Seminars and Colloquia by Series

A Lorentzian manifold-with-boundary where causality breaks down due to shock singularities

Series
Geometry Topology Seminar
Time
Monday, October 7, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Leo Abbrescia – Georgia Tech

We present a novel example of a Lorentzian manifold-with-boundary featuring a dramatic degeneracy in its deterministic and causal properties known as “causal bubbles” along its boundary. These issues arise because the regularity of the Lorentzian metric is below Lipschitz and fit within the larger framework of low regularity Lorentzian geometry. Although manifolds with causal bubbles were recently introduced in 2012 as a mathematical curiosity, our example comes from studying the fundamental equations of fluid mechanics and shock singularities which arise therein. No prior knowledge of Lorentzian geometry or fluid mechanics will be assumed for this talk.

Data-driven model discovery meets mechanistic modeling for biological systems

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 7, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Niall M Mangan – Northwestern University –
Abstract: Building models for biological, chemical, and physical systems has traditionally relied on domain-specific intuition about which interactions and features most strongly influence a system.  Alternatively, machine-learning methods are adept at finding novel patterns in large data sets and building predictive models but can be challenging to interpret in terms of or integrate with existing knowledge. Our group balances traditional modeling with data-driven methods and optimization to get the best of both worlds.  Recently developed for and applied to dynamical systems, sparse optimization strategies can select a subset of terms from a library that best describes data, automatically interfering potential model structures from a broad but well-defined class. I will discuss my group's application and development of data-driven methods for model selection to 1) recover chaotic systems models from data with hidden variables,  2) discover models for metabolic and temperature regulation in hibernating mammals, and 3) model selection for differential-algebraic-equations. I'll briefly discuss current preliminary work and roadblocks in developing new methods for model selection of biological metabolic and regulatory networks.
 
Short Bio: Niall M. Mangan received the Dual BS degrees in mathematics and physics, with a minor in chemistry, from Clarkson University, Potsdam, NY, USA, in 2008, and the PhD degree in systems biology from Harvard University, Cambridge, MA, USA, in 2013. Dr. Mangan worked as a postdoctoral associate in the Photovoltaics Lab at MIT from 2013-2015 and as an Acting Assistant Professor at the University of Washington, Seattle from 2016-2017. She is currently an Assistant Professor of engineering sciences and applied mathematics with Northwestern University, where she works at the interface of mechanistic modeling, machine learning, and statistical inference. Her group applies these methods to many applications including metabolic and regulatory networks to accelerate engineering.

Lie Groups and Applications to Multi-Orientation Image Analysis

Series
Algebra Seminar
Time
Monday, October 7, 2024 - 11:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nicky J. van den Berg – Eindhoven University of Technology

Please Note: There will be a pre-talk at 10:55 am in Skiles 005.

Retinal images are often used to examine the vascular system in a non-invasive way. Studying the behavior of the vasculature on the retina allows for noninvasive diagnosis of several diseases as these vessels and their behavior are representative of the behavior of vessels throughout the human body. For early diagnosis and analysis of diseases, it is important to compare and analyze the complex vasculature in retinal images automatically.

During this talk, we will talk about a geodesic tracking approach that is better able to handle difficult structures, like high curvature and crossings. Additionally, we discuss how one can identify connected components in images that allow for small interruptions within the same component. Both methods takes place in the lifted space of positions and orientations SE(2), which allows us to differentiate between crossings and bifurcations.

CANCELLED - Tight minimum colored degree condition for rainbow connectivity

Series
Graph Theory Seminar
Time
Friday, October 4, 2024 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Xiaofan Yuan – Arizona State University

Let $G = (V,E)$ be a graph on $n$ vertices, and let $c : E \to P$, where $P$ is a set of colors. Let $\delta^c(G) = \min_{v \in V} \{ d^{c}(v) \}$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$ of $G$.  In 2011, Fujita and Magnant showed that if $G$ is a graph on $n$ vertices that satisfies $\delta^c(G)\geq n/2$, then for every two vertices $u, v$ there is a properly-colored $u,v$-path in $G$.
In this paper, we show that the same bound for $\delta^c(G)$ implies that any two vertices are connected by a rainbow path. This is joint work with Andrzej Czygrinow.


This is to note that the graph theory seminar for Friday the 4th has been CANCELLED. This is due to the cancellation of the AMS sectional meeting due to Hurricane Helene. I apologize for any inconvenience. We intend to reschedule the talk for next semester.

Shock formation in weakly viscous conservation laws

Series
PDE Seminar
Time
Friday, October 4, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Cole Graham – University of Wisconsin–Madison –

The compressible Euler equations readily form shocks, but in 1D the inclusion of viscosity prevents such singularities. In this talk, we will quantitatively examine the interaction between generic shock formation and viscous effects as the viscosity tends to zero. We thereby obtain sharp rates for the vanishing-viscosity limit in Hölder norms, and uncover universal viscous structure near shock formation. The results hold for large classes of viscous hyperbolic conservation laws, including compressible Navier–Stokes with physical rather than artificial viscosity. This is joint work with John Anderson and Sanchit Chaturvedi.

Minimum Norm Interpolation Meets The Local Theory of Banach Spaces

Series
Stochastics Seminar
Time
Thursday, October 3, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gil Kur – ETH –
Minimum-norm interpolators have recently gained attention primarily as an analyzable model to shed light on the double descent phenomenon observed for neural networks.  The majority of the work has focused on analyzing interpolators in Hilbert spaces, where typically an effectively low-rank structure of the feature covariance prevents a large bias. More recently, tight vanishing bounds have also been shown for isotropic high-dimensional data  for $\ell_p$-spaces with $p\in[1,2)$, leveraging the sparse structure of the ground truth.
This work takes a first step towards establishing a general framework that connects generalization properties of the interpolators to well-known concepts from high-dimensional geometry, specifically, from the local theory of Banach spaces.

Harmonic Functions and Beyond

Series
School of Mathematics Colloquium
Time
Thursday, October 3, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yanyan Li – Rutgers University –

A harmonic function of two variables is the real or imaginary part of an analytic function. A harmonic function of $n$ variables is a function $u$ satisfying

$$

\frac{\partial^2 u}{\partial x_1^2}+\ldots+\frac{\partial^2u}{\partial x_n^2}=0.

$$

We will first recall some basic results on harmonic functions: the mean value property, the maximum principle, the Liouville theorem, the Harnack inequality, the Bocher theorem, the capacity and removable singularities. We will then present a number of more recent results on some conformally invariant elliptic and degenerate elliptic equations arising from conformal geometry. These include results on Liouville theorems, Harnack inequalities, and Bocher theorems.

Introduction to Bergman geometry

Series
Geometry Topology Seminar
Time
Wednesday, October 2, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jihun Yum – Gyeongsang National University in South Korea

The Poincaré metric on the unit disc $\mathbb{D} \subset \mathbb{C}$, known for its invariance under all biholomorphisms (bijective holomorphic maps) of $\mathbb{D}$, is one of the most fundamental Riemannian metrics in differential geometry.

In this presentation, we will first introduce the Bergman metric on a bounded domain in $\mathbb{C}^n$, which can be viewed as a generalization of the Poincaré metric. We will then explore some key theorems that illustrate how the curvature of the Bergman metric characterizes bounded domains in $\mathbb{C}^n$ and more generally, complex manifolds. Finally, I will discuss my recent work related to these concepts. 

Smooth Discrepancy and Littlewood's Conjecture

Series
Analysis Seminar
Time
Wednesday, October 2, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Niclas Technau – University of Bonn –

Given x in $[0,1]^d$, this talk is about the fine-scale distribution of the Kronecker sequence $(n x mod 1)_{n\geq 1}$.
After a general introduction, I will report on forthcoming work with Sam Chow.
Using Fourier analysis, we establish a novel deterministic analogue of Beck’s local-to-global principle (Ann. of Math. 1994),
which relates the discrepancy of a Kronecker sequence to multiplicative diophantine approximation.
This opens up a new avenue of attack for Littlewood’s conjecture.

Proof of the Goldberg-Seymour conjecture (Guantao Chen)

Series
Graph Theory Seminar
Time
Tuesday, October 1, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Guantao Chen – Georgia State University

The Goldberg-Seymour Conjecture asserts that if the chromatic index $\chi'(G)$ of a loopless multigraph $G$ exceeds its maximum degree $\Delta(G) +1$, then it must be equal to another well known lower bound $\Gamma(G)$, defined as

$\Gamma(G) = \max\left\{\biggl\lceil  \frac{ 2|E(H)|}{(|V (H)|-1)}\biggr\rceil \ : \  H \subseteq G \mbox{ and } |V(H)| \mbox{ odd }\right\}.$

 

In this talk, we will outline a short proof,  obtained recently with  Hao, Yu, and Zang. 

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