Dinesh thakur mathematician biography
Home / Scientists & Inventors / Dinesh thakur mathematician biography
He also addressed elementary methods for prime counting in function fields.[30][20]These advances connect multizeta values to transcendence questions in positive characteristic. The Goss zeta function, defined as ζA(s)=∑f∈A∣f∣−s for the polynomial ring A=Fq with ∣f∣=qdegf, serves as a central object, interpolating the degrees of monic polynomials and exhibiting a functional equation analogous to the Riemann zeta function but with poles and zeros shifted due to the characteristic p.
These are generalized by Drinfeld modules, higher-rank structures over Fq that incorporate the Frobenius endomorphism and enable the definition of L-functions attached to motives or Galois representations in characteristic p. Thakur and collaborators defined these for Fq and conjectured restrictions on their weight sequences, such as non-decreasing orders si≤si+1.[24] If a multizeta value is zeta-like, its tail is Eulerian, providing structural insights into their arithmetic nature.[24] Additionally, connections to the leading coefficients of the Goss zeta function at negative integers link finite variants Z℘(s) to ζ(−(qD−1−s)) for appropriate degrees D.[24]Joint work with Böckle examined the leading coefficient of the Goss zeta value and its relation to p-ranks of Jacobians in Carlitz cyclotomic covers.
He shares his profound passion for mathematics with students at all levels, teaching a wide variety of courses that span from foundational introductory calculus to highly specialized advanced graduate seminars. In 2024, Thakur further explored Fermat-Wilson supercongruences and arithmetic derivatives, revealing strange factorizations in function fields.[28][29][20]In recent work, Thakur proposed heuristics for expected Mordell-Weil ranks of elliptic curves over function fields, leveraging Sato-Tate distributions and Birch-Swinnerton-Dyer conjectures.
in mathematics in 1987.[1][7]Thakur's doctoral work was supervised by John Tate, a prominent number theorist.[4] His dissertation, titled "Gamma Functions and Gauss Sums for Function Fields and Periods of Drinfeld Modules," focused on developing gamma functions in characteristic p settings and applying them to the study of Gauss sums within function field arithmetic.[4][8] These investigations laid early groundwork for his contributions to arithmetic geometry over function fields.
Professional career
Early appointments
Following the completion of his PhD at Harvard University in 1987 under John Tate, Dinesh Thakur began his professional career with a postdoctoral membership at the Institute for Advanced Study in Princeton, serving from 1987 to 1989 and accumulating a total of 3.5 years across multiple stays there.[9][5]He then held visiting positions at the Tata Institute of Fundamental Research in Bombay from 1989 to 1991, where he contributed to research in arithmetic geometry.[1]In 1991, Thakur joined the University of Minnesota as an assistant professor, a role he maintained until 1992.[1] The following year, from 1992 to 1993, he served as an assistant professor at the University of Michigan, continuing to build his expertise in number theory during these early academic appointments.[1]Mid-career developments
In 1993, Dinesh Thakur joined the University of Arizona as an assistant professor in the Department of Mathematics, where he advanced through the ranks to become a full professor in the early 2000s.[1] His tenure at Arizona, spanning 1993–2014, marked a period of significant institutional contributions in arithmetic geometry.[1]Thakur was a founding member of the Southwest Center for Arithmetic Geometry, established in 1997 with support from an NSF Group Infrastructure Grant, alongside colleagues including Douglas Ulmer and Felipe Voloch.[10] He played a key role in organizing the center's activities, which fostered collaboration among researchers in number theory and arithmetic geometry across the southwestern United States.[10]From the late 1990s onward, Thakur served as an organizer for the Arizona Winter School, an annual intensive program hosted at the University of Arizona that trained graduate students and postdocs in advanced topics; he contributed to its organization for approximately 15 years.[11][12][10] During this period, he also undertook secondary visits, including a year at the Tata Institute of Fundamental Research in Mumbai from 1998 to 1999 and a year at the Institute for Advanced Study in Princeton from 2000 to 2001.[1]Current position and service
Since the 2013–2014 academic year, Dinesh Thakur has served as a Professor of Mathematics at the University of Rochester.[9] Prior to this appointment, he held a professorship at the University of Arizona, providing foundational experience in number theory research and mentorship that informed his transition to Rochester.[2]Thakur contributes to the mathematical community through editorial service on several prominent journals.He has been a member of the editorial board of the Journal of Number Theory,[13] the International Journal of Number Theory,[14] and p-adic Numbers, Ultrametric Analysis and Applications.[15] These roles involve reviewing and guiding submissions in areas such as arithmetic geometry and p-adic analysis.In mentorship, Thakur has advised 10 PhD students, according to the Mathematics Genealogy Project.[4] Among them is Javier Diaz-Vargas, who completed his 1996 dissertation under Thakur's supervision at the University of Arizona, examining zeros of characteristic p zeta functions.[16]Thakur remains active in professional outreach via lectures and seminars on function field arithmetic.
He moved to University of Arizona in 1993.
He joined University of Rochester in July 2013. Holding a PhD from Harvard University, which he earned in 1987, Professor Thakur has significantly enriched the landscape of mathematics, particularly through his deep engagement with number theory and arithmetic geometry.
in mathematics from the University of Bombay in 1981 and his Ph.D. Multizeta monomials are transcendental over the constants, and sets including analogs of π and odd zeta values exhibit algebraic independence, underscoring the depth of arithmetic relations in function fields.[24]
Publications and recognition
Key books and monographs
Thakur's seminal monograph, Function Field Arithmetic, published in 2004 by World Scientific, spans 404 pages and delivers a comprehensive exposition of function field arithmetic, with particular emphasis on zeta and L-functions, Drinfeld modules, and their special values.[18][31] The book integrates classical results with recent advances, providing a unified framework for arithmetic properties in positive characteristic analogous to those in number fields.[18]In collaboration with David Savitt, Thakur co-edited p-adic Geometry: Lectures from the 2007 Arizona Winter School in 2008, part of the American Mathematical Society's University Lecture Series (volume 45).This wonderfully interdisciplinary approach allows him to forge meaