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11:00 AM - 12:00 PM
380 Soda |
Matrix Computations & Scientific Computing Seminar, Detecting dependencies among equality constraints in large scale linear and geometric programs, Mark Hoemmen, CS, UC Berkeley
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01:10 PM - 02:00 PM
891 Evans |
Combinatorics Seminar, A new proof of Stanley's reciprocity theorem, Mike Develin, American Institute of Mathematics
We begin by presenting a miraculous identity involving one-variable Laurent series. This incantation is actually the simplest case of a remarkable theorem due to Richard Stanley regarding generating functions for polyhedral cones: if f(x1,...,xn) is the generating function for the points inside a rational polyhedral cone in Rn pointed at zero, then f(1/x1,...,1/xn) is the generating function for the interior points in this cone. We present a new, elegant proof of Stanley’s theorem using elementary techniques in contour integration. No prior knowledge will be assumed. This is joint work with Matthias Beck (SFSU) and Sinai Robins (Temple).
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02:00 PM - 04:00 PM
939 Evans |
Operator Algebras Seminar, Noncommutative random walks II, Dennis Courtney, UC Berkeley
We have seen that the Poisson boundary associated to a "noncommutative random walk" on a countable discrete group Gamma is a crossed product of the Poisson boundary of the corresponding classical walk by a natural Gamma-action. It is known that this relation persists when Gamma is assumed merely second countable and locally compact; we discuss the issues involved in establishing this.
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03:10 PM - 04:00 PM
334 Evans |
Probability Seminar, Markov Chains in metric spaces and their applications, Assaf Naor, Microsoft Research
In the past 15 years the behavior of Markov chains in metric spaces has found many surprising geometric applications. The notion of Markov type, introduced by K. Ball, is an important bi-Lipschitz invariant which measures the rate of wandering of time reversible Markov chains in a metric space. This notion is useful for extensions of Lipschitz and Holder functions, and in the theory of bi-Lipschitz embeddings. In this talk we will present some of the main results and problems in this directions, with emphasis on the use of Markov chains for proving non-embeddability results.
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03:10 PM - 05:00 PM
959 Evans |
Recursion Theory Seminar, K-triviality, Leo Harrington, UC Berkeley
There are several equivalent definition of when a set of integers A is Kolmogoroff-trivial:
1. A is K-low: there is a c such that for all y, KA(y) < K(y) + c.
2. A is low for ML-random: for all R, if R is 1-random then R is 1-random relative to A.
3. A is recursive in an R which is 1-random relative to A.
4. (the official definition of K-trivial): there is a d such that for all n, K(A|n) < n + d.
The collection of all K-trivial sets has the properties:
– it is countable and consists only of sets recursive in 0’;
– it consists just of low sets;
– it is closed under join and closed downward under recursive-in;
– every K-trivial set is recursive in a r.e. K-trivial set;
– there is a non-recusive K-trivial set.
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04:00 PM - 06:00 PM
939 Evans |
Combinatorics, Geometry & Representation Theory Seminar, The Hodge theorem on infinite dimensional manifolds, Ramanujan sums, and Kostka polynomials, Ian Grojnowski, Cambridge/UCB
I’ll explain that the Hodge theorem is false for even the simplest infinte dimensional manifolds—there is a "combinatorial obstruction" to it holding.
The correct statement implies new combinatorial identities generalising Ramanjuan 1psi1 sums. These turn out to be equivalent to the Macdonald constant term conjectures (giving a new proof of a theorem of Cherednik).
Finally, I’ll explain how all this generalises to define a type of Kostka polynomial, and that these are positive.
This is joint work with S. Fishel and C. Teleman.
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04:10 PM - 05:00 PM
891 Evans |
Applied Mathematics Seminar, Modeling DNA Unknotting by Type II Topoisomerases, Mariel Vazquez, UC Berkeley
Type II topoisomerases (Topo II) are essential enzymes common to all organisms. Their cellular functions include maintaining the levels of chromosome compaction and ensuring proper segregation at cell division. Topo II performs strand passage on its substrate DNA in a reaction that has been well characterized at the molecular level. When acting on knotted DNA molecules TopoII is known to unknot DNA very efficiently. Different biophysical models have been proposed to explain this phenomenon. We here propose a novel method to model the enzymatic action that uses knot theory and Monte Carlo simulations.
Here we address the question of whether the knot crossings acted on by topoII are selected at random or not. Our study is based on Monte-Carlo computer simulations of DNA unknotting. We model the enzymatic action as a finite state Markov chain in which each state is a knot type (with crossing number less than 8) and whose transition probabilities are estimated by Monte-Carlo computer simulations of random strand-passage on knotted molecules of fixed length. DNA molecules are modeled as a polygonal chain in the simple cubic lattice and their state space is sampled using the BFACF algorithm. Strand passage simulations are performed symbolically at the Dowker-code level (an integer-entry vector whereby each entry is assigned to a crossing on a fixed knot projection). To each knot is assigned a set of Dowker codes with weights given by the BFACF simulation.
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04:10 PM - 05:00 PM
330 Evans |
Special Seminar, Poisson inversion formula on SLn, Serge Lang
I shall follow up my colloquium talk with two lectures giving a more technical and detailed description of the analogue of the Poisson Inversion Formula on SL2(C), then conjecturally on SLn(C), and eventually other semisimple Lie groups. The talks will be self contained and should be accessible to graduate students and undergraduate majors.
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