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Series: School of Mathematics Colloquium

TBA

Series: School of Mathematics Colloquium

Series: School of Mathematics Colloquium

The colloquium will be the second lecture of the TRIAD Distinguished Lecture Series by Prof. Sara van de Geer. For further information please see http://math.gatech.edu/events/triad-distinguished-lecture-series-sara-van-de-geer-0.

Series: School of Mathematics Colloquium

Given two complex polynomials, we can try to mathematically paste them
together to obtain a rational function through a procedure known as
mating the polynomials. In this talk, we will begin by trying to
understand the "shape" of complex polynomials in general. We will then
discuss the mating of two quadratic polynomials: we explore examples
where the mating does exist, and examples where it does not. There will
be lots of movies and exploration in this talk.

Series: School of Mathematics Colloquium

[CV: Prof. Oded Margalit has a PhD in computer science from Tel Aviv University under the supervision of Prof. Zvi Galil. He has worked at IBM Research – Haifa in the areas of machine learning, constraint satisfaction, verification, and more. Currently, he is the CTO of the IBM Cybersecurity Center of Excellence in Beer Sheva, Israel. Oded helps organize several computer science competitions, like the international IEEEXtreme and the Israeli national CodeGuru competition. He loves riddles and authors the IBM Research monthly challenge corner Ponder This.]

For the sake of puzzle-lovers worldwide, IBM Research offers a monthly mathematical challenge known as Ponder This. Every month, a new challenge is posted together with the solution for the previous month's riddle. Prof. Oded Margalit has served as the Ponder This puzzlemaster for the last decade. In this talk, he’ll survey some of most interesting riddles posted over the years, and tell some anecdotes about various challenges and regular solvers, such as one person who sent in his solution from an intensive care unit. Several challenges have led to conference and journal papers, such as a PRL paper born from a riddle on random walks, and an ITA 2014 paper on a water hose model (using quantum entanglement to break location-based encryption). Other monthly challenges have riffed on games such as 2048, Kakuro, an infinite chess game, the probability of backgammon ending with a double, Fischer Random Chess, and more. Other challenges have been more purely mathematic, focusing on minimal hash functions, combinatorial test design, or finding a natural number n such that round ((1+2 cos(20))^n) is divisible by 10^9.
The talk will present a still-open question about a permutation-firing cannon. The talk will be self contained.

Series: School of Mathematics Colloquium

Evolution of random systems as well as dynamical systems with chaotic (stochastic) behavior traditionally (and seemingly naturally) is described by studying only asymptotic in time (when time tends to infinity) their properties. The corresponding results are formulated in the form of various limit theorems (CLT, large deviations, etc). Likewise basically all the main notions (entropy, Lyapunov exponents, etc) involve either taking limit when time goes to infinity or averaging over an infinite time interval. Recently a series of results was obtained demonstrating that finite time predictions for such systems are possible. So far the results are on the intersection of dynamical systems, probability and combinatorics. However, this area suggests some new analytical, statistical and geometric problems to name a few, as well as opens up possibility to obtain new types of results in various applications. I will describe the results on (extremely) simple examples which will make this talk quite accessible.

Series: School of Mathematics Colloquium

Associated to a planar cubic graph, there is a closed surface in R^5, as defined by Treumann and Zaslow. R^5 has a canonical geometry, called a contact structure, which is compatible with the surface. The data of how this surface interacts with the geometry recovers interesting data about the graph, notably its chromatic polynomial. This also connects with pseudo-holomorphic curve counts which have boundary on the surface, and by looking at the resulting differential graded algebra coming from symplectic field theory, we obtain new definitions of n-colorings which are strongly non-linear as compared to other known definitions. There are also relationships with SL_2 gauge theory, mathematical physics, symplectic flexibility, and holomorphic contact geometry. During the talk we'll explain the basic ideas behind the various fields above, and why these various concepts connect.

Series: School of Mathematics Colloquium

Traditional Erdos Magic (a.k.a. The Probabilistic Method) proves the existence of an object with certain properties
by showing that a random (appropriately defined) object will have those properties with positive probability. Modern Erdos Magic analyzes a random process, a random (CS take note!) algorithm. These, when successful, can find a "needle in an exponential haystack" in polynomial time.
We'll look at two particular examples, both involving a family of n-element sets under suitable side conditions. The Lovasz Local Lemma finds a coloring with no set monochromatic. A result of this speaker finds a coloring with low discrepency. In both cases the original proofs were not implementable but Modern Erdos Magic finds the colorings in polynomial times.
The methods are varied. Basic probability and combinatorics. Brownian Motion. Semigroups. Martingales. Recursions ... and Tetris!

Series: School of Mathematics Colloquium

Series: School of Mathematics Colloquium

The study of nonconventional sums $S_{N}=\sum_{n=1}^{N}F(X(n),X(2n),\dots,X(\ell n))$, where $X(n)=g \circ T^n$ for a measure preserving transformation $T$, has a 40 years history after Furstenberg showed that they are related to the ergodic theory proof of Szemeredi's theorem about arithmetic progressions in the sets of integers of positive density. Recently, it turned out that various limit theorems of probabilty theory can be successfully studied for sums $S_{N}$ when $X(n), n=1,2,\dots$ are weakly dependent random variables. I will talk about a more general situation of nonconventional arrays of the form $S_{N}=\sum_{n=1}^{N}F(X(p_{1}n+q_{1}N),X(p_{2}n+q_{2}N),\dots,X(p_{\ell}n+q_{\ell}N))$ and how this is related to an extended version of Szemeredi's theorem. I'll discuss also ergodic and limit theorems for such and more general nonconventional arrays.