Matthew Badger received a $410,000 NSF CAREER award for his work on “Analysis and Geometry of Measures”. Measures are an abstract generalization of “length”, “area”, or “volume” that assign a “size” to every mathematical set. The five-year grant (2017–2022) will support research by Dr. Badger and UConn postdocs and graduate students into fundamental questions about the structure of measures in Euclidean spaces in relation to canonical families of lower-dimensional sets. On the educational front, the award also supports two conferences for junior scientists working under the umbrella of nonsmooth analysis, including a workshop for postdocs in Fall 2017 and a conference with mini-courses for graduate students in Spring 2019.
Damir Dzhafarov has been awarded a Simons collaboration grant to support his research on computable combinatorics and reverse mathematics, with a focus on computationally weak variants of Ramsey’s theorem. The basic objective is to characterize how close a mathematical problem is to being algorithmically solvable, and to understand the axioms and techniques needed to prove various results concerning it. This has proven immensely powerful in shedding light on the basic combinatorial and logical properties underlying different areas of mathematics, and in revealing new connections between them.
Maria Gordina‘s 2017 NSF grant, entitled “Probabilistic Methods in Geometry and Analysis”, supports research in several directions combining probability, geometry, analysis and representation theory. One of the directions of research is to study Cameron-Martin type quasi-invariance in elliptic and subelliptic settings, and its applications to functional inequalities, smoothness of probability laws for subelliptic and singular diffusions, and unitary representations of infinite-dimensional groups such as path groups. Another direction is applying coupling techniques to hypoelliptic stochastic processes, including gradient estimates and connections with geometric and analytic techniques for hypoelliptic diffusions. Some of the proposed research is motivated by physics, especially the quantum field theory (QFT), as infinite-dimensional spaces such as loop groups and path spaces appear in the QFT.
Emiliano Valdez, Jeyaraj Vadiveloo, and Guojun Gan have been awarded a Center of Actuarial Excellence (CAE) research grant for $157,300 from the Society of Actuaries. The grant will support a three-year (2017-2020) research project on “Applying Data Mining Techniques in Actuarial Science” which aims to examine and evaluate data mining tools and approaches for analyzing data in actuarial science and insurance. In particular, they will focus on tools and methods that will effectively demonstrate on how actuaries can use them to preform predictive analytics in three specific areas: claims tracking and monitoring in life insurance, understanding policyholder behavior in general insurance, and model efficiency for variable annuity products.
Vasileios Chousionis has been awarded a Simons collaboration grant to support his research on sub-Riemannian analysis and dynamics. Sub-Riemannian geometry, or “geometry of constrained motion”, provides mathematical models for any physical situation in which allowed motion is subject to a priori nonholonomic constraints. Chousionis’ research will focus on geometric harmonic analysis and conformal dynamics in local models of sub-Riemannian geometry, which are poorly described by Euclidean language.
Damir Dzhafarov received a follow-up seed grant from the Connecticut Institute for the Brain and Cognitive Sciences (IBACS) to support the UConn Logic Group. The UConn Logic Group is an active interdisciplinary research hub with over forty faculty and graduate student members from mathematics, philosophy, linguistics, psychology, and law. Logic is a subject that concerns language, computation, reasoning and problem-solving. As such, it is an important area of interest in many disciplines. This project aims to enhance the Groups’ profile and activities, furthering UConn’s reputation as a center for excellence in research and scholarship in logic and formal methods.
Guojun Gan and Emiliano Valdez have been awarded a grant from the Society of Actuaries to support their project “Regression Modeling for the Valuation of Large Variable Annuity (VA) Portfolios” starting in 2016. They will investigate the potential use of GB2 (generalized beta of the second kind) distributions with four parameters to model the fair market values of VA guarantees. The findings from this project can help insurance companies to reduce significantly the processing time of the Monte Carlo simulation model commonly used in practice for VA valuation.
