Seminars and Colloquia by Series

Nonnegativity and Real-Rootedness

Series
Algebra Student Seminar
Time
Friday, March 11, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 or ONLINE
Speaker
Kevin Shu – Georgia Tech

There are many interesting classes of polynomials in real algebraic geometry that are of modern interest. A polynomial is nonnegative if it only takes nonnegative values on R^n. A univariate polynomial is real-rooted if all of its complex roots are real, and a hyperbolic polynomial is a multivariate generalization of a real-rooted polynomial. We will discuss connections between these two classes of polynomials. In particular, we will discuss recent ideas of Saunderson giving new ways of proving that a polynomial is nonnegative beyond showing that it is sum-of-squares.

Teams link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1646885419648?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%2206706002-23ff-4989-8721-b078835bae91%22%7d

Bootstrap Percolation with Drift

Series
Stochastics Seminar
Time
Thursday, March 10, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Daniel Blanquicett – University of California, Davis –

We will motivate this talk by exhibiting recent progress on (either general or symmetric anisotropic) bootstrap percolation models in $d$-dimensions. Then, we will discuss our intention to start a deeper study of non-symmetric models for $d\ge 3$. It looks like some proportion of them could be related to first passage percolation models (in lower dimensions).

This talk will be online at https://bluejeans.com/216376580/6460

Matrix Concentration and Synthetic Data

Series
Job Candidate Talk
Time
Thursday, March 10, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/405947238/3475
Speaker
March Boedihardjo – UC Irvine –

Classical matrix concentration inequalities are sharp up to a logarithmic factor. This logarithmic factor is necessary in the commutative case but unnecessary in many classical noncommutative cases. We will present some matrix concentration results that are sharp in many cases, where we overcome this logarithmic factor by using an easily computable quantity that captures noncommutativity. Joint work with Afonso Bandeira and Ramon van Handel.

Due to privacy, access to real data is often restricted. Data that are not completely real but resemble certain properties of real data become natural substitutes. Data of this type are called synthetic data. I will talk about the extent to which synthetic data may resemble real data under privacy and computational complexity restrictions. Joint work with Thomas Strohmer and Roman Vershynin.

The link to the online talk:  https://bluejeans.com/405947238/3475

L^2-boundedness of gradients of single layer potentials for elliptic operators with coefficients of Dini mean oscillation-type

Series
Analysis Seminar
Time
Wednesday, March 9, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE (Zoom link in abstract)
Speaker
Carmelo Puliatti – University of the Basque Country, Spain –

We consider a uniformly elliptic operator $L_A$ in divergence form associated with an $(n+1)\times(n+1)$-matrix  $A$ with real, bounded, and possibly non-symmetric coefficients. If a proper {$L^1$-mean oscillation} of the coefficients of $A$ satisfies suitable Dini-type assumptions, we prove the following: if $\mu$ is a compactly supported Radon measure in $\mathbb{R}^{n+1}$, $n \geq 2$,   and

$$T_\mu f(x)=\int \nabla_x\Gamma_A (x,y)f(y)\, d\mu(y)$$

denotes the gradient of the single layer potential associated with $L_A$, then

$$1+ \|T_\mu\|_{L^2(\mu)\to L^2(\mu)}\approx 1+ \|\mathcal R_\mu\|_{L^2(\mu)\to L^2(\mu)},$$

where $\mathcal R_\mu$ indicates the $n$-dimensional Riesz transform. This makes possible to obtain direct generalization of some deep geometric results, initially obtained for $\mathcal R_\mu$, which were recently extended to  $T_\mu$ under a H\"older continuity assumption on the coefficients of the matrix $A$.

This is a joint work with Alejandro Molero, Mihalis Mourgoglou, and Xavier Tolsa.

Multiscale Modeling of Prion Aggregate Dynamics in Yeast

Series
Mathematical Biology Seminar
Time
Wednesday, March 9, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Mikahl Banwarth-Kuhn – University of California, Merced

Please Note: Meeting Link: https://bluejeans.com/426529046/8775

Prion proteins are responsible for a variety of fatal neurodegenerative diseases in mammals but are harmless to Baker's yeast (S. cerevisiae)- making it an ideal system for investigating the protein dynamics associated with prion diseases. Most mathematical frameworks for modeling prion aggregate dynamics either focus on protein dynamics in isolation, absent from a changing cellular environment, or modeling prion aggregate dynamics in a population of cells by considering the "average" behavior. However, such models are unable to reproduce in vivo properties of different yeast prion strains.

In this talk, I will show some results from recent individual-based simulations where we study how the organization of a yeast population depends on the division and growth properties of the colonies. Each individual cell has their own configuration of prion aggregates, and we study how the population level phenotypes are a natural consequence of the interplay between the cell cycle, budding cell division and aggregate dynamics. We quantify how common experimentally observed outcomes depend on population heterogeneity.

