Danger2021/3 DESCRIPTION:Title: Deep learning over the moduli space of quiver representations\nby Siu-Cheong Lau (Boston University) as part of DANGER: Data\, Numbers \, and Geometry\n\n\nAbstract\nIt is interesting to observe that neural ne twork in machine learning has a similar basic setup as quiver representati on theory. In this talk\, I will build an algebro-geometric formulation of a computing machine\, which is well-defined over the moduli space of repr esentations. I will also explain a uniformization between spherical\, Euc lidean and hyperbolic moduli of framed quiver representations\, and constr uct a learning algorithm over these moduli spaces.\n LOCATION:https://stable.researchseminars.org/talk/Danger2021/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Riccardo Finotello (CEA Paris-Saclay) DTSTART:20210826T120000Z DTEND:20210826T130000Z DTSTAMP:20260404T094310Z UID:Danger2021/4 DESCRIPTION:Title: Algebraic geometry and computer vision: inception neural network for Calabi-Yau manifolds\nby Riccardo Finotello (CEA Paris-Saclay) as part of DANGER: Data\, Numbers\, and Geometry\n\n\nAbstract\nComputing to pological properties of Calabi-Yau manifolds is\, in general\, a challengi ng mathematical task: traditional methods lead to complicated algorithms\, without expressions in closed form in most cases. At the same time\, rece nt years have witnessed the rising use of deep learning as a method for ex ploration of large sets of data\, to learn their patterns and properties. This is specifically interesting when it comes to unravel complicated geom etrical structures\, as it is a central issue both in mathematics and theo retical physics\, as well as in the development of trustworthy AI methods. Motivated by their distinguished role in string theory for the study of c ompactifications\, we compute the Hodge numbers of Complete Intersection C alabi-Yau (CICY) manifolds using deep neural networks. Specifically\, we i ntroduce new regression architectures\, inspired by Google's Inception net work and multi-task learning\, which leverage the theoretical knowledge on the inputs with recent advancements in AI. This shows the potential of d eep learning to learn from geometrical data\, and it proves the versatilit y of architectures developed in different contexts\, which may therefore f ind their way in theoretical physics and mathematics for exploration and i nference.\n LOCATION:https://stable.researchseminars.org/talk/Danger2021/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Kyu-Hwan Lee (University of Connecticut) DTSTART:20210826T133000Z DTEND:20210826T143000Z DTSTAMP:20260404T094310Z UID:Danger2021/5 DESCRIPTION:Title: Applications of machine learning to data from number theory\ nby Kyu-Hwan Lee (University of Connecticut) as part of DANGER: Data\, Num bers\, and Geometry\n\n\nAbstract\nIn this talk\, we apply machine learnin g techniques to various data from the L-functions and modular forms databa se (LMFDB) and show that a machine can be trained to distinguish objects i n number theory according to their standard invariants. The applications i n this talk will include class numbers of quadratic number fields\, ranks of elliptic curves\, Sato-Tate groups of genus 2 curves. This is joint wor k with Yang-Hui He and Thomas Oliver.\n LOCATION:https://stable.researchseminars.org/talk/Danger2021/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Roozbeh Yousefadeh (Yale University) DTSTART:20210826T143000Z DTEND:20210826T153000Z DTSTAMP:20260404T094310Z UID:Danger2021/6 DESCRIPTION:Title: Deep learning generalization\, extrapolation\, over-parameteriza tion and decision boundaries\nby Roozbeh Yousefadeh (Yale University) as part of DANGER: Data\, Numbers\, and Geometry\n\n\nAbstract\nDeep neura l networks have achieved great success\, most notably in learning to class ify images. Yet\, the phenomenon of learning images is not well understood \, and generalization of deep networks is considered a mystery. Recent stu dies have explained the generalization of deep networks within the framewo rk of interpolation. In this talk\, we will see that the task of classifyi ng images requires extrapolation capability\, and interpolation by itself is not adequate to understand functional task of deep networks. We study i mage classification datasets in the pixel space\, the internal representat ions of images learned throughout the layers of trained networks\, and als o in the low-dimensional feature space that one can derive using wavelets/ shearlets. We show that in all these spaces\, image classification remains an extrapolation task to a moderate (yet considerable) degree outside the convex hull of training set. From the mathematical perspective\, a deep l earning image classifier is a function that partitions its domain and assi gns a class to each partition. Partitions are defined by decision boundari es and so is the model. Therefore\, the extensions of decision boundaries outside the convex hull of training set are crucial in model's generalizat ion. From this perspective\, over-parameterization is a necessary conditio n for the ability to control the extensions of decision boundaries\, a nov el way of explaining why deep networks need to be over-parameterized. I wi ll also present a homotopy algorithm for computing points on the decision boundaries of deep networks\, and finally\, I will explain how we can leve rage the decision boundaries to audit and debug ML models used in social a pplications.\n LOCATION:https://stable.researchseminars.org/talk/Danger2021/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Ruriko Yoshida (Naval Postgraduate School) DTSTART:20210826T160000Z DTEND:20210826T170000Z DTSTAMP:20260404T094310Z UID:Danger2021/7 DESCRIPTION:Title: Tree topologies along a tropical line segment\nby Ruriko Yos hida (Naval Postgraduate School) as part of DANGER: Data\, Numbers\, and G eometry\n\n\nAbstract\nTropical geometry with the max-plus algebra has bee n applied to statistical learning models over the spaces of phylogenetic t rees because geometry with the tropical metric over tree spaces has some n ice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical m etric. This talk focuses on combinatorics of tree topologies along a trop ical line segment\, an intrinsic geodesic with the tropical metric\, betwe en two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically\, we show that a probability of a tropical line segment of two randomly chosen trees goi ng through the origin (the star tree) is zero and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move\, the n the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\n LOCATION:https://stable.researchseminars.org/talk/Danger2021/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Minhyong Kim (Warwick) DTSTART:20210825T121500Z DTEND:20210825T131500Z DTSTAMP:20260404T094310Z UID:Danger2021/8 DESCRIPTION:Title: How hard is it to learn a mathematical structure?\nby Minhyo ng Kim (Warwick) as part of DANGER: Data\, Numbers\, and Geometry\n\nAbstr act: TBA\n LOCATION:https://stable.researchseminars.org/talk/Danger2021/8/ END:VEVENT END:VCALENDAR