Most recently, Adamus's research interests concentrated on semialgebraic and subanalytic geometry, as well as algebraic approximation of singularities. Together with his postdoc, Hadi Seyedinejad, he 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. Articles containing these results were published in Mathematische Annalen. In a joint paper with his PhD student Aftab Patel, in Journal of Singularities, Adamus showed that every real or complex analytic Cohen-Macaulay singularity may be arbitrarily well approximated by a topologically equivalent Nash singularity which is also Cohen-Macaulay and has the same Hilbert-Samuel function. This result is aimed at proving that every analytic singularity admits a resolution which is topologically equivalent to a Nash object.
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 work on homotopy theory in a broad sense, including algebraic topology, stable homotopy theory, model categories, higher categories, and homotopy type theory.
My most recent work has been in homotopy type theory, with a focus on using it to prove results for general ∞-topoi. In collaboration with Opie, Rijke and Scoccola, I have studied modalities and more general reflective subuniverses in homotopy type theory. These include localizations at primes, truncations, and other important examples. In related work, I have shown that in certain cases, one can localize at a "large" collection of morphisms, extending work of Casacuberta-Scevenels-Smith. Scoccola and I have proven the Hurewicz theorem in homotopy type theory, an important calculational tool. Much of my work has been formalized in the proof assistant Coq, and I am frequent contributor to the Coq-HoTT library.
Kapulkin and I run the Homotopy Type Theory Electronic Seminar Talks, an online seminar started in 2019.
In other recent work, Frankland and I have been studying morphisms between exact triangles in a triangulated category, comparing Neeman's good morphisms with a new class of Verdier good morphisms.
(Updated June 2021)
Matt Davison does research in stochastic optimal control theory applied in two main areas.
Many pieces of energy infrastructure act on random inputs, controlled in certain ways, to create cash flows. So for example a hydro dam takes (random) water inflows and, depending on a control variable, either turns it into electricity, which is sold at a (random) price, or saves the water behind the dam for future conversion and sale. If one hopes to, on average, maximize financial return, there will be an optimal control strategy. The resulting problems can be cast in terms of (nonlinear) Hamilton Jacobi Bellman (HJB) problems or, in some cases, as linear diffusion PDES where the nonlinearity arises with endogenously defined “moving” boundaries. I have been working on a sequence of problems in this general area for the last two decades, most recently in the area of differential games, in which the actions of one producer change the prices experienced by other producers.
More recently I have begun to apply the same HJB technology to problems in personal finance as part of the industry supported Financial Wellness Lab, which combines financial mathematics thinking with the data science analysis of large volumes of relevant data.
A list of my research publications may be found at my Google Scholar page: https://scholar.google.ca/citations?user=pEdDqE0AAAAJ
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I am interested in mathematics that is close to the overlap of combinatorics, topology and geometry. I like to think about matroids and their linear realizations (also known as hyperplane arrangements) using tools either borrowed from or inspired by complex algebraic geometry. Some of my recent projects have been the following:
(1) "Lagrangian geometry of matroids" together with Federico Ardila (San Francisco State) and June Huh (Stanford and KIAS), we introduce the conormal fan of a matroid. This is a Lagrangian analogue of the Bergman fan of a matroid (which is also known as a tropical linear space.) We show that a combinatorial invariant of the matroid, the h-vector of its broken circuit complex, is reflected in the intersection theory of the conormal fan. We further develop the ideas of "tropical Hodge theory" in order to prove that the h-vector of a matroid is log-concave. This resolves conjectures of Dawson and Brylawski from the 1980's.
(2) "A Leray model for the Orlik-Solomon algebra" with Christin Bibby (Louisiana State) and Eva Maria Feichtner (Bremen): we construct a Leray model for a matroid, providing a combinatorial generalization of the Leray or Morgan model of a complex hyperplane arrangement complement.
