Seminars and Colloquia by Series

The Combinatorial Nullstellensatz and its applications

Series
Graph Theory Working Seminar
Time
Thursday, September 5, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Youngho Yoo – Georgia Tech

Please Note:

In 1999, Alon proved the “Combinatorial Nullstellensatz” which resembles Hilbert’s Nullstellensatz and gives combinatorial structure on the roots of a multivariate polynomial. This method has numerous applications, most notably in additive number theory, but also in many other areas of combinatorics. We will prove the Combinatorial Nullstellensatz and give some of its applications in graph theory.

 

Outliers in spectrum of sparse Wigner matrices

Series
Stochastics Seminar
Time
Thursday, September 5, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Konstantin Tikhomirov – Georgia Tech –

We study the effect of sparsity on the appearance of outliers in the semi-circular law. As a corollary of our main results, we show that, for the Erdos-Renyi random graph with parameter p, the second largest eigenvalue is (asymptotically almost surely) detached from the bulk of the spectrum if and only if pn

On the QQR codes in coding theory

Series
High Dimensional Seminar
Time
Wednesday, September 4, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jing Hao – Georgia Tech –

In this talk I will briefly talk about coding theory and introduce a specific family of codes called Quasi-quadratic residue (QQR) codes. These codes have large minimum distances, which means they have good error-correcting capabilities. The weights of their codewords are directly related to the number of points on corresponding hyperelliptic curves. I will show a heuristic model to count the number of points on hyperelliptic curves using a coin-toss model, which in turn casts light on the relation between efficiency and the error-correcting capabilities of QQR codes. I will also show an interesting phenomenon we found about the weight enumerator of QQR codes. Lastly, using the bridge between QQR codes and hyperelliptic curves again, we derive the asymptotic behavior of point distribution of a family of hyperelliptic curves using results from coding theory.

0-Concordance of 2-Knots

Series
Geometry Topology Student Seminar
Time
Wednesday, September 4, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anubhav Mukherjee

 A 2-knot is a smooth embedding of S^2 in S^4, and a 0-concordance of 2-knots is a concordance with the property that every regular level set of the concordance is just a collection of S^2's. In his thesis, Paul Melvin proved that if two 2-knots are 0-concordant, then a Gluck twist along one will result in the same smooth 4-manifold as a Gluck twist on the other. He asked the following question: Are all 2-knots 0-slice (i.e. 0-concordant to the unknot)? I will explain all relevant definitions, and mostly follow the paper by Nathan Sunukjian on this topic.

Bases of exponentials and tilings

Series
Analysis Seminar
Time
Wednesday, September 4, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mihalis Kolountzakis – University of Crete –

Mathematicians have long been trying to understand which domains admit an orthogonal (or, sometimes, not) basis of exponentials of the form , for some set of frequencies (this is the spectrum of the domain). It is well known that we can do so for the cube, for instance (just take ), but can we find such a basis for the ball? The answer is no, if we demand orthogonality, but this problem is still open when, instead of orthogonality, we demand just a Riesz basis of exponentials.

 
This question has a lot to do with tiling by translation (i.e., with filling up space with no overlaps by translating around an object). Fuglede originally conjectured that an orthogonal exponential basis exists if and only if the domain can tile space by translation. This has been disproved in its full generality but when one adds side conditions, such as, for instance, a lattice set of frequencies, or the space being a group of a specific type, or many other natural conditions, the answer is often unknown, and sometimes known to be positive or known to be negative. A major recent  development is the proof (2019) by Lev and Matolcsi of the truth of the Fuglede conjecture for convex bodies in all dimensions.
 
This is a broad area of research, branching out by varying the side conditions on the domain or the group in which the domain lives, or by relaxing the orthogonality condition or even allowing time-frequency translates of a given function to serve as basis elements (Gabor, or Weyl-Heisenberg, bases). When working with both exponential bases and tiling problems the crucial object of study turns out to be the zero set of the Fourier Transform of the indicator function of the domain we care about. In particular we want to know how large structured sets this zero set contains, for instance how large difference sets it contains or what kind of tempered distributions it can support.
 
In this talk I will try to show how these objects are tied together, what has been done recently, and indicate specific open problems.

An Introduction to Quantum Topology

Series
Research Horizons Seminar
Time
Wednesday, September 4, 2019 - 12:20 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Wade Bloomquist – Georgia Tech

We will explore some of the basic notions in quantum topology.  Our focus will be on introducing some of the foundations of diagrammatic algebra through the lens of the Temperley-Lieb algebra.  We will attempt to show how these diagrammatic techniques can be applied to low dimensional topology.  Every effort will be made to make this as self-contained as possible.  If time permits we will also discuss some applications to topological quantum computing.

Construction of unstable quasi-periodic solutions for a system of coupled NLS equations.

Series
CDSNS Colloquium
Time
Wednesday, September 4, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Victor Vilaça Da Rocha – Georgia Tech –

The systems of coupled NLS equations occur in some physical problems, in particular in nonlinear optics (coupling between two optical waveguides, pulses or polarized components...).

From the mathematical point of view, the coupling effects can lead to truly nonlinear behaviors, such as the beating effect (solutions with Fourier modes exchanging energy) of Grébert, Paturel and Thomann (2013). In this talk, I will use the coupling between two NLS equations on the 1D torus to construct a family of linearly unstable tori, and therefore unstable quasi-periodic solutions.

The idea is to take profit of the Hamiltonian structure of the system via the construction of a Birkhoff normal form and the application of a KAM theorem. In particular, we will see of this surprising behavior (this is the first example of unstable tori for a 1D PDE) is strongly related to the existence of beating solutions.

This is a work in collaboration with Benoît Grébert (Université de Nantes).

Some combinatorics of RNA branching

Series
Mathematical Biology Seminar
Time
Wednesday, September 4, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christine Heitsch – Georgia Tech

Understanding the folding of RNA sequences into three-dimensional structures is one of the fundamental challenges in molecular biology.  For example, the branching of an RNA secondary structure is an important molecular characteristic yet difficult to predict correctly.  However, recent results in geometric combinatorics (both theoretical and computational) yield new insights into the distribution of optimal branching configurations, and suggest new directions for improving prediction accuracy.

Positively Hyperbolic Varieties

Series
Algebra Seminar
Time
Tuesday, September 3, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Josephine Yu – Georgia 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.

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