<div dir="ltr"><div dir="ltr" style="font-size:12.8px">Dear All,<div>We will have the second of our IMBM Model Theory Meetings <span class="" tabindex="0"><span class="">on </span></span><span style="font-size:12.8px">18 April </span><span class="" tabindex="0" style="font-size:12.8px"><span class="">Monday, starting at 13:30.</span></span><span style="font-size:12.8px"> I attach below the program.</span></div><div><span style="font-size:12.8px">We thank IMBM for their hospitality,</span><br></div><div><span style="font-size:12.8px"><br></span></div><div><span style="font-size:12.8px">Özlem Beyarslan</span></div><div>----------------------------------------------------------------------------<br></div><div><div style="font-size:12.8px">IMBM Model Theory Meetings, April 18</div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px">David Pierce (Mimar Sinan) <span class="" tabindex="0"><span class="">13:30</span></span> -- <span class="" tabindex="0"><span class="">15:00</span></span></div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px">Title: Spaces and fields</div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px">Abstract: Using areas, Euclid proved results that today we consider</div><div style="font-size:12.8px">as algebraic. We consider them so, because Descartes justified</div><div style="font-size:12.8px">algebra by showing how it could be considered as geometry.</div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px">Such observations can be understood as resulting from the equivalence</div><div style="font-size:12.8px">of certain categories.</div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px">Models of a given first-order theory T are the objects of two</div><div style="font-size:12.8px">different categories:</div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px">* Mod(T), in which the morphisms are embeddings, and</div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px">* Mod*(T), in which the morphisms are elementary embeddings.</div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px">The latter category is closed under direct limits; if the former is</div><div style="font-size:12.8px">likewise closed, then T has universal-existential axioms (and</div><div style="font-size:12.8px">conversely). T is called model-complete if the two categories are the</div><div style="font-size:12.8px">same. T is called companionable if it includes a model-complete</div><div style="font-size:12.8px">theory, called the model companion of T, in a model of which each</div><div style="font-size:12.8px">model of T embeds.</div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px">A vector space here is a pair (K,V), where K is a field, V is an</div><div style="font-size:12.8px">abelian group, and K acts on V. The theory of vector spaces in this</div><div style="font-size:12.8px">sense has a model companion, which is theory of one-dimensional vector</div><div style="font-size:12.8px">spaces.</div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px">If T is the theory of vector spaces of dimension at least two, and U</div><div style="font-size:12.8px">is the theory of abelian groups with an appropriate notion of</div><div style="font-size:12.8px">parallelism, then Mod(T) and Mod(U) are equivalent. If S is field</div><div style="font-size:12.8px">theory, and T_n is the theory of n-dimensional vector spaces (where</div><div style="font-size:12.8px">n>0) with a given basis, then Mod(S) and Mod(T_n) are equivalent.</div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px">In a vector space (K,V), V may also act on K as a Lie ring of</div><div style="font-size:12.8px">derivations; then (K,V) becomes a Lie--Rinehart pair. Such pairs can</div><div style="font-size:12.8px">be given universal-existential axioms, using only the signature of</div><div style="font-size:12.8px">abelian groups for each of K and V, along with a symbol for the action</div><div style="font-size:12.8px">of each on the other. In his 2010 dissertation, Özcan Kasal showed</div><div style="font-size:12.8px">that the resulting theory is not companionable, although if predicates</div><div style="font-size:12.8px">for certain definable relations are introduced, the theory becomes</div><div style="font-size:12.8px">companionable, and the model companion is not stable. It turns out</div><div style="font-size:12.8px">that like the theory of the integers as a group, the model companion</div><div style="font-size:12.8px">even has the so-called tree property.</div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px">Gönenç Onay (Mimar Sinan) <span class="" tabindex="0"><span class="">15:30</span></span> -- <span class="" tabindex="0"><span class="">17:00</span></span></div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px">Title: On the elementary theory of F_p((t))</div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px">Abstract: </div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px">The famous Ax-Kochen and Ershov theorem gives in the complete </div><div style="font-size:12.8px">theory of the fields of p-adics numbers Q_p for every prime p and hence proves that this theory is decidable. While seems to be so similar, the elementary theory of F_p((t)) is widely unknown and it is one of the challenging questions of contemporary model theory and arithmetics. In this talk I will sketch historical approaches to this problem and give my humble contribution. </div></div></div><div class="" style="margin:2px 0px 0px;font-size:12.8px"></div><div><br></div>-- <br><div class="gmail_signature">Özlem Beyarslan<br><br>Bogazici Universitesi Matematik Bolumu <br>Bebek, Istanbul 34342<br>Tel: 90 212 359 6535</div>
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