(3) Configuration polynomials: together with Delphine Pol, Mathias Schulze and Uli Walther, we continue to investigate the singular hypersurface defined by the Kirchhoff polynomial of a graph and, more generally, by a
matroid realization. The topology of the corresponding hypersurface complements continues to be of interest from the point of view of mathematical physics.
With my former student, Dinesh Valluri, I recently extended some results on essential dimension to orbifold curves. The result required developing some relevant deformation theory and Riemann-Roch machinery first. There are a number of tangential projects in this direction that I am exploring. Some of these are about cohomological invariants, Brauer groups and essential dimension of torsors over other classical algebraic groups.
Currently I am working with my PhD, Sayantan Roy-Chowdhury, on extending known results on the derived category of coherent sheaves of homogeneous varieties to positive characteristic. Specifically, we are looking at exceptional collections constructed by Kapranov, Kuznetsov, Polishchuck and others.
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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.
Chris Hall: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 current research is primarily in Topological Data Analysis (TDA), in collaboration with members of the Tutte Institute and my postdoc Katharine Adamyk.
The basic thrust of this work is an exploration of homotopy theoretic objects related to the Vietoris-Rips filtration for a data set, and its associated sets of clusters, or path components. The Vietoris-Rips filtration can be recast as a simplicial fuzzy set or, most usefully, as a diagram of partially ordered sets (posets). The homotopy theory of posets leads to fast conceptual proofs of the basic stability theorems of TDA. The poset technique extends to distances defined in extended pseudo metric spaces, and then one finds a conceptual and more general version of the Healy-McInnes UMAP algorithm.
The usual constructions of TDA do not scale to very large data sets. The Vietoris-Rips construction does not satisfy excision, so there are no local to global patching arguments for such objects. If one works instead with small enough finite subcomplexes of Vietoris-Rips complexes, then patching arguments are available. These subcomplexes are defined by their generating sets of simplices, which are hypergraphs, and patching results can be attractively expressed as observations about hypergraphs. Hypergraphs (or graphs) consist of sets of simplices (or edges), which can be randomly chosen, and one expects a probabilistic and/or sheaf theoretic approach to large scale calculations to arise from this point of view.
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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-Bicocca), 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.
The first step toward solving this problem is to consider actions of the multiplicative group G_m 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 ones that comes from geometric invariant theory.
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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.
Gord Sinnamon:By finding the operator norm of the discrete Cesaro averaging operator on the little-lp sequence spaces, I settled a recent conjecture and answered a 25-year-old question. The paper was placed on the arxiv in 2021.
I introduced a normal form for weighted Hardy operators. This begins by introducing a very large class called the Abstract Hardy Operators and a very small class called the Normal Form Hardy Operators. The main result shows that every operator in the large class can be associated in a natural way with one in the small class that shares many of its important properties. In addition, known results are given very simple proofs and new results are proved using the advantages of normal form. The paper was placed on the arxiv in 2021.
With J. Rastegari (former 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 appeared in the Journal of Fourier Analysis and Application in 2018.
With M. Mastylo (Poznan): We identified two fundamental pairs of Banach spaces as Calderon-Mityagin couples. This gives a complete description of their interpolation spaces, a very large class of spaces that share bounded operators with the original couple. The paper appeared in the Journal of Functional Analysis in 2017.
With W. Grey (former 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.
(Updated June 2021)
I'm interested in research questions at the intersection of theoretical population genetics and microbial evolution. Current projects include the evolution of the mutation spectrum (with D. Agashe, pre-print on the biorXiv), the use of mosaic vaccination strategies in the face of pathogen evolution (with D. McLeod and N. Mideo, also on the biorXiv), and the counter-intuitive increase in extinction risk that may be caused by adaptive sweeps (with M. Tanaka).
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I am interested in modeling and analysing interplays of those identified important factors in population dynamics, transmission dynamics of infectious diseases, evolutionary dynamics, and temporal-spatial dynamics in population spread and invasion. These factors include but are not limited to nonlinearities, spatial diffusions and time delays.
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