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Series: Algebra Seminar

Matroids are basic combinatorial objects arising from graphs and vector configurations. Given a vector configuration, I will introduce a “matroid Schubert variety” which shares various similarities with classical Schubert varieties. I will discuss how the Hodge theory of such matroid Schubert varieties can be used to prove a purely combinatorial conjecture, the “top-heavy” conjecture of Dowling-Wilson. I will also report an on-going project joint with Tom Braden, June Huh, Jacob Matherne, Nick Proudfoot on the cohomology theory of non-realizable matroids.

Monday, February 4, 2019 - 13:55 ,
Location: Skiles 005 ,
Ashwin Renganathan ,
GT AE ,
Organizer: Sung Ha Kang

In the design of complex engineering systems like aircraft/rotorcraft/spacecraft, computer experiments offer a cheaper alternative to physical experiments due to high-fidelity(HF) models. However, such models are still not cheap enough for application to Global Optimization(GO) and Uncertainty Quantification(UQ) to find the best possible design alternative. In such cases, surrogate models of HF models become necessary. The construction of surrogate models requires an offline database of the system response generated by running the expensive model several times. In general, the training sample size and distribution for a given problem is unknown apriori and can be over/under predicted, which leads to wastage of resources and poor decision-making. An adaptive model building approach eliminates this problem by sequentially sampling points based on information gained in the previous step. However, an approach that works for highly non-stationary response is still lacking in the literature. Here, we use Gaussian Process(GP) models as surrogate model. We employ a novel process-convolution approach to generate parameterized non-stationary
GPs that offer control on the process smoothness. We show that our approach outperforms existing methods, particularly for responses that have localized non-smoothness. This leads to better performance in terms of GO, UQ and mean-squared-prediction-errors for a given budget of HF function calls.

Monday, February 4, 2019 - 13:55 ,
Location: Skiles 005 ,
Ashwin Renganathan ,
GT AE ,
Organizer: Sung Ha Kang
In the design of complex engineering systems like aircraft/rotorcraft/spacecraft, computer experiments offer a cheaper alternative to physical experiments due to high-fidelity(HF) models. However, such models are still not cheap enough for application to Global Optimization(GO) and Uncertainty Quantification(UQ) to find the best possible design alternative. In such cases, surrogate models of HF models become necessary. The construction of surrogate models requires an offline database of the system response generated by running the expensive model several times. In general, the training sample size and distribution for a given problem is unknown apriori and can be over/under predicted, which leads to wastage of resources and poor decision-making. An adaptive model building approach eliminates this problem by sequentially sampling points based on information gained in the previous step. However, an approach that works for highly non-stationary response is still lacking in the literature. Here, we use Gaussian Process(GP) models as surrogate model. We employ a novel process-convolution approach to generate parameterized non-stationary
GPs that offer control on the process smoothness. We show that our approach outperforms existing methods, particularly for responses that have localized non-smoothness. This leads to better performance in terms of GO, UQ and mean-squared-prediction-errors for a given budget of HF function calls.

Series: Research Horizons Seminar

In this talk, we will discuss various ways to describe three-manifolds by decomposing them into pieces that are (maybe) easier to understand. We will use these descriptions as a way to measure the complexity of a three-manifold.

Series: Analysis Seminar

We prove sparse bounds for the spherical maximal operator of Magyar,Stein and Wainger. The bounds are conjecturally sharp, and contain an endpoint esti-mate. The new method of proof is inspired by ones by Bourgain and Ionescu, is veryefficient, and has not been used in the proof of sparse bounds before. The Hardy-Littlewood Circle method is used to decompose the multiplier into major and minor arccomponents. The efficiency arises as one only needs a single estimate on each elementof the decomposition.

Wednesday, February 6, 2019 - 14:00 ,
Location: Skiles 006 ,
Surena Hozoori ,
Georgia Institute of Technology ,
shozoori3@gatech.edu ,
Organizer: Surena Hozoori

In this series of (3-5) lectures, I will talk about different aspects of a class of contact 3-manifolds for which geometry, dynamics and topology interact subtly and beautifully. The talks are intended to include short surveys on "compatibility", "Anosovity" and "Conley-Zehnder indices". The goal is to use the theory of Contact Dynamics to show that conformally Anosov contact 3-manifolds (in particular, contact 3-manifolds with negative α-sectional curvature) are universally tight, irreducible and do not admit a Liouville cobordism to the tight 3-sphere.

Series: High Dimensional Seminar

We study delocalization properties of null vectors and eigenvectors of matrices with i.i.d. subgaussian entries. Such properties describe quantitatively how "flat" is a vector and confirm one of the universality conjectures stating that distributions of eigenvectors of many classes of random matrices are close to the uniform distribution on the unit sphere. In particular, we get lower bounds on the smallest coordinates of eigenvectors, which are optimal as the case of Gaussian matrices shows.
The talk is based on the joint work with Konstantin Tikhomirov.

