Aula Magna (Dip. Matematica)

Earnshaw's Theorem in Electrostatics

Tipo Seminario: 
Relatore: 
Jeffrey Rauch

Earnshaw's Theorem. (Trans. Cambridge Phil. Soc(1842)).

A test charge in an electrostatic field can never be in a position of stable equilibrium.
This theorem is still not proved but much progress has been made. Some seemingly more difficult results asserted by Maxwell have been proved.

Data Seminario: 
Friday, October 9, 2015
Aula: 
Aula Magna (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 15:00
Affiliazione: 
University of Michigan
Ora Inizio: 
Saturday, January 1, 0000 - 14:00

Twisted Whitney towers and concordance of links

Tipo Seminario: 
Relatore: 
Marco Castronovo

The Whitney trick can be used to remove intersection points in generic immersed concordances of links. Due to the fact that Whitney disks may intersect the concordance and each other, sometimes one only trades old with new intersections and these organize in "towers" that admit a combinatorial description in terms of trees modulo some relations. These relations show up in other areas of topology like those of quantum and finite type invariants. For example, there is a connection with Milnor invariants and the Kontsevich integral.

Data Seminario: 
Wednesday, July 15, 2015
Aula: 
Aula Magna (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 16:00
Affiliazione: 
Unversità di Pisa
Ora Inizio: 
Saturday, January 1, 0000 - 15:00

TQFT non semi-semplici

Tipo Seminario: 
Relatore: 
Francesco Costantino

Dopo aver rapidamente ricordato cos'è una TQFT e come costruirne alcune, ricorderemo alcuni degli aspetti salienti della teoria delle rappresentazioni del gruppo quantico U_q(sl_2) per poi spiegare come questi aspetti sono rispecchiati nella topologia. 

Discuteremo quindi la costruzione di invarianti "non semisemplici" di 3-varietà basati su questa teoria delle rappresentazioni e della costruzione delle TQFT associate a questi invarianti. 

Se il tempo lo permette discuteremo alcune proprietà di tali TQFT e le connetteremo ai recenti lavori di Bonahon e Wong.  

Data Seminario: 
Tuesday, June 16, 2015
Aula: 
Aula Magna (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 11:00
Affiliazione: 
Université Paul Sabatier de Toulouse
Ora Inizio: 
Saturday, January 1, 0000 - 10:00

Hypersurfaces with almost constant mean curvature

Relatore: 
Francesco Maggi

Motivated by the study of capillarity problems we investigate
smooth boundaries with almost constant mean curvature, and quantify their distance from compounds of tangent balls with equal radii in terms of the oscillation of their mean curvature. This is a joint work with Giulio Ciraolo (U. Palermo). 

Data Seminario: 
Friday, June 5, 2015
Aula: 
Aula Magna (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 15:00
Affiliazione: 
University of Texas at Austin, USA
Ora Inizio: 
Saturday, January 1, 0000 - 14:00

Comparing the real, complex and Zilber exponentials from a logical point of view

Tipo Seminario: 
Relatore: 
Angus Macintyre

Tarski's work from the 1930's, on the structure of first-order definitions in the real and complex fields, eventually became a standard tool in semi-algebraic geometry. Tarski (who knew that the complex exponential was "wild" in the sense of interpreting arithmetic) posed the issue of analyzing effectively the real exponential along the lines of his analysis for the real field. This problem took well over 50 years to solve, and general ideas isolated during its solution (the theory of 0-minimality) have proved applicable to deep matters in Lie theory and, more recently, in diophantine geometry.  The decisive result was proved by Wilkie in 1991 (depending on Hovanski's work) , and Tarski's original problem on effectivity was solved by Wilkie and Macintyre in 1992, ASSUMING  the truth of Schanuel's Conjecture in transcendental number theory .

Now, even Schanuel's Conjecture cannot make the complex exponential decidable, but very deep work of Zilber has led to the unconditional construction of exponential fields ("Zilber fields") which have an amazingly structured model theory, satisfy Schanuel's Conjecture, and share quite a few properties of the complex exponential. One striking feature is that the properties in question are usually proved for the complexes by difficult analysis, and for the Zilber fields by relatively simple algebra. Zilber has conjectures that the complex exponential field is a Zilber field. This cannot be true unless Schanuel's Conjecture is true, but if one assumes that then the conjecture yields many new insights about complex analysis of exponential functions.

     It turns out that Zilber's model theory is deeply connected to issues in diophantine geometry studied initially by Bombieri, Masser and Zannier. It turns out too that there are connections to old problems in complex analysis, for example Shapiro's Conjecture on common zeros of exponential functions. I will discuss how this is related to Schanuel's Conjecture.

    A big mystery, where the best we know is due to Vincenzo Mantova, is how to detect in Zilber fields something like the Euclidean topology. I will discuss this.

Data Seminario: 
Thursday, February 26, 2015
Aula: 
Aula Magna (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 15:30
Affiliazione: 
Queen Mary University of London (U.K.)
Ora Inizio: 
Saturday, January 1, 0000 - 14:30

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