User:Mfrasca

2011-11-30T09:43:04Z

Mfrasca: Profile added

== Marco Frasca ==

I am a theoretical physicist working on perturbation theory and solutions of PDEs, quantum field theory and specially quantum chromodynamics, Yang-Mills equations both in classical and quantum cases.

I manage a [http://marcofrasca.wordpress.com/ blog] about my fields of interest in research.

Here is a [http://arxiv.org/find/grp_physics/1/au:+frasca_marco/0/1/0/all/0/1 list] of papers of mine on arXiv.

== Some selected publications ==

* [http://prc.aps.org/abstract/PRC/v84/i5/e055208 Chiral symmetry in the low-energy limit of QCD at finite temperature], [http://arxiv.org/abs/1105.5274 arXiv], May 2011.

* [http://pos.sissa.it/archive/conferences/117/039/FacesQCD_039.pdf Mapping theorem and Green functions in Yang-Mills theory], [http://arxiv.org/abs/1011.3643 arXiv], November 2010.

* [http://www.worldscinet.com/jnmp/18/1802/S1402925111001441.html Exact solutions of classical scalar field equations], [http://arxiv.org/abs/0907.4053 arXiv], July 2009.

* [http://jmp.aip.org/resource/1/jmapaq/v50/i10/p102904_s1?isAuthorized=no Dual Lindstedt series and Kolmogorov–Arnol’d–Moser theorem], [http://arxiv.org/abs/0905.4886 arXiv], May 2009.

* [http://www.worldscinet.com/mpla/24/2430/S021773230903165X.html Mapping a Massless Scalar Field Theory on a Yang-Mills Theory: Classical Case], [http://arxiv.org/abs/0903.2357 arXiv], March 2009.

* [http://www.worldscinet.com/ijmpe/18/1803/S0218301309012781.html Infrared QCD], [http://arxiv.org/abs/0803.0319 arXiv], March 2008.

* [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVN-4TPHRKP-3&_user=10&_coverDate=12%2F04%2F2008&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=d27b76dd23274d0794c2cd821adfc89e&searchtype=a Infrared gluon and ghost propagators], [http://arxiv.org/abs/0709.2042 arXiv], September 2007.

* [http://www.worldscinet.com/ijmpa/23/2302/S0217751X08038160.html Green functions and nonlinear systems: Short time expansion], [http://arxiv.org/abs/0704.1568 arXiv], April 2007.

* [http://www.worldscinet.com/ijmpa/22/2229/S0217751X07037986.html Spectrum in the broken phase of a $\lambda\phi^4$ theory], [http://arxiv.org/abs/hep-th/0703203 arXiv], March 2007.

* [http://www.worldscinet.com/mpla/22/2218/S0217732307023705.html Green Function Method for Nonlinear Systems], [http://arxiv.org/abs/hep-th/0702056 arXiv], February 2007.

* [http://rspa.royalsocietypublishing.org/content/463/2085/2195 A strongly perturbed quantum system is a semiclassical system], [http://arxiv.org/abs/hep-th/0603182 arXiv], March 2006.

* [http://prd.aps.org/abstract/PRD/v73/i2/e027701 Strongly coupled quantum field theory], [http://arxiv.org/abs/hep-th/0511068 arXiv], November 2005.

* [http://www.worldscinet.com/ijmpd/15/1509/S0218271806009091.html Strong coupling expansion for general relativity], [http://arxiv.org/abs/hep-th/0508246 arXiv], August 2005.

* [http://www.worldscinet.com/mplb/20/2017/S0217984906011578.html A Quantum Many-Body Instability in the Thermodynamic Limit], [http://arxiv.org/abs/quant-ph/0403111 arXiv], March 2004.

* [http://link.aps.org/doi/10.1103/PhysRevA.58.3439 Duality in perturbation theory and the quantum adiabatic approximation],[http://arxiv.org/abs/hep-th/9801069 arXiv], January 1998.

Other proposed projects

2011-11-30T09:30:31Z

Mfrasca: Proposal on Yang-Mills problem added

This page is a repository for any polymath proposals which are not fleshed out enough to have their own separate posts for the proposal. Contributions are welcome.

== Beating the trivial subset sum algorithm ==

This problem was proposed by Ernie Croot, though he would not have the time to run a project based on this problem.

