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Events happening on February 28, 2017

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Feb. 28, 2017 11am (MATH 350)	Number Theory Pedro Berrizbeitia (ULAM Visitor) X Let $x$ and $n$ be integers, $n >1$ . If $xn-1$ is prime, then it is a Mersenne Prime! The 6 largest known primes are Mersenne primes, and so are 10 of the 11 largest primes. We start the talk reviewing the properties of Mersenne Numbers which contribute to explain this fact. The desire of finding other sequences of natural number with similar properties led eventually to the notion of ``Algebraic Mersenne Primes," which I define as prime numbers that are the absolute norm of $αn -1$ for some algebraic integer $α$ and some $n >2$ . Simple necessary conditions on $α$ and $n$ are given without which the sequence $Mα,n = \| NQ(α)/ Q(αn -1)\|$ contains no prime. The first interesting examples of sequence we studied are the Gaussian-Mersenne Primes, $M1-i,p= 2p ± 2p+1/2+1$ and the Eisenstein-Primes, $M1-ω,p=3p ± 3p+1/2 +1$ . The use of the Biquadratic Reciprocity Law and the Cubic Reciprocity Law are the key ingredients for the primality test for this sequence obtained in a joint paper with Boris Iskra. We then will show the results obtained by Miguel Rodriguez in his master thesis in 2004. He studied the sequence $M2-(ζ5 + ζ5-1), p =5p - Fp 5p+1/2 + 1$ , where $Fp = p$ -th Fibonnaci number, for which the Proth Primality test can be used. In 2006 V\'{\i}ctor Ram\'{\i}rez gave a deterministic primality test based on variants of AKS, for an infinite set of sequences, which we named Cyclotomic Mersenne Integers, $M1+ζm,p$ , $m ≥ 5$ . For the test, he made use of a reciprocity Law due to Hasse and extended my variant of AKS for its purpose. We also present recent results obtained by Euro Lucena in his mater's thesis in 2015. In his thesis he studies the properties of $M3+3,p$ and uses the Reciprocity Law of order 6 to give the primality test. He also searches for primes of the form $αn + 1$ and obtains interesting new results, one of which is a primality test based on the cubic test! The talk will have a historic flavor and we will include related results from Lehmer and collaborators and recent results of Palimar and Shankra on ``Mersenne primes in Real Quadratic Fields". Algebraic Mersenne Numbers
Feb. 28, 2017 12:10pm (Math …	Kempner Timothy Trujillo (Colorado School of Mines) X The two-dimensional Ramsey theorem states that if the edges of a complete graph with countably infinitely many vertices are colored with two colors then there is a complete subgraph with countably infinitely many vertices all whose edges are monochromatic. The theorem can be extended to all finite dimensions. However, the naive infinite-dimensional Ramsey conjecture is false in ZFC. Nonetheless the collection of all sets that satisfy the conclusion of the infinite-dimensional Ramsey theorem form a sigma-algebra of sets. The Ellentuck theorem characterizes this sigma-algebra as the sigma-algebra of sets with the Baire property ( that is, a symmetric difference of an open set and a meager set ) with respect to a topology that extends the natural metric topology on the collection of infinite subsets of the natural numbers. We will discuss the historical development of the Ellentuck theorem and use it to motivate and introduce topological Ramsey theory and the abstract Ellentuck theorem. Time permitting, we will discuss some new examples of topological Ramsey spaces and their applications. Topological Ramsey theory
Feb. 28, 2017 1pm (MATH 220)	Jeffrey Shriner (CU Boulder) NP-hardness Results for the Subpower Membership Problem, Part 2
Feb. 28, 2017 2pm (Math 220)	Jakub Bulin (CU) X Continuation. Blockers, loop lemmata and tree formulas, 2
Feb. 28, 2017 3pm (MATH 350)	Topology Cliff Blakestad (CU Boulder) Quillen's Theorem: MU and Formal Group Laws
Feb. 28, 2017 3pm (Math 220)	Functional Analysis Leonard Huang (CU Boulder) X The uniqueness of the Heisenberg Commutation Relations is usually stated in the language of unitary operators on a Hilbert space. This is done by first using the Baker-Campbell-Hausdorff Formula to integrate the Heisenberg Commutation Relations to the Weyl Commutation Relations and then applying the Stone-von Neumann Theorem. The question thus remains whether or not the uniqueness result can be stated purely in the original language of self-adjoint unbounded operators on a Hilbert space. Using the concept of analytic vectors created by Edward Nelson in the 1950s and a result about such vectors due to Moshé Flato and his collaborators, we attempt to answer the question in the affirmative. Uniqueness of the Heisenberg Commutation Relations from the Viewpoint of Unbounded Operator Theory

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