Series: Graph Theory Working Seminar

Strong edge coloring of a graph $G$ is a coloring of the edges of the
graph such that each color class is an induced subgraph. The strong
chromatic index of $G$ is the smallest number $k$ such that $G$ has a
$k$-strong edge coloring. Erd\H{o}s
and Ne\v{s}et\v{r}il conjecture that the strong chromatic index of a
graph of max degree $\Delta$ is at most $5\Delta^2/4$ if $\Delta$ is
even and $(5\Delta^2-2\Delta + 1)/4$ if $\Delta$ is odd. It is known for
$\Delta=3$ that the conjecture holds, and in
this talk I will present part of Anderson's proof that the strong
chromatic index of a subcubic planar graph is at most $10$

Series: Job Candidate Talk

A major challenge in clinical and biomedical research is on translating in-vitro and in- vivo model findings to humans. Translation success rate of all new compounds going through different clinical trial phases is generally about 10%. (i) This field is challenged by a lack of robust methods that can be used to translate model findings to humans (or interpret preclinical finds to accurately design successful patient regimens), hence providing a platform to evaluate a plethora of agents before they are channeled in clinical trials. Using set theory principles of mapping morphisms, we recently developed a novel translational framework that can faithfully map experimental results to clinical patient results. This talk will demonstrate how this method was used to predict outcomes of anti-TB drug clinical trials. (ii) Translation failure is deeply rooted in the dissimilarities between humans and experimental models used; wide pathogen isolates variation, patient population genetic diversities and geographic heterogeneities. In TB, bacteria phenotypic heterogeneity shapes differential antibiotic susceptibility patterns in patients. This talk will also demonstrate the application of dynamical systems in Systems Biology to model (a) gene regulatory networks and how gene programs influence Mycobacterium tuberculosis bacteria metabolic/phenotypic plasticity. (b) And then illustrate how different bacteria phenotypic subpopulations influence treatment outcomes and the translation of preclinical TB therapeutic regimens. In general, this talk will strongly showcase how mathematical modeling can be used to critically analyze experimental and patient data.

Series: School of Mathematics Colloquium

Interpolative decomposition is a simple and yet powerful tool for approximating low-rank matrices. After discussing the theory and algorithms, I will present a few new applications of interpolative decomposition in numerical partial differential equations, quantum chemistry, and machine learning.

Series: Stochastics Seminar

I will present joint work with Elena Kosygina and Ofer Zeitouni in which we prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.

Series: Intersection Theory Seminar

We continue the discussion of Chapter 8 in 3264 and All That. We will discuss complete quadrics, Hilbert schemes and Kontsevich spaces.

Friday, February 8, 2019 - 11:00 ,
Location: Skiles 005 ,
Prof. Lexing Ying ,
Stanford University ,
Organizer: Molei Tao

We will go to lunch together after the talk with the graduate students.

We introduce methods from convex optimization to solve the multi-marginal transport type problems arise in the context of density functional theory. Convex relaxations are used to provide outer approximation to the set of N-representable 2-marginals and 3-marginals, which in turn provide lower bounds to the energy. We further propose rounding schemes to obtain upper bound to the energy.

Series: ACO Student Seminar

The popularity of machine learning is increasingly growing in
transportation, with applications ranging from traffic engineering to
travel demand forecasting and pavement material modeling, to name just a
few. Researchers often find that machine learning achieves higher
predictive accuracy compared to traditional methods. However, many
machine-learning methods are often viewed as “black-box” models, lacking
interpretability for decision making. As a result, increased attention
is being devoted to the interpretability of machine-learning results. In
this talk, I introduce the application of machine learning to study
travel behavior, covering both mode prediction and behavioral
interpretation. I first discuss the key differences between machine
learning and logit models in modeling travel mode choice, focusing on
model development, evaluation, and interpretation. Next, I apply the
existing machine-learning interpretation tools and also propose two new
model-agnostic interpretation tools to examine behavioral heterogeneity.
Lastly, I show the potential of using machine learning as an
exploratory tool to tune the utility functions of logit models. I
illustrate these ideas by examining stated-preference travel survey
data for a new mobility-on-demand transit system that integrates
fixed-route buses and on-demand shuttles. The results show that the
best-performing machine-learning classifier results in higher predictive
accuracy than logit models as well as comparable behavioral outputs. In
addition, results obtained from model-agnostic interpretation tools
show that certain machine-learning models (e.g. boosting trees) can
readily account for individual heterogeneity and generate valuable
behavioral insights on different population segments. Moreover, I show
that interpretable machine learning can be applied to tune the utility
functions of logit models (e.g. specifying nonlinearities) and to
enhance their model performance. In turn, these findings can be used to
inform the design of new mobility services and transportation policies.

Series: Combinatorics Seminar

Abstract: Reiher, R\"odl, Ruci\'nski, Schacht, and Szemer\'edi proved,
via a modification of the absorbing method, that every 3-uniform
$n$-vertex hypergraph, $n$ large, with minimum vertex degree at least
$(5/9+\alpha)n^2/2$ contains a tight Hamiltonian cycle. Recently, owing
to a further modification of the method, the same group of authors
joined by Bjarne Schuelke, extended this result to 4-uniform hypergraphs
with minimum pair degree at least, again, $(5/9+\alpha)n^2/2$. In my
talk I will outline these proofs and point to the crucial ideas behind
both modifications of the absorbing method.