Recording link: https://bluejeans.com/s/lbpACr_YZ0N

Evasiveness conjecture and topological methods in graph theory III

Series
Graph Theory Working Seminar
Time
Tuesday, March 8, 2022 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ruilin Shi – Georgia Institute of Technology –

In the third talk of this seminar series, we continue to follow the manuscript of Carl Miller. We will begin with a quick review of chain complexes and simplicial isomorphisms and then we will detour to discuss the geometric interpretation of homology groups in lower dimensions. This work can help us understand the structure of simplicial complexes with boundary maps and their homology groups. Then we go back to abstract homological algebra which is the study of homology groups without reference to simplicial complexes. We will introduce the Snake Lemma without proof. Finally, we will apply this lemma to prove the goal of this chapter: collapsibility for a simplicial complex implies its homology groups are trivial which is called acyclicity.

Degree bounds for sums of squares of rational functions

Series
Algebra Seminar
Time
Tuesday, March 8, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Grgoriy Blekhermany – Georgia Tech –

Hilbert’s 17th problem asked whether every nonnegative polynomial is a sum of squares of rational functions. This problem was solved affirmatively by Artin in the 1920’s, but very little is known about degree bounds (on the degrees of numerators and denominators) in such a representation. Artin’s original proof does not yield any upper bounds, and making such techniques quantitative results in bounds that are likely to be far from optimal, and very far away from currently known lower bounds. Before stating the 17th problem Hilbert was able to prove that any globally nonnegative polynomial in two variables is a sum of squares of rational functions, and the degree bounds in his proof have been best known for that two variable case since 1893. Taking inspiration from Hilbert’s proof we study degree bounds for nonnegative polynomials on surfaces. We are able to improve Hilbert’s bounds and also give degree bounds for some non-rational surfaces. I will present the history of the problem and outline our approach. Joint work with Rainer Sinn, Greg Smith and Mauricio Velasco.

Trees in graphs and hypergraphs

Series
Job Candidate Talk
Time
Tuesday, March 8, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Maya Stein – University of Chile

Graphs are central objects of study in Discrete Mathematics. A graph consists of a set of vertices, some of which are connected by edges. Their elementary structure makes graphs widely applicable, but the theoretical understanding of graphs is far from complete. Extremal graph theory aims to find connections between global parameters and substructure. A key topic is how a large average or minimum degree of a graph can force certain subgraphs (where the degree is the number of edges at a vertex). For instance, Erdős and Gallai proved in the 1960's that any graph of average degree at least $k$ contains a path of length $k$. Some of the most intriguing open questions in this area concern trees (connected graphs without cycles) as subgraphs. For instance, can one substitute the path from the previous paragraph with a tree? We will give an overview of open problems and recent results in this area, as well as their possible extensions to hypergraphs.

An invariant for families of contact structures in monopole Floer homology

Series
Geometry Topology Seminar
Time
Monday, March 7, 2022 - 14:00 for
Location
Skiles 006
Speaker
Juan Muñoz-Echániz – Columbia University

The contact invariant, introduced by Kronheimer and Mrowka,
is an element in the monopole Floer homology of a 3-manifold which is
canonically attached to a contact structure. I will describe an
application of monopole Floer homology and the contact invariant to
study the topology of spaces of contact structures and
contactomorphisms on 3-manifolds. The main new tool is a version of
the contact invariant for families of contact structures.
 

Symmetry-preserving machine learning for computer vision, scientific computing, and distribution learning

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 7, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
https://gatech.zoom.us/j/96551543941 (note: Zoom, not Bluejeans)
Speaker
Prof. Wei Zhu – UMass Amherst

Please Note: Note the talk will be hosted by Zoom, not Bluejeans any more.

Symmetry is ubiquitous in machine learning and scientific computing. Robust incorporation of symmetry prior into the learning process has shown to achieve significant model improvement for various learning tasks, especially in the small data regime.

In the first part of the talk, I will explain a principled framework of deformation-robust symmetry-preserving machine learning. The key idea is the spectral regularization of the (group) convolutional filters, which ensures that symmetry is robustly preserved in the model even if the symmetry transformation is “contaminated” by nuisance data deformation.
 
In the second part of the talk, I will demonstrate how to incorporate additional structural information (such as group symmetry) into generative adversarial networks (GANs) for data-efficient distribution learning. This is accomplished by developing new variational representations for divergences between probability measures with embedded structures. We study, both theoretically and empirically, the effect of structural priors in the two GAN players. The resulting structure-preserving GAN is able to achieve significantly improved sample fidelity and diversity—almost an order of magnitude measured in Fréchet Inception Distance—especially in the limited data regime. 
 

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