2023
Andreas Lietz, University of Münster and Scott Mutchnik, University of California, Berkeley
Andreas Lietz received his Ph.D. in 2023 under the guidance of Ralf Schindler at the University of Munster. His dissertation is titled “Forcing ‘NSω_1 is ω1-dense‘ from large cardinals – a journey guided by the stars”. Lietz solved a long-standing problem as to whether it was possible to force over a ZFC model the assertion that the non-stationary ideal over omega_1 could be omega_1-dense. This was a prominent assertion in set theory and had been used by Woodin and others in the so-called core-model induction. It is equiconsistent with the assertion that there exist infinitely many Woodin cardinals, and hence also with the Axiom of Determinacy. Woodin had obtained this assertion by forcing over models of determinacy. Nevertheless, it had been an open question for some 30 years as to whether the assertion could be forced, in particular, by using semi-proper forcing, over a model of ZFC. Lietz answered this question affirmatively by demonstrating that starting from a model with a cardinal kappa that is an inaccessible limit of <-kappa-supercompacts, one could obtain this property in the generic extension by a stationary set-preserving forcing. Moreover using the novel techniques he invented, Lietz went on to also isolate a whole variety of axioms that are variants of MM, Martin’s Maximum.
Scott Mutchnik received his Ph.D. in 2023 under the guidance of Tom Scanlon at the University of California, Berkeley. His dissertation is titled “On the Classification of Unstable First-Order Theories”. Mutchnik introduced fundamentally new ways to understand unstable theories, revealing structure in a surprising range of theories. In the process, he answered a central and longstanding open problem by showing that the classes of NSOP$_1$ theories and of NSOP$_2$ theories coincide. His arguments combine techniques from several different parts of model theory in technically powerful and original ways. His work has also opened up promising areas for further study.
2022
Francesco Gallinaro, Leeds University and Patrick Lutz, University of California, Berkeley
Francesco Gallinaro received his Ph.D. in 2022 from Leeds University under the joint supervision of Vincenzo Mantova and Dugald Macpherson. His thesis “Around exponential-algebraic closedness” provides further evidence towards the quasi-minimality property of the field of complexes enriched with the exponential map, conjectured by Boris Zilber. For a family of varieties (defined by equations that are dimensionally likely to have solutions), Francesco Gallinaro shows that the Exponential Algebraic Closedness-property holds in the field of complexes, then considers the analogous problem for abelian varieties with their associated exponential maps and finally in the upper half plane endowed with other analytic functions such as the elliptic modular function. His novel approaches also demonstrate a mastery of quite different techniques and are strongly expected to enable further progress.
Patrick Lutz received his Ph.D. in 2021 from the University of California, Berkeley under the supervision of Theodore A. Slaman. His dissertation “Results on Martin’s Conjecture” contains some of the most substantial progress in decades on Martin’s Conjecture, including a proof of the first part of Martin’s Conjecture for order-preserving functions, and of its analog for regressive functions on the hyperarithmetic degrees. Its methods also open up new possibilities in the study of Martin’s Conjecture. The proofs involve several novel ideas, and a powerful combination of methods from set theory and computability theory, with applications beyond Martin’s Conjecture, including the most significant advance in decades on a question of Sacks about embeddability of continuum-sized partial orders into the Turing degrees.
2021
Marcos Mazari Armida, Carnegie Mellon University
Mazari Armida received his Ph.D. in 2021 from Carnegie Mellon University under the supervision of Rami Grossberg. His thesis “Remarks on classification theory for abstract elementary classes with applications to abelian group theory and ring theory” provides strong evidence that abstract elementary classes can impact traditional mathematics in interesting ways. Armida shows various natural classes of abelian groups to be AEC and proves a family of theorems characterizing well-known classes of rings (e.g. left Noetherian, left perfect) in terms of the superstability of an associated AEC of modules. This leads to the solution below ℵω of a 1970 question of Fuchs asking in which cardinals there is a universal abelian p-group for purity. His versatility is indicated by important work on neo-stablility and categoricity in the context of AEC.