Guojun Gan (co-PI) and Emiliano Valdez (PI) have been awarded a research grant from the Society of Actuaries to support their research project “Fat-tailed Regression Modeling with Spliced Distributions.” This project aims to examine and demonstrate the use of spliced distributions to model actuarial data that exhibit extreme tail behavior. The results of this project will benefit insurance companies in the areas of pricing, financial reporting and risk management.
Zhongyang Li has been awarded an NSF grant in 2016 in the amount of $100,000 for her work on developing new theory concerning the phase transition of certain lattice models, including the constrained percolation model, the Ising model and the self-avoiding walk.
Alexander Teplyaev was awarded a $150,000 NSF grant in 2016 to support his project “Random, Stochastic, and Self-Similar Equations”. The main goal of the project is to develop robust tools for stochastic, spectral and vector analysis on highly non-smooth spaces such as fractals, and to establish connections with mathematical physics and other sciences.
Damin Wu‘s three-year NSF grant of $223,400 awarded in 2016 entitled “Positivity in Complex Geometry” is to support his research on the study of canonical bundle from viewpoints of complex geometry, algebraic geometry, and fully nonlinear partial differential equations.
Matthew Badger‘s three-year NSF grant of $120,000 awarded in 2015 supports his project “Geometry of sets and measures”. His research is in the field of geometric measure theory, which has its origins in the 1920s and 1930s in order to describe nonsmooth phenomena such as the formation of corners in soap bubble clusters.
Fabrice Baudoin received a grant in 2015 for $300,000 from the NSF. The project focuses on different aspects of the theory of diffusion processes and diffusion semigroups. The PI will investigate applications to sub-Riemannian geometry where diffusion methods turn out to be very fruitful to study generalized Ricci curvature lower bounds. Undergraduate and graduate students will be involved in the project.
Damir Dzhafarov received a seed grant from the Connecticut Institute for the Brain and Cognitive Sciences (IBACS) to support the UConn Logic Group. The UConn Logic Group is an active interdisciplinary research hub with over forty faculty and graduate student members from mathematics, philosophy, linguistics, psychology, and law. Logic is a subject that concerns language, computation, reasoning and problem-solving. As such, it is an important area of interest in many disciplines. This project aims to enhance the Groups’ profile and activities, furthering UConn’s reputation as a center for excellence in research and scholarship in logic and formal methods.
Zhongyang Li has been awarded a Simons foundation grant in 2015 to work on her project “Phase transitions in lattice models and conformal invariance at criticality”. She is studying the phase transitions in lattice models including the Ising model, dimer model, self-avoiding walk, and the 1-2 model. Once the critical parameter is identified, one can try to prove conformal invariance of certain observables at criticality.
Dmitriy Leykekhman received a three-year NSF grant of $99,999, also awarded in 2015, to work on the project “Point and state constrained optimal control parabolic problems”. The optimal control problems he is studying in this project are classical and have a wide range of applications, for instance in water waste treatment, river pollution, calcium waves in a heart cell, and noise control. The finite element method is the most widely used method to solve such problems numerically, but there are very few results in this area on a priori error estimates.
Oleksii Mostovyi‘s NSF grant of $115,237 (awarded in 2015) is titled “Utility based Pricing and Hedging in Incomplete Markets with Stochastic Preferences in a Unifying Framework of Admissibility.” He is studying problems of pricing and hedging of financial instruments, which are of fundamental importance from both the theoretical and practical sides of mathematical finance.
Ovidiu Munteanu received a three-year NSF grant of $166,545 in 2015 for his project “The Geometry of Ricci Solitons.” This research is in differential geometry, a branch of mathematics that studies the shapes of geometric objects. The goal of his project is to understand the structure and properties of Ricci solitons in arbitrary dimensions.
