Upcoming Seminars

A Class of Universal Space-Time Gaussian Random Fields

Speaker: Edward Waymire

A new class of universal space-time Gaussian random fields on a separable Hilbert space is introduced and investigated. The universality derives from a central limit theorem for ergodic Markov processes due to Bhattacharya (1982). A number of interesting problems are motivated by companion questions and mathematical examples arising in other Gaussian random field theories to be described. The mathematical framework is largely that of functional analysis, and the research goals are curiosity driven at this stage. Read more.

A Class of Universal Space-Time Gaussian Random Fields

Speaker: Edward Waymire

A new class of universal space-time Gaussian random fields on a separable Hilbert space is introduced and investigated. The universality derives from a central limit theorem for ergodic Markov processes due to Bhattacharya (1982). A number of interesting problems are motivated by companion questions and mathematical examples arising in other Gaussian random field theories to be described. The mathematical framework is largely that of functional analysis, and the research goals are curiosity driven at this stage. Read more.

A Class of Universal Space-Time Gaussian Random Fields

Speaker: Edward Waymire

A new class of universal space-time Gaussian random fields on a separable Hilbert space is introduced and investigated. The universality derives from a central limit theorem for ergodic Markov processes due to Bhattacharya (1982). A number of interesting problems are motivated by companion questions and mathematical examples arising in other Gaussian random field theories to be described. The mathematical framework is largely that of functional analysis, and the research goals are curiosity driven at this stage. Read more.

Extrapolation for eigenvalue problems: Upcycling data with dynamic momentum

Speaker: Sara Pollock

ABSTRACT: We will discuss accelerating convergence to numerical solutions of eigenvalue problems using a simple post-processing step applied to standard eigensolver techniques. We will motivate extrapolation-based acceleration approaches which reuse previous steps of an iterative process to define the next term in an approximation sequence. We will introduce using a heavy-ball momentum type of extrapolation and review recent results using both static and dynamically assigned momentum parameters to accelerate the power and inverse iterations. The theory will be illustrated with numerical results. BIO: Sara Pollock is an Associate Professor in the Department of Mathematics at the University of Florida. She obtained her Ph.D. in Mathematics with a specialization in Computational Science from UC San Diego in 2012, an MS in Applied Mathematics from the University of Washington in 2008 and a BS in Mathematics from the University of New Mexico in 2007. Her research is focused on the design… Read more.

Extrapolation for eigenvalue problems: Upcycling data with dynamic momentum

Speaker: Sara Pollock

ABSTRACT: We will discuss accelerating convergence to numerical solutions of eigenvalue problems using a simple post-processing step applied to standard eigensolver techniques. We will motivate extrapolation-based acceleration approaches which reuse previous steps of an iterative process to define the next term in an approximation sequence. We will introduce using a heavy-ball momentum type of extrapolation and review recent results using both static and dynamically assigned momentum parameters to accelerate the power and inverse iterations. The theory will be illustrated with numerical results. BIO: Sara Pollock is an Associate Professor in the Department of Mathematics at the University of Florida. She obtained her Ph.D. in Mathematics with a specialization in Computational Science from UC San Diego in 2012, an MS in Applied Mathematics from the University of Washington in 2008 and a BS in Mathematics from the University of New Mexico in 2007. Her research is focused on the design… Read more.

Computing magnitude homology of graphs

Speaker: Chad Giusti

Magnitude is a parameterized measure of the structure of a metric space that roughly counts "how many points" are in the space at different levels of resolution. Because it is defined in analogy to a generalized notion of Euler characteristic, it is natural to ask whether this measurement arises as the Euler characteristic of a homology theory, and this turns out to be true. The corresponding dual-graded magnitude homology has garnered interest recently as a method for extracting information about the structure of finite metric spaces. In spaces where convexity may be defined, classes in magnitude homology seem to correspond, very roughly, to failures in local convexity. In this talk, I will define all of the relevant elements, and then discuss some recent joint work with Giuliamaria Menara on computing magnitude homology of graphs equipped with the shortest-path distance, including some results on the magnitude homology of ER and geometric random graphs. Read more.

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Avoiding vacuum in two models of superfluidity

Speaker: Pranava Jayanti

At low pressures and very low temperatures, Helium-4 is composed of two interacting phases: the superfluid and the normal fluid. We discuss some recent mathematical results in the analysis of two distinct models of superfluidity.Micro-scale model: The nonlinear Schrödinger equation is coupled with the incompressible inhomogeneous Navier-Stokes equations through a bidirectional nonlinear relaxation mechanism that facilitates mass and momentum exchange between phases. For small initial data, we construct solutions that are either global or almost-global in time, depending on the strength of the superfluid's self-interactions. The primary challenge lies in controlling inter-phase mass transfer to prevent vacuum formation within the normal fluid. Two approaches are employed: one based on energy estimates alone, and another combining energy estimates with maximal regularity. These results are part of joint work with Juhi Jang and Igor Kukavica.Macro-scale model: Both phases are governed by… Read more.

