This Week's Seminars and Colloquia

Cayley-Bacharach Relations and Sums of Squares

Series
Student Algebraic Geometry Seminar
Time
Monday, November 11, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Kevin ShuGeorgia Tech (grad student)

This talk is based on a paper by Grigoriy Blekherman. In most cases, nonnegative polynomials differ from positive polynomials. We will discuss precisely what equations cause these differences, and relate them to the well known Cayley-Bacharach theorem for low degree polynomials.

Boundary control of optimal mixing via fluid flows

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 11, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Weiwei HuUniversity of Georgia

We discuss the problem of optimal mixing of an inhomogeneous distribution of a scalar field via an active control of the flow velocity, governed by the Stokes or the Navier-Stokes equations, in a two dimensional open bounded and connected domain.  We consider the velocity field steered by a control input that acts tangentially on the boundary of the domain through the  Navier slip boundary conditions. This is motivated by mixing  within a cavity or vessel  by moving the walls or stirring at the boundaries. Our main objective is to design an optimal Navier slip boundary control  that optimizes mixing at a given final time. Non-dissipative scalars, both passive and active, governed by the transport equation will be discussed.  In the absence of diffusion, transport and mixing occur due to pure advection.  This essentially leads to a nonlinear control problem of a semi-dissipative system. We shall provide a rigorous proof of the existence of an optimal controller, derive the first-order necessary conditions for optimality, and present some preliminary results on the numerical implementation.

An inverse problems approach to some questions arising in harmonic analysis

Series
School of Mathematics Colloquium
Time
Tuesday, November 12, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Betsy StovallUniversity of Wisconsin

 One strategy for developing a proof of a claimed theorem is to start by understanding what a counter-example should look like.  In this talk, we will discuss a few recent results in harmonic analysis that utilize a quantitative version of this approach.  A key step is the solution of an inverse problem with the following flavor.  Let $T:X \to Y$ be a bounded linear operator and let $0 < a \leq \|T\|$.  What can we say about those functions $f \in X$ obeying the reverse inequality $\|Tf\|_Y \geq a\|f\|_X$?  

Positively Hyperbolic Varieties

Series
Algebra Seminar
Time
Tuesday, November 12, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Josephine YuGeorgia Tech

A multivariate complex polynomial is called stable if any line in any positive direction meets its hypersurface only at real points.  Stable polynomials have close relations to matroids and hyperbolic programming.  We will discuss a generalization of stability to algebraic varieties of codimension larger than one.  They are varieties which are hyperbolic with respect to the nonnegative Grassmannian, following the notion of hyperbolicity studied by Shamovich, Vinnikov, Kummer, and Vinzant. We show that their tropicalization and Chow polytopes have nice combinatorial structures related to braid arrangements and positroids, generalizing some results of Choe, Oxley, Sokal, Wagner, and Brändén on Newton polytopes and tropicalizations of stable polynomials. This is based on joint work with Felipe Rincón and Cynthia Vinzant.

Network reconstruction using computational algebra and gene knockouts

Series
Mathematical Biology Seminar
Time
Wednesday, November 13, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Matthew MacauleyClemson University

I will discuss an ongoing project to reconstruct a gene network from time-series data from a mammalian signaling pathway. The data is generated from gene knockouts and the techniques involve computational algebra. Specifically, one creates an pseudomonomial "ideal of non-disposable sets" and applies a analogue of Stanley-Reisner theory and Alexander duality to it. Of course, things never work as well in practice, due to issue such as noise, discretization, and scalability, and so I will discuss some of these challenges and current progress.

Variational models, PDEs, numerical analysis and applications

Series
Research Horizons Seminar
Time
Wednesday, November 13, 2019 - 12:20 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Sung Ha KangGeorgia Tech

Starting from mathematical approaches for image processing, we will discuss different models, analytic aspects of them, and numerical challenges.  If time permits we will consider numerical applications to data understanding. A few other applications may be presented.

Small deviation estimates for norms of Gaussian vectors

Series
Analysis Seminar
Time
Wednesday, November 13, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Konstantin TikhomirovGeorgia Tech
Let |.| be a norm in R^n, and let G be the standard Gaussian vector.
We are interested in estimating from above the probabilities
P{|G|<(1-t)E|G|} in terms of t. For 1-unconditional norms
in the L-position, we prove small deviation estimates which match those for the
ell-infinity norm: in a sense, among all 1-unconditional norms in the L-position,
the left tail of |G| is the heaviest for ell-infinity. Results for general norms are also obtained.
The proof is based on an application of the hypercontractivity property combined with
certain transformations of the original norm.
Joint work with G.Paouris and P.Valettas.

