Over the last year my research interests concentrated on semialgebraic and subanalytic geometry. Together with my postdoc, Hadi Seyedinejad, I obtained important results in one of the most active areas of contemporary real algebraic geometry.
We proved a long standing conjecture of Kurdyka on the geometry of zeroes of semialgebraic arc-analytic functions, and a theorem stating that every semialgebraic arc-analytic function admits an arc-analytic extension to the ambient Euclidean space (an arc-analytic analogue of Cartan's Theorem A). A paper containing these results was published in Mathematische Annalen.
Barron's research identifies new synergies between geometry, analysis, and mathematical physics, and exhibits how the geometric properties of submanifolds are reflected in the analytic properties of sections of line bundles or vector bundles. In the context of quantum mechanics, this framework gives insight to the behavior of particles, such as entangled particles or antiparticles.
Barron and her student Pollock proved that the Bohr-Sommerfeld quantum states of a certain type have maximal entanglement entropy. These states have been studied for geometric reasons. Also, when entangled particles are used for information transmission, the entropy of entanglement is used to quantify the amount of information that can be transmitted. The Barron-Pollock result will appear in Reports on Mathematical Physics in 2017.
Barron and her students study line bundles on compact Kahler manifolds and Toeplitz operators. Barron recently authored a book on Toeplitz operators on Kahler manifolds. Barron and her student Serajelahi extended the Poisson bracket-commutator correspondence for Kahler manifolds to Toeplitz operators on multisymplectic and hyperkahler manifolds. These results appeared in Glasgow Math. J. in 2017.
I supervised a group of PhD students at an AMS summer school, and the team has developed the theory of localization in homotopy type theory. Two papers resulting from this will be finished soon.
Enxin Wu (former PhD student, at Shantou University, China) and I developed the theory of smooth bundles and smooth classifying spaces, allowing one to study differential forms on classifying spaces. The resulting paper has been submitted, and is on the arXiv.
Andries Brouwer (Centrum Wiskunde & Infomatica, Netherlands) and I found a counterexample to David Gale's conjecture in game theory, and used the same techniques to count the number of linear extensions of the poset of subsets of a set of size 7. This work resulted in a preprint that is on the arXiv, and has been published online in Order.
I co-authored two papers about generic (cohomological) vanishing. The first sets out some general theory and introduces the "abelian duality" property. The second specializes the theory to certain complex hypersurface complements, where we establish an easy-to-check sufficient condition for duality and abelian duality.
1) "Abelian duality and propagation of resonance", with Alex Suciu (Northeastern) and Sergei Yuzvinsky (Oregon), Selecta Mathematica, 2017.
2) "Local systems on arrangements of smooth, complex algebraic hypersurfaces", with Alex Suciu, preprint on the arXiv.
A paper with Christin Bibby (Western) and Eva Feichtner (Bremen) is nearing completion, in which we construct a combinatorial generalization of the Leray or Morgan model for a compactified arrangement complement.
In a project with Federico Ardila (San Francisco State) and June Huh (Princeton), we show that the h-vector of the broken circuit complex of a matroid is log-concave. The approach involves some toric geometry, matroid combinatorics, and an extension of the ideas of tropical Hodge theory.
Christin Bibby was my postdoc at Western during 2015-17. She is now a postdoc at the University of Michigan.
In joint work with my postdoc Pal Zsamboki we generalised a classical theorem of Hilbert that tells us that certain principal bundles are in fact trivial.
Matthias Franz and his student Jianing Huang have made significant progress to proving the main outstanding conjecture about big polygon spaces. These spaces exhibit a special kind of symmetry that previously was not known to exist. These results will appear in Huang's PhD thesis.
The paper "A quotient criterion for syzygies in equivariant cohomology" was published in Transformation Groups in 2017. This paper gives a hands-on criterion for determining syzygies in equivariant cohomology, a subtle feature of spaces with rotational symmetries. Using it in an ongoing project, Franz and his student Jianing Huang have made significant progress to proving the main outstanding conjecture about big polygon spaces. These spaces exhibit a special kind of symmetry that previously was not known to exist.
We analyzed a family of 'superelliptic curves' over Fq(t) and showed their Jacobians have 'large' rank. We also showed the (analogue of the) Birch-Swinnerton-Dyer was true and calculated 'everything' involved in the finest statement of that conjecture. This is a five-year eight-coauthor paper, entitled "Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields", which will appear in the Memoirs of the AMS and is available on the arXiv.
The paper "Ramanujan Coverings of Graphs" has been accepted for publication in Advances in Mathematics. It is available on the arXiv.
In 2016-17, Lumsdaine (Stockholm) and I gave a formal statement of the conjectures on internal languages of higher categories (arXiv:1610.00037), thus relating infinity-categories and (homotopy) type theory. The first of these conjectures was later proven in a joint paper (arXiv:1709.09519) with Szumilo (Western).
With Voevodsky (IAS Princeton), I worked on a cubical approach to Lurie's straightening/unstraightening adjunction and the resulting paper should be coming out soon.
