College of Liberal Arts and Sciences

# Department of Mathematics

## Faculty Grants

### 2018

Iddo Ben-Ari received a grant from the NSA ($67,336) for an REU on Markov Chains to be held in the summer of 2018. Guojun Gan (PI) and Emiliano Valdez (Co-PI) have been awarded a grant of$15,000 from the Society of Actuaries to support their project “Valuation of Large Variable Annuity Portfolios with Rank Order Kriging” starting in June 2018. They will investigate various aspects of the rank order kriging method to address the computational issues arising from the variable annuity area. The findings from this project can help insurance companies to reduce significantly the valuation time of large variable annuity portfolios.

Ovidiu Munteanu received a three-year NSF grant of $158,151 in 2018 for his project “Analysis of Singularities of the Ricci flow.” This research is in differential geometry, a branch of mathematics that studies the shapes of geometric objects called manifolds. Ricci flow is a parabolic partial differential equation defined on a manifold which starts with a given metric (i.e. shape) and evolves it in time into an improved one, for example spherical. Being a nonlinear equation, Ricci flow often develops singularities after finite time. The goal of his project is to understand the structure and properties of such singularities. Vyron Vellis received a three-year NSF grant of$101,610 in 2018 for his project “Parametrization, Embedding and Extension Problems in Metric Spaces.” This research is in metric geometry, a branch of mathematics where first order differential calculus and geometric measure theory are extended from the classical Euclidean setting to the realm of spaces without a priori smooth structure. One direction of research is to study which metric spaces admit a “nice” (e.g. Lipschitz, quasiconformal) parametrization by the unit n-sphere. Problems related to Lipschitz or Holder parametrizations have played a pivotal role in geometric measure theory. Another direction is to study which metric spaces (specifically sub-Riemannian manifolds) can be nicely embedded in Euclidean spaces.

Jerzy Weyman got a three year NSF grant for the project ”Applications of Representation Theory to Commutative Algebra” for $282,000. This research is in Commutative Algebra which, roughly speaking, studies sets of points given by polynomial equations. The goal of the project is to study properties of free resolutions of coordinate rings of such sets. Weyman has results on the structure of such resolutions for sets of codimension three and he plans to develop these results further. ### 2017 Fabiana A. Cardetti with Manuela Wagner, Professor of Foreign Language Education at LCL, have been awarded a grant from the UCHI through the Humility and Conviction in Public Life project at UCONN. The PIs will work with school teachers, renowned expert in intercultural competence, Michael Byram (Duram University), and intellectual humility expert philosopher, Heather Battaly (California State University, Fullerton) to investigate the opportunities to engage math and languages students in intellectually humble discourse and becoming interculturally competent. (Oct. 2017 – Aug 2018,$33,000).

Fabiana A. Cardetti with Manuela Wagner, Professor of Foreign Language Education at LCL, have been awarded funding to conduct an interdisciplinary project to study the development of Intercultural Competence in international teaching assistants (ITA) and their students, and its impact on teaching and learning of science at the undergraduate level. (May 2017 – Aug 2018, $25,000) 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 for $210,000, 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. Sean Li received an NSF grant for$120,000 entitled “Analysis and geometry of metric measure spaces”. The main focus of the project is to study the structure and analytic properties of metric measure spaces, particularly when the space is non-Euclidean. A special focus will be put on the setting of sub-Riemannian geometries. Goals of the project include studying many phenomena like rectifiability, curvature, differentiability, and boundedness of singular integrals, which all have been well developed in the classical Euclidean/Riemannian settings.

Luke Rogers and Alexander Teplyaev were awarded a $315,000 REU Site grant to continue our department’s successful summer research program for undergraduates in 2017-2020. During this period students will work on projects in Differential Geometry, Mathematical Physics, Mathematical Finance, and Mathematics Education under the guidance of faculty members Fabrice Baudoin, Fabiana Cardetti, Masha Gordina, Luke Rogers and Alexander Teplyaev. 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.

### 2016

Fabiana A. Cardetti with Manuela Wagner, Professor of Foreign Language Education at LCL, have been awarded a fellowship from the UCHI through the Humility and Conviction in Public Life project. They will investigate interconnections between theories of intellectual humility and intercultural competence and citizenship, specifically as applied to education. (Spring 2017, $25,000). 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.

Ambar Sengupta was awarded a $25,295 NSA grant in 2016 to support his project “Geometric and Probabilistic Problems”. The project develops geometry and probability on infinite dimensional spaces and explores their interaction with areas of potential application. A centerpiece of the project is the interplay between Gaussian measure on infinite dimensional linear spaces and on subspaces. The project also includes problems such as those of a geometric nature inspired by gauge theories in physics. 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. ### 2015 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).

### 2014

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. Fabiana A. Cardetti with colleagues Megan Staples, Tutita Casa, and Dorothea Anagnostopoulous from the Neag School of Education have been awarded a Math and Science Partnership grant from the State Department of Education to carry out Phase I of the project Bridging Practices Among Connecticut Mathematics Educators. As part of the project, math teachers from three Connecticut school districts are working with UConn faculty members and graduate students to develop skills and resources to enable them to meet new teaching and assessment standards and improve their students’ learning in math. (Feb 2014 – Sept. 2016,$428,592). Read more about the program, its goals, and members, on UConn Today, or in the CLAS Newsletter.

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.

### 2013

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.