Hard-core models on triangular and square lattices

Series
High Dimensional Seminar
Time
Wednesday, November 13, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Izabella StuhlPenn State

One of the outstanding open problems of statistical mechanics is about the hard-core model which is a popular topic in mathematical physics and has applications in a number of other disciplines. Namely, do non-overlapping hard disks of the same diameter in the plane admit a unique Gibbs measure at high density? It seems natural to approach this question by requiring the centers to lie in a fine lattice; equivalently, we may fix the lattice, but let the Euclidean diameter D of the hard disks tend to infinity. In two dimensions, it can be a unit triangular lattice A_2 or a unit square lattice Z^2. The randomness is generated by Gibbs/DLR measures with a large value of fugacity which corresponds to a high density. We analyze the structure of high-density hard-core Gibbs measures via the Pirogov-Sinai theory. The first step is to identify periodic ground states, i.e., maximal-density disk configurations which cannot be locally `improved'. A key finding is that only certain `dominant' ground states, which we determine, generate nearby Gibbs measures. Another important ingredient is the Peierls bound separating ground states from other admissible configurations. In particular, number-theoretic properties of the exclusion diameter D turn out to be important. Answers are provided in terms of Eisenstein primes for A_2 and norm equations in the cyclotomic ring Z[ζ] for Z^2, where ζ is the primitive 12th root of unity. Unlike most models in statistical physics, we find non-universality: the number of high-density hard-core Gibbs measures grows indefinitely with D but
non-monotonically. In Z^2 we also analyze the phenomenon of 'sliding' and show it is rare.
This is a joint work with A. Mazel and Y. Suhov.

Zero-free regions and central limit theorems

Series
Stochastics Seminar
Time
Thursday, November 14, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Marcus MichelenUniversity of Illinois, Chicago

Let X be a random variable taking values in {0,...,n} and f(z) be its probability generating function.  Pemantle conjectured that if the variance of X is large and f has no roots close to 1 in the complex plane, then X must be approximately normal. We will discuss a complete resolution of this conjecture in a strong quantitative form, thereby giving the best possible version of a result of Lebowitz, Pittel, Ruelle and Speer. Additionally, if f has no roots with small argument, then X must be approximately normal, again in a sharp quantitative form. These results also imply a multivariate central limit theorem that answers a conjecture and completes a program of Ghosh, Liggett and Pemantle.  This talk is based on joint work with Julian Sahasrabudhe.

Faster Width-dependent Algorithm for Mixed Packing and Covering LPs

Series
ACO Student Seminar
Time
Friday, November 15, 2019 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Digvijay BoobISyE, Georgia Tech

In this talk, we provide the details of our faster width-dependent algorithm for mixed packing-covering LPs. Mixed packing-covering LPs are fundamental to combinatorial optimization in computer science and operations research. Our algorithm finds a $1+\eps$ approximate solution in time $O(Nw/ \varepsilon)$, where $N$ is number of nonzero entries in the constraint matrix, and $w$ is the maximum number of nonzeros in any constraint. This algorithm is faster than Nesterov's smoothing algorithm which requires $O(N\sqrt{n}w/ \eps)$ time, where $n$ is the dimension of the problem. Our work utilizes the framework of area convexity introduced in [Sherman-FOCS’17] to obtain the best dependence on $\varepsilon$ while breaking the infamous $\ell_{\infty}$ barrier to eliminate the factor of $\sqrt{n}$. The current best width-independent algorithm for this problem runs in time $O(N/\eps^2)$ [Young-arXiv-14] and hence has worse running time dependence on $\varepsilon$. Many real life instances of mixed packing-covering problems exhibit small width and for such cases, our algorithm can report higher precision results when compared to width-independent algorithms. As a special case of our result, we report a $1+\varepsilon$ approximation algorithm for the densest subgraph problem which runs in time $O(md/ \varepsilon)$, where $m$ is the number of edges in the graph and $d$ is the maximum graph degree.

On a class of sums with unexpectedly high cancellation, and its applications

Series
Combinatorics Seminar
Time
Friday, November 15, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hamed MousaviGeorgia Tech

We report on the discovery of a general principle leading to the unexpected cancellation of oscillating sums. It turns out that sums in the
class we consider are much smaller than would be predicted by certain probabilistic heuristics. After stating the motivation, and our theorem,
we apply it to prove a number of results on integer partitions, the distribution of prime numbers, and the Prouhet-Tarry-Escott Problem. For example, we prove a "Pentagonal Number Theorem for the Primes", which counts the number of primes (with von Mangoldt weight) in a set of intervals very precisely. In fact the result is  stronger than one would get using a strong form of the Prime Number Theorem and also the Riemann Hypothesis (where one naively estimates the \Psi function on each of the intervals; however, a less naive argument can give an improvement), since the widths of the intervals are smaller than \sqrt{x}, making the Riemann Hypothesis estimate "trivial".

Based on joint work with Ernie Croot.