2020
James Walsh, University of California Berkeley
Walsh received his Ph.D. in 2020 from the University of California at Berkeley under the supervision of Paolo Mancosu and Antonio Montalbán. His thesis, “Reflection Principles and Ordinal Analysis”, presents significant new results on the philosophical and conceptual foundations of proof theory. It identifies natural classes of theories that rule out pathological behaviors that can otherwise arise in ordinal analysis, and it makes important progress towards explaining why proof-theoretic results tend to be robust with respect to the kinds of theories that occur naturally in mathematics.
2019
Gabriel Goldberg, Harvard University
Goldberg received his Ph.D. in 2019 at Harvard University under the supervision of W. Hugh Woodin. In his thesis, “The Ultrapower Axiom”, he isolates and develops a powerful new structural hypothesis for inner models of set theory. The Ultrapower Axiom holds in all inner models built using the current methodology. On the other hand, Goldberg shows it has sweeping structural consequences: the linearity of the Mitchell order on countably complete ultrafilters, a characterization of strong compactness in terms of supercompactness, and GCH above a strongly compact cardinal. The Prize Committee noted the transformational quality of Goldberg’s thesis work within the Inner Model Program.
2018
Danh (Danny) Nguyen Luu, University of California, Los Angeles
Danny Nguyen received his Ph.D. in 2018 from the University of California, Los Angeles, under the direction of Igor Pak. His thesis, “The Computational Complexity of Presburger Arithmetic”, contains stunning results on the complexity of the decision problem for the linear theory of the integers. For example, whereas it has been known since the 70’s that the full decision procedure has doubly-exponential lower bounds, Nguyen’s thesis shows that even very restricted fragments have high complexity. Other results deal with VC-dimension of PA formulas and the complexity of the counting problem for various PA-definable sets. The dissertation is a tour de force, combining methods from number theory, discrete geometry, model theory, and computational complexity.
2017
Matthew Harrison-Trainor, Victoria University, Wellington, and Sebastien Vasey, Harvard University
Matthew Harrison-Trainor received his Ph.D. in 2017 from the University of California, Berkeley under the direction of Antonio Montalbán. In his thesis, “The complexity of countable structures”, Harrison-Trainor established many very strong theorems in computable structure theory. Of these results two stand out. His full description of the Scott spectrum of a theory was a very surprising general result whose proof settled several open problems including ones raised by Marker, Sacks and Montalbán. The second provides a thorough analysis of degree spectra and degrees of categoricity on cones. It shows that the behaviors of these notions are natural in the sense of relativizing to all degrees above some fixed one.
Sebastien Vasey received his Ph.D. in 2017 from Carnegie Mellon University under the direction of Rami Grossberg. In his thesis, “Superstability and categoricity in abstract elementary classes”, Vasey undertook a deep and sustained study of classification theory for abstract elementary classes. Among the many theorems he proved, his eventual categoricity theorem for universal classes is recognized as a landmark achievement towards Shelah’s conjecture generalizing Morley’s theorem on uncountable categoricity to abstract elementary classes. A second remarkable result is his classification of the stability spectrum for tame AEC’s, which may well pave the way for connections with, and applications to, other areas of mathematics.
2016
Will Johnson, University of California, Berkeley, and Ludovic Patey, Université Paris VII
Johnson received his Ph.D. in 2016 from the University of California, Berkeley, under the supervision of Tom Scanlon. Johnson’s thesis, Fun with Fields, contains a number of outstanding results in the model theory of fields, including the classification of the fields $K$ whose theories have the property of “dp-minimality”, a strong form of “not the independence property”. Johnson’s main breakthrough is the construction of a definable topology on $K$, when $K$ is not algebraically closed, introducing vastly new ideas and techniques into the subject.
Patey received his Ph.D. in 2016 from the the Université Paris VII under the supervision of Laurent Bienvenu and Hugo Herbelin. In his thesis, The Reverse Mathematics of Ramsey-type Theorems, he solved a large number of problems in the reverse-mathematical and computability-theoretic analysis of combinatorial principles. In doing so, he combined great technical ability with a powerful eye for unification, isolating several notions that have helped systematize the area.
2015
Omer Ben-Neria, University of California, Los Angeles, and Martino Lupini, California Institute of Technology
Ben-Neria received his Ph.D. in 2015 from Tel Aviv University under the supervision of Moti Gitik. In his thesis, The Possible Structure of the Mitchell Order, he proved the remarkable result that, under suitable large cardinal assumptions on the cardinal $\kappa$, every well-founded partial order of cardinality $\kappa$ can be realized as the Mitchell order of $\kappa$ in some forcing extension. The Prizes and Awards Committee noted that the proof is a tour de force combination of sophisticated forcing techniques with the methods of inner model theory.