Can zigzag persistence be computed as efficiently as the standard version?

Speaker: Tao Hou

Zigzag persistence has proven a critical extension of the standard non-zigzag persistence. However, its computation is in general considered to be harder than the non-zigzag version. In this talk, I will review some of our recent efforts on bridging the efficiency gap between computing the zigzag and non-zigzag persistence. Specifically, I will talk about algorithms for the following problems: (1) computing the barcode from a zigzag filtration; (2) computing zigzag barcodes for graph filtrations; (3) updating zigzag persistence over local changes; (4) updating zigzag persistence for graph filtrations; (5) computing representatives for zigzag.This is joint work with Tamal K. Dey, Dmitriy Morozov, and Salman Parsa. Read more.

A generalization of Franklin's partition identity and a Beck-type companion identity

Speaker: Holly Swisher

Euler's classic partition identity states that the number of partitions of n into odd parts equals the number of partitions of n into distinct parts. We develop a new generalization of this identity, which yields a previous generalization of Franklin as a special case, and provide both a q-series and bijective proof. We further establish an accompanying Beck-type companion identity which gives the excess in the total number of parts of partitions of one kind over the other.This is joint work with Gabriel Gray, David Hovey, Brandt Kronholm, Emily Payne, and Ren Watson. Read more.

Snakes & Spiders, Robots & Geometry

Speaker: Ross Hatton

ABSTRACT: Locomotion and perception are a common thread between robotics and biology. Understanding these phenomena at a mechanical level involves nonlinear dynamics and the coordination of many degrees of freedom. In this talk, I will discuss geometric approaches to organizing this information in two problem domains: Undulatory locomotion of snakes and swimmers, and vibration propagation in spider webs. In the first section, I will discuss how differential geometry and Lie group theory provide insight into the locomotion of undulating systems through a vocabulary of lengths, areas, and curvatures. In particular, a tool called the Lie bracket combines these geometric concepts to describe the effects of cyclic changes in the locomotor's shape, such as the gaits used by swimming or crawling systems. Building on these results, I will demonstrate that the geometric techniques are useful beyond the "clean" ideal systems on which they have traditionally been developed, and can provide… Read more.

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Speaker: Elie Abdo

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Speaker: Sergio Zamora Barrera

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Speaker: Justin Bloom

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Speaker: Maurice Fabien

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Speaker: Radu Dascaliuc

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Descartes’s Revenge: or, support varieties and what they’re good for

Speaker: Jon Kujawa

Since Descartes, we’ve known that algebra and geometry are intertwined. Support varieties are a modern incarnation of this observation. Information about algebraic objects (e.g. representations of finite groups) can be revealed by introducing algebraic geometry via cohomology. This will be a very high level introduction focused on the big themes of the subject along with concrete examples. Read more.

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Speaker: Malgorzata Peszynska

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Speaker: Patrik Nabelek

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Speaker: Dev Sinha

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Oscillating asymptotics and conjectures of Andrews

Speaker: Amanda Folsom

In the 1980s, Andrews studied certain q-hypergeometric series from Ramanujan’s ``Lost" Notebook, and made several conjectures on their Fourier coefficients, which encode partition theoretic information. The corresponding conjectures on the coefficients of Ramanujan’s $\sigma(q)$ were resolved by Andrews, Dyson and Hickerson, who related them to the arithmetic of $\mathbb Q(\sqrt{6})$. Cohen also made connections to Maass waveforms. In this work, by new methods, we prove additional conjectures of Andrews, e.g., we blend novel techniques inspired by Garoufalidis' and Zagier's recent work on asymptotics of Nahm sums with classical techniques such as the Circle Method in Analytic Number Theory.This is joint work with Joshua Males, Larry, Rolen, and Matthias Storzer. Read more.

Convex duality and glacier momentum balance

Speaker: Daniel Shapero

ABSTRACT: The momentum balance equation for glacier flow can be expressed as a convex optimization problem for the ice velocity. The geometry of real glaciers, however, evolves in time as the terminus advances and retreats. The orthodox approaches to solving this free boundary problem are to either (1) set a minimum thickness and accept the resulting mass balance errors, or (2) apply the level-set method or related approaches that "turn off" the physics where the ice thickness is zero. Both of these approaches have their drawbacks. Here, I'll show an alternative formulation of the momentum balance equation that I derived using the principle of convex duality. This dual form adds the stresses as unknowns, but it remains solvable even when the ice thickness is equal to 0. Solvability at zero thickness allows us to simulate terminus advance and retreat -- including iceberg calving -- in a simple way. I'll show some numerical demonstrations and discuss what solution methods were viable. I… Read more.