It is a problem that is easy to state, probably will succumb to elementary methods, and probably could be solved if enough people contributed ideas. Here goes: consider the usual subset sum (or is it knapsack?) problem where you are given a list of positive integers N_1, …, N_k, and a target number T, and you must decide whether there is some subset of N_1, …, N_k that sums to T. The problem is to beat the “trivial algorithm”, which I shall describe presently.

The first thing to realize is that there is a subset sum equal to T iff there is one equal to S-T, where S=N_1 + … + N_k. Furthermore, subsets of size t summing to T correspond uniquely to subsets of size k-t summing to S-T. In this way, you only need to consider subsets of size at most k/2 (and check whether they sum of T or S-T) to solve the problem. But now you can use the usual “collision technique” to reduce the problem to subsets of size at most k/4, by forming a table of all subsets of at most this size, along with their sum of elements, until you find a disjoint pair of subsets that sums to either T or S-T. The running time of this procedure should be comparable to (k choose k/4) = c^(k+o(k)), for a certain constant c that is easy to work out. This is what I mean by the “trivial algorithm”. Now, the problem is find an algorithm — any at all — that runs in time at most d^k, where d < c. To my knowledge no such algorithm is know to exist!

== Filters on posets and generalizations ==

I suggest to collaboratively finish writing my draft "Filters on posets and generalizations".

See [http://filters.wikidot.com/ this wiki].

Note that this manuscript contains [http://portonmath.wordpress.com/2009/07/31/complementive-complete-lattice/ this conjecture] which can be separated into an other smaller polymath project.

== Coupling determinantal processes ==

Any n-dimensional subspace V of a Euclidean space <math>{\Bbb R}^N</math> gives rise to a random subset A_V of {1,...,N}, with the probability that <math>A_V = \{i_1,\ldots,i_k\}</math> being the square of the magnitude of the projection of <math>e_{i_1} \wedge \ldots \wedge e_{i_n}</math> to V. This is known as the determinantal process associated to V.

If V is a subspace of W, it is known that one can couple <math>A_V</math> to <math>A_W</math> in such a way that the former set is a subset of the latter, but no "natural" way of doing this is known. One problem in this project would be to find such a natural way.

A related problem: if V, W are orthogonal, is it always possible to couple <math>A_V, A_W, A_{V+W}</math> together in such a way that <math>A_{V + W} = A_V \cup A_W</math>?

These questions are raised in page 38 of [http://uk.arxiv.org/PS_cache/math/pdf/0204/0204325v4.pdf this paper of Lyons], and also discussed at [http://terrytao.wordpress.com/2009/08/23/determinantal-processes/ this blog post].

== Stable commutator length in the free group ==

Let G be the free group on two generators, and let [G,G] be the commutator subgroup. Given any g in [G,G], the commutator length cl(g) is the least number of commutators needed to express g, and the stable commutator length scl(g) is the lim of cl(g^n)/n.

It is known that scl(g) >= 1/2 for any non-trivial g. Find a combinatorial proof of this fact.

It is conjectured that { scl(g): g in [G,G] } has an isolated point at 1/2. Prove this.

Reference: scl, Danny Calegari

== Quantum cellular automata ==

The proposal is tackling [http://mathoverflow.net/questions/78707/are-all-quantum-cellular-automata-invertible-representable these 2 questions].

== Yang-Mills existence and mass gap ==

Quite recently, [http://pro.osu.edu/profiles/dynin.1/ Alexander Dynin], a mathematician at Ohio State University with a reputable CV, proposed a solution of the Millennium prize problem on Yang-Mills existence and mass gap (see [http://arxiv.org/abs/1110.4682 here] and [http://arxiv.org/abs/0903.4727 here]). I have discussed this in my blog (see [http://marcofrasca.wordpress.com/2011/11/07/a-millenium-problem-issue/ here]) but I am a physicist and I cannot be sure about the correctness of all the mathematical arguments by the author. Of course, I am involved in this line of research as a physicist and, in our area, significant progress seems to have been made (e.g., besides my works, see Alexander (Sasha) Migdal's papers [http://arxiv.org/abs/1109.1623 here] and refs. therein and [http://marcofrasca.wordpress.com/2011/11/28/yang-mills-mass-gap-scenario-further-confirmations/ my blog entry]). So, it would be of paramount importance to have such a question addressed by the community of mathematicians at large, much in the same way as happened last year with "N vs. NP" question for [http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper Deolalikar's paper], in order to fix the avenues to pursue for the community of theoretical physicists.--[[User:Mfrasca|Marco Frasca]] 09:30, 30 November 2011 (UTC)

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User:Mfrasca

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