Carlton, Essex (Western), and I introduced a new cryptosystem, based on subgroups of prime power order, and used it to construct a secure comparison protocol. I have also been working on constructing cryptographic multilinear maps, using the techniques from algebraic topology.
I was recently appointed to the Editorial Board of Hales' FAbstracts project, which aims to give formal statements (in the proof assistant Lean) of the main theorems, definitions, and constructions from many important mathematical papers.
My student Nicholas Meadows is completing his thesis on local higher category theory. He has constructed sheaf theoretic analogues of the extant flavours of higher categories, namely quasicategories, Segal spaces and simplicial categories, and has constructed equivalences between these theories. Meadows' local theory comes equipped with a theory of cocycles and torsors, which is of interest in its own right, and is applicable to Simpson's descent theory for n-categories.
In separate work I found a graph theoretic interpretation of the theory of clusters, which is applicable to higher dimensional settings. There may also be a corresponding way to interpret peresistent homology theory. This work has been written up, in a paper entitled "Cluster graphs", which is on my website and will be submitted for publication soon.
I recently completed a project on Twisted Grassmannians and Torsion in codimension 2 Chow groups, joint work with my postdoctoral fellow, Caroline Junkins, and Danny Krashen (University of Georgia).
This summer, I graduated a doctoral student, Armin Jamshidpey who worked on rationality problems for algebraic tori. I was Armin's principal supervisor, but he was also cosupervised by Eric Schost, in Computer Science at the University of Waterloo. Armin is now a postdoctoral fellow at the University of Waterloo. Armin, Eric and I have a preprint on computational problems about algebraic tori.
With collaborators (including Pierre Guillot (Strasbourg), N.D. Tan (Vietnam Acdemy of Science and Technology, Hanoi), Ido Efrat (Ben Gurion University), Minac published a series of papers on Massey products in Galois cohomology, during 2016-17. Some of these papers appeared in Journal of the European Mathematical Society, Advances in Mathematics, Transactions of the AMS, and Journal of Number Theory. These invariants of absolute Galois groups give new information about their structure, and lead to new conjectures.
A series of further joint papers is in preparation. The coauthors/collaborators include Adam Topaz (Oxford), Olivier Wittenberg (ENS, Paris), Andrew Schultz (Wellesley), N.D. Tan, Sunil Chebolu (Illinois State), Claudio Quadrelli (Milano-Boccoca), Cihan Okay (UBC), Federico Pasini (Western) and Marina Palaisti (Western).
Invariant theory has been used for many years as a method for constructing quotient spaces. The overarching theme of my current research is to find more "indigenous" quotient spaces than the ones provided by geometric invariant theory. This program is made possible by the results of the following papers:
1) L. Renner, Local invariants and exceptional divisors of group actions, Journal of Algebra, 2016.
2) L. Renner, Almost Local Properties of Group Actions, 2017 preprint.
My PhD student Chris Dugdale is working toward solving this problem for actions of the multiplicative group Gm on smooth affine varieties. The key observation in this important special case is that the entire problem will become transparent in terms of semi-invariants. Furthermore, the resulting quotient space can be compared with the one that comes from geometric invariant theory.
Riley published three papers in 2016-17, with his students M. Shahada and Chuluundorj Ben-Ochir, on various aspects of ring theory and algebra. Two of these papers appeared in the Journal of Algebra, and one appeared in Communications in Algebra.
He also completed three other manuscripts, on polynomial identities, with the students M. Shahada, Chris Plyley and Chris Henry, and with his long-time collaborator Eric Jespers (Vrije Universiteit Brussel).
In 2016-17 R. Shafikov co-authored 4 research papers submitted or already accepted in peer-reviewed journals. This includes a longer
survey paper on holomorphic mapping, which is to appear in Proceedings of the Steklov Institute of Mathematics. Other papers are to appear in the Proceedings of the Steklov Institute and in Izvestiya: Mathematics. The two papers in the Proceedings of the Steklov Institute were invited submissions.
Some of the recent papers were co- authored with Purvi Gupta, his post-doc from 2015-17. She now holds a Hill Assistant Professor
position at Rutgers University.
With E. Kikianty (Pretoria): We introduced a notion of equivalence of norms on Banach spaces that is finer than the usual, topological, equivalence. The paper appeared in the Journal of Mathematical Analysis and Applications in 2017.
With J. Rastegari (PhD student): We pointed out an error in the Fourier inequality literature with the effect that a certain case believed closed since 2003 was reopened. We provided a counterexample, and went on to provide comparable (but correct) results. The paper was accepted by the Journal of Fourier Analysis and Application in 2017.
With W. Grey (PhD student): We investigated the embedding problem between mixed-norm Lebesgue spaces, giving a complete solution in simple terms that is valid in the great majority of cases. In certain very special cases, we showed that the problem is quite intractable by relating it to a known NP-complete problem. The appeared in the Transactions of the American Mathematical Society in 2016.
With M. Mastylo (Poznan): We identify two fundamental pairs of Banach spaces as Calderon-Mityagin couples. This gives a complete description of their interpolation spaces, a vast class of spaces that share bounded operators with the original couple. The paper appeared in the Journal of Functional Analysis in 2017.