Lupini received his Ph.D. in 2015 from York University, Toronto under the supervision of Ilijas Farah. His thesis, Operator Algebras and Abstract Classification, includes a beautiful result establishing a fundamental dichotomy in the classification problem for the automorphisms of a separable unital $C^*\/$-algebra up to unitary equivalence, as well as a proof that the Gurarij operator space is unique, homogeneous, and universal among separable 1-exact operator spaces. The Prizes and Awards Committee noted that his thesis exhibits a high level of originality, as well as technical sophistication, in a broad spectrum of areas of logic and operator algebras.
2014
No prize was awarded.
2013
Artem Chernikov, Université Paris 7-Diderot and MSRI and Nathanaël Mariaule, Seconda Università di Napoli
Chernikov received his Ph.D. in 2012 from Université Claude Bernard–Lyon 1 under the supervision of Itaï Ben Yaacov. His thesis, Sur les théories sans la propriété de l’arbre du second type (On theories without the tree property of the second kind\/), is a fundamental contribution to the development of model theory in unstable theories. He develops a good theory of forking in $\hbox{NTP}_2\/$-theories, a class containing all simple theories and all NIP theories, and shows that the ultraproduct of $p\/$-adic fields, as $p$ varies, is $\hbox{NTP}_2\/$.
Mariaule received his Ph.D. in 2013 from Manchester University under the direction of Alex Wilkie. In his thesis, On the decidability of the $p\/$-adic exponential ring, Mariaule solved a long-standing problem in the model theory of the ring of $p\/$-adic integers with exponentiation. He proved an effective model completeness theorem and, using a appropriate version of Schanuel’s Conjecture, showed decidability.
2012
Pierre Simon, Hebrew University of Jerusalem
Simon received his Ph.D. in October 2011 from the Universit� Paris-Sud, under the supervision of Elisabeth Bouscaren. The Prizes and Awards Committee notes that his thesis, Ordre et stabilit� dans les th�ories NIP: “provides new model theoretic tools and concepts for the study of NIP structures, a generalization of stability that also encompasses o-minimal structures, algebraically closed valued fields, and the p-adics. The thesis makes substantial progress at the full generality of NIP theories, and at the same time obtains new results and new proofs in classical settings. It stands out for its depth, originality, and elegance in seeking the appropriate tools and dividing lines for the subject.”
2011
Mingzhong Cai, University of Wisconsin, Madison and Adam Day, University of California at Berkeley
Cai received his Ph.D. in 2011 from Cornell University, under the supervision of Richard Shore. The Prizes and Awards Committee notes that his thesis, Elements of Classical Recursion Theory: Degree-Theoretic Properties and Combinatorial Properties, introduces powerful new techniques that are used to solve several longstanding open problems, among the most striking of which are the definability of array nonrecursiveness, and a solution to an old problem of Lerman’s about the relation between minimal covers of minimal degrees and the standard jump classes.
Day received his Ph.D. in 2011 from Victoria University of Wellington under the supervision of Rodney Downey. The Committee cited that his thesis, Randomness and Computability, draws on the Russian and Western developments in algorithmic randomness, unifies them, and comes to new results straddling both points of view, among them a recursion theoretic analysis of Levin’s neutral measure (done jointly with J. Miller), and a sharpening of one of the most famously difficult results in the theory of Kolmogorov complexity, Gács’ theorem that monotone complexity and a priori entropy differ.
2010
Uri Andrews, University of Wisconsin-Madison
Andrews received his Ph.D. in 2010 from the University of California, Berkeley, under the supervision of Thomas Scanlon. The Prizes and Awards Committee notes that in his thesis, Amalgamation Constructions in Recursive Model Theory, he combines deep methods from model theory and computability to solve some problems posed by Goncharov that had resisted solution by specialists in computability theory.
2009
Isaac Goldbring and Grigor Sargsyan, both of the University of California, Los Angeles
Goldbring received his Ph.D. in 2009 from the University of Illinois at Urbana-Champaign, under the supervision of Lou van den Dries. The Prizes and Awards Committee notes that in his thesis, Nonstandard Methods in Lie Theory, he applies model theory to a fundamental problem from topological group theory and that the main result replaces an incorrect proof in a widely cited paper from 1957 using totally new ideas.