Methods in studying Hecke traces

Speaker: Liubomir Chiriac

Hecke operators play a central role in the theory of modular forms, and appear in a number of important conjectures. In this talk I will discuss some methods used to study the traces of these operators acting on spaces of cusp forms. We will explore techniques from combinatorics, p-adic analysis and diophantine approximation, highlighting their applicability in broader contexts. Read more.

Intrusion of Tsunami-Like Waves into a Channel with Opposing Current

Speaker: Harry Yeh

ABSTRACT:The primary objective is to identify the physical mechanisms associated with the intrusion process by performing the controlled laboratory experiments. We use solitary waves and an undular bore for the study. The opposing current amplifies and slows the waves locally on the current, but the propagation speed is faster than the local Doppler effect due to the influence of the wave propagating in the flank of the current. At the channel mouth, the wave amplitude is enhanced due to the waveform altered by the current together with diffraction effects of the reflected waves from the adjacent shore. The leading wave of the undular bore is impacted by the opposing flow and transition similarly to the solitary waves. In contrast, the subsequent waves of the undular bore have a complex phase interference on the current that causes disconnection in the lateral wave formation across the breadth of the current. At the transition, the subsequent waves exhibit greater amplification than… Read more.

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Speaker: Brandon Young

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Speaker: Halley Fritze

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Machine learning invariants of elliptic curves

Speaker: Angelica Babei

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Data-Driven Kernel Matrix Computations: Geometric Analysis and Scalable Algorithms

Speaker: Cai Difeng

ABSTRACT: Dense kernel matrices arise in a broad range of disciplines, such as potential theory, molecular biology, statistical machine learning, etc. To reduce the computational cost, low-rank or hierarchical low-rank techniques are often used to construct an economical approximation to the original matrix. In this talk, we consider general kernel matrices associated with possibly high dimensional data. We perform analysis to provide a straightforward geometric interpretation that answers a central question: what kind of subset is preferable for skeleton low-rank approximations. Based on the theoretical findings, we present scalable and robust algorithms for black-box dense kernel matrix computations. The efficiency and robustness will be demonstrated through experiments for various datasets, kernels, and dimensions, including benchmark comparison to the state-of-the-art packages for N-body simulations.BIO: Difeng Cai received his BS in math from University of Science and Technology… Read more.

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Speaker: Nicholas Zitzelberger

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Modularity and Resurgence

Speaker: Eleanor McSpirit

The study of asymptotics as q approaches roots of unity is central to the theories of mock and quantum modular forms. In a collection of works, Gukov, Pei, Putrov, and Vafa proposed a candidate for a q-series invariant of closed 3-manifolds coming from physics. Many of these invariants are known to be mock and quantum modular forms, and this modularity has been integral to their study. Resurgent analysis is a natural tool to study this invariant from the perspective of physics, and is a theory centrally concerned with the relationship of functions to their asymptotic series. This has led to several questions on the interrelationship of resurgence and modularity. While this has been discussed across the subject, many questions remain. This talk will discuss ongoing work to make this connection explicit and natural from the perspective of number theory. Read more.

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Speaker: John Harvey

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Modular functions and the monstrous exponents

Speaker: Holly Swisher

In 1973 Ogg initiated the study of monstrous moonshine with the observation that the prime divisors of the monster group are exactly those for which the Fricke quotient X_0(p)+p of the modular curve X_0(p) has genus zero. Here, motivated by Deligne's theorem on the p-adic rigidity of the elliptic modular j-invariant, we present a modular function-based approach to explaining some of the exponents that appear in the prime decomposition of the order of the monster.This is joint work with John Duncan. Read more.

Sink or Soar: the interplay between buoyant bubbles and sinking sediments inenergizing turbulence near the ice-ocean boundary

Speaker: Megan Wengrove

ABSTRACT: At the terminus of tidewater glaciers an interplay of connected processes result in the melt of ice. From both field and laboratory observations, it has been suggested that both bubbles and sediments could be important yet neglected contributors to ice melt at the submarine tidewater glacier terminus. In the laboratory it has been shown that as glacier ice melts, air trapped in pores inside of the ice is released creating flow transpiration at the boundary and buoyant bubble rise at the ice-ocean interface, leading to increased melt (Wengrove et al.,2023). Additionally, during separate laboratory experiments, sediments entrained in the subglacial discharge plume are shown to increase the entrainment of warm ocean water toward the ice leading to higher melt rates (McConnochie andCenedese, 2023). In July 2024, we made the first ever video observations of both bubbles rising and sediments mixing and falling from a stationary-bolted platform to an Alaskan tidewater glacier… Read more.

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Speaker: Derek Garton

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Speaker: Elizabeth Carlson

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