Luke Rogers (PI) and Alexander Teplyaev (co-PI) received a $324,000 NSF REU Site grant titled “Mathematics REU at UConn”. The main goal of the REU is to engage a diverse group of undergraduates from primarily non-PhD granting institutions in research that results in publications, produces conference talks and posters, and encourages the students to pursue graduate studies and careers in mathematics and mathematics education. In the last three years the program included research groups “Analysis on Fractals” mentored by Ulysses Andrews, Antoni Brzoska, Joe Chen, Dan Kelleher, Luke Rogers, Sasha Teplyaev; “Math Education” mentored by Fabiana Cardetti, Kyle Evans, Gabriel Feinberg; “Representation Theory: Maximal Green sequences” mentored by Ralf Schiffler, Khrystyna Serhiyenko; “Stochastic stabilization of Planar flows” mentored by Masha Gordina, Fan Ny Shum.
Liang Xiao‘s 2015 NSF grant of $135,000 is “Special fibers of modular varieties”. Modular varieties help establish relations between number theory and harmonic analysis. These are used in the Langlands program, which is aimed at establishing deep relations between number theory (arithmetic information about integers, for example their factorization into products of primes) and harmonic analysis (for example, harmonic functions on certain spaces with additional symmetries).
Fabiana A. Cardetti (jointly with Manuela Wagner, Professor of Foreign Language Education at LCL) has been awarded a $137,000 grant “Prototype of P-20 and Interdisciplinary Collaboration and P-20 Articulation”. This grant is funded by UConn’s College of Liberal Arts and Sciences (CLAS) and the Neag School of Education runs from May 2014 to conduct an interdisciplinary project to study the intersections of mathematics and foreign languages education under the framework of Intercultural Competence.
Damir Dzhafarov‘s 2014 NSF grant of $150,000 is called “New directions in reverse mathematics and applied computability theory”. His research focuses on reverse mathematics, including Ramsey’s theorem, various equivalents of the axiom of choice, and principles arising from certain problems in cognitive science. This will be facilitated by the application of methods from computability theory and proof theory, and by the addition of ideas from various collaborations across a number of areas of pure and applied mathematics, as well as interactions with members of the multidisciplinary University of Connecticut logic group.
Maria Gordina‘s 2014 NSF grant of $288,000 is called “Stochastic analysis and related topics”. This project is focused on elliptic and subelliptic diffusions in infinite-dimensional curved spaces, such as infinite-dimensional groups, loop groups and path spaces. Her research connects diverse fields such as stochastic analysis, geometric analysis, representation theory and mathematical physics.
Lan-Hsuan Huang received a $400,648 NSF Career grant for her work titled “Geometric Problems in General Relativity”. Her research aims to better understand globally conserved quantities in general relativity and their connections to geometric structure. This is geometric mathematics used to describe the shape of the universe.
Kyu-Hwan Lee has been awarded a five-year collaboration grant by the Simons Foundation in 2014 to work on his project “Topics on Hyperbolic Kac-Moody Algebras and Groups”. He will continue his research in representation theory on connections of the Kac-Moody algebras to automorphic forms. Some of the topics include a construction of an automorphic correction of hyperbolic Kac-Moody algebra, a study of Eisenstein series on rank 2 hyperbolic Kac-Moody groups over the real field, and description of products coming from certain p-adic integrals as sums over crystals.
Jerzy Weyman, the Stuart and Joan Sidney Professor, was awarded $288,666 by the NSF in 2014 to work on the project ” Free resolutions and representation theory”. This research is related to two branches of algebra: commutative algebra and representations of quivers. Results of this research might lead to better algorithms to solve linear algebra problems, which have many applications in other areas of mathematics as well as in other sciences.
Iddo Ben-Ari has been awarded a Simons collaboration grant in 2013 to work on his project “Analysis of Markov processes”. The project consists of two major parts: ergodicity for diffusions with redistribution, and analysis of mathematical models for biological evolution. The research involves Iddo’s PhD and undergraduate students.
Ralf Schiffler received a $400,000 NSF Career grant to work the project on “Cluster algebras, combinatorics and representation theory” in 2013. Cluster algebras are commutative algebras with a special combinatorial structure, which are related to various fields in mathematics and physics. The cluster algebras provide a mathematical framework for fundamental patterns which occur throughout representation theory. These patterns are also observed in various other branches of science which, a priori, are not related to representation theory.