Sargsyan received his Ph.D. in 2009 from the University of California, Berkeley under the supervision of John Steel. The Committee cited that his thesis, A Tale of Hybrid Mice, contains “uncountably many new ideas” in inner model theory.
2008
Inessa Epstein, California Institute of Technology and Dilip Raghavan, University of Toronto
Epstein received her Ph.D. in 2008 from the University of California, Los Angeles, under the supervision of Greg Hjorth. The Prizes and Awards Committee citation notes that in her thesis, Some results on orbit inequivalent actions of non-amenable groups, she “solves one of the most important problems in measurable group theory, the resolution of which involves a combination of depp results from different branches of mathematics.”
Raghavan received his Ph.D. in 2008 from the University of Wisconsin at Madison under the supervision of Bart Kastermans and Ken Kunen. The Committee cited that his thesis, Madness and set theory, “uses modern methods associated with independence proofs to obtain, just using Z F C, results on almost disjoint (MAD) families, that in particular solve a twenty-year old problem of Van Douwen.”
2007
Adrien Deloro, Rutgers University and Wojciech Moczydlowski, Cornell University
Deloro received his Ph.D. in 2007 from the Université Paris 7, under the supervision of Eric Jaligot. The Prizes and Awards Committee citation notes that his thesis, Groupes simples connexes minimaux de type impair, “deals with the Cherlin-Zilber conjecture, according to which every simple group of finite Morley rank is isomorphic to an algebraic group over an algebraically closed field. In particular, the thesis removes the assumption that there are no bad fields from the classification of minimal counterexamples. Deloro operates with an impressive mastery of a wide range of techniques, which must be interwoven in a very delicate way, and contributes to them in important ways. The result is significant progress on a basic open problem.”
Moczydlowski received his Ph.D. in 2007 from Cornell University under the supervision of Robert Constable. The Committee cited that his thesis, Investigations on sets and types, “contains ground breaking results on constructive set theory and its relation to type theory. Among other things, Moczydlowski proves weak normalization for the theory IZF_R, Intuitionistic Zermelo-Fraenkel Set Theory with Replacement rather than Collection. He also introduces IZF_D, a novel combination of type theory and set theory which has the proof-theoretic power of ZFC, and proves normalization for this theory.”
2006
Matteo Viale, University of Torino and the University of Paris 7
Viale received his Ph.D. in 2006 from the University of Torino and the University of Paris 7, under the supervision of Alessandro Andretta and Boban Velickovic. The Committee’s citation reads: “Viale’s thesis makes fundamental contributions to our understanding of the consequences of forcing axioms in the combinatorics of singular cardinals. In particular, it solves a well-known problem, by showing that the Proper Forcing Axiom implies the Singular Cardinals Hypothesis.”
2005
Antonio Montalbán, University of Chicago
Montalbán received his Ph.D. in 2005 from Cornell University, under the supervision of Richard Shore. The Committee on Prizes and Awards’ citation reads: “The thesis, entitled Beyond the Arithmetic, contains deep and major contributions to an impressively broad array of areas in logic, including computability theory, reverse mathematics, and effective mathematics. It uses a wide arsenal of techniques from set theory, computability theory, proof theory and combinatorics including the development of a new class of invariants for countable linear orderings.”
2004
Joseph Mileti, University of Chicago, and Nathan Segerlind, University of Washington
Mileti received his Ph.D. in 2004 from the University of Illinois at Urbana-Champaign, under the supervision of Carl Jockusch. His thesis, Partition theorems and computability theory, was cited by the Prizes and Awards Committee as containing a “penetrating computability theoretical analyses of Ramsey-type theorems, an important feature of which is an ingeneous completely new proof of the Canonical Ramsey Theorem whose ideas allowed a deep effective analysis of this theorem.”
Segerlind received his Ph.D. in 2004 from the University of California, San Diego, under the supervision of Sam Buss and Russell Impagliazzo. The Committee noted that his thesis, New Separations in Propositional Proof Complexity, “extends switching lemmas, one of the most primary tools in the area, in a very unexpected way, that, among other things allowed him to take, in a single step, one important proof system from an almost complete mystery to being almost completely understood.”
2003
Itay Ben Yaacov, Massachusetts Institute of Technology
Ben Yaacov received his Ph.D. in 2002 from the University of Paris VII, under the supervision of Daniel Lascar. His thesis, Théories simples : constructions de groupes et interprétabilité généralisée, was cited by the Committee as, “a major contribution to pure model theory on two intimately related fronts: an extension of simplicity theory beyond the first order context, and the interpretability of groups in simple theories.”
2002
No prize was awarded.
2001
Matthias Aschenbrenner, University of California, Berkeley
Aschenbrenner received his Ph.D. in 2001 from the University of Illinois at Urbana-Champaign, under the direction of Lou van den Dries. His thesis solves a long-standing problem concerning the complexity of the ideal membership problem in the polynomial ring over the integers. Aschenbrenner shows that the complexity of this problem is doubly exponential in the number of variables, which is optimal since single exponential complexity was known to be impossible.
2000
Eric Jaligot, Universite Claude Bernard (Lyon-1)
Jaligot received his Ph.D. in 1999 from the Institut Girard Desargues, Universite Claude Bernard (Lyon-1), under the direction of Tuna Altinel and Bruno Poizat. His thesis concerned the Cherlin-Zil’ber Conjecture, which asserts that infinite simple groups of finite Morley rank are algebraic groups over algebraically closed fields.
1999
Denis Hirschfeldt, Cornell University and Rene Schipperus, University of Colorado
This was the first year the Prize was awarded under the auspices of the ASL. Denis Hirschfeldt received his Ph.D. in 1999 from Cornell University, under the guidance of Richard Shore. In his thesis, he introduced a new technique for building constructive models which solves an open question which had evaded solution despite attempts by some of the top people in the field. These techniques have since been applied to solve other questions. He also introduced a new class of problems which have since become a focal point for investigations by others. The committee felt that this thesis was marked by ingenuity, insight, originality, and considerable technical prowess. Rene Schipperus received his Ph.D. in 1999 from the University of Colorado, under the guidance of Richard Laver. In his thesis, he introduced new techniques to solve a problem of Erdös on ordinal partition relations which had been open for about 30 years. The thesis has stimulated work which led to some generalizations of Schipperus’s result. The committee felt that this result required a quite intricate proof and the thesis introduced a game-theoretic technique that is quite different from what had been tried before in that area.
1998
No Prize was awarded.
1997
Ilijas Farah, University of Toronto and Tom Scanlon, Mathematical Sciences Research Institute in Berkeley
Farah received his Ph.D. in June, 1997, at the University of Toronto under the direction of Stevo Todorcevic. His thesis contained remarkable results concerning the structure of analytic ideals and their quotients. Scanlon received his Ph.D. form Harvard University in June, 1997, under the direction of Ehud Hrushovski. His thesis provided a model completion for the theory of differential fields connected by a specialization, together with striking applications of the model theory of valued differential fields to diophantine geometry.
1996
Dr. Byunghan Kim, Fields Institute and Dr. Itay Neeman, Harvard University
Dr. Kim received his Ph.D. under the direction of Professor Anand Pillay at the University of Notre Dame in August, 1996. His thesis included ground-breaking work in the area of Stability Theory, concerning the study of simple theories. Dr. Neeman wrote his thesis with John Steel at UCLA, earning his degree in June, 1996. In it, he established some striking results in the area of Determinacy in set theory.
1995
Dr. Slawomir Solecki, Caltech
Dr. Solecki received his Ph.D. under the direction of Professor Alexander Kechris of Caltech in June, 1995. His doctoral dissertation was entitled “Applications of descriptive set theory to topology and analysis” and was notable for its surprising results connecting modern descriptive set theory with other areas of mathematics such as ergodic theory and harmonic analysis.
1994
Gregory Hjorth, California Institute of Technology
The first Sacks Prize was awarded to Professor Gregory Hjorth of the California Institute of Technology, as author of the best dissertation in the field of logic during 1993 and 1994. Hjorth completed his Ph.D. in 1993 under the direction of Professor W. Hugh Woodin at the University of California at Berkeley. His thesis research in descriptive set theory was singled out by the selection committee for its surprising consequences concerning the relationship between projective sets and large