Miha E. Habič successfully defended his dissertation under my supervision at the CUNY Graduate Center on April 7th, 2017, earning his Ph.D. degree in May 2017.

It was truly a pleasure to work with Miha, who is an outstanding young mathematician with enormous promise. I shall look forward to seeing his continuing work.

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Miha E. Habič, “Joint Laver diamonds and grounded forcing axioms,”  Ph.D. dissertation for The Graduate Center of the City University of New York, May, 2017 (arxiv:1705.04422).

Abstract. In chapter 1 a notion of independence for diamonds and Laver diamonds is investigated. A sequence of Laver diamonds for $\kappa$ is joint if for any sequence of targets there is a single elementary embedding $j$ with critical point $\kappa$ such that each Laver diamond guesses its respective target via $j$. In the case of measurable cardinals (with similar results holding for (partially) supercompact cardinals) I show that a single Laver diamond for $\kappa$ yields a joint sequence of length $\kappa$, and I give strict separation results for all larger lengths of joint sequences. Even though the principles get strictly stronger in terms of direct implication, I show that they are all equiconsistent. This is contrasted with the case of $\theta$-strong cardinals where, for certain $\theta$, the existence of even the shortest joint Laver sequences carries nontrivial consistency strength. I also formulate a notion of jointness for ordinary $\diamondsuit_\kappa$-sequences on any regular cardinal $\kappa$. The main result concerning these shows that there is no separation according to length and a single $\diamondsuit_\kappa$-sequence yields joint families of all possible lengths.

 

In chapter 2 the notion of a grounded forcing axiom is introduced and explored in the case of Martin’s axiom. This grounded Martin’s axiom, a weakening of the usual axiom, states that the universe is a ccc forcing extension of some inner model and the restriction of Martin’s axiom to the posets coming from that ground model holds. I place the new axiom in the hierarchy of fragments of Martin’s axiom and examine its effects on the cardinal characteristics of the continuum. I also show that the grounded version is quite a bit more robust under mild forcing than Martin’s axiom itself.

Miha will shortly begin a post-doctoral research position at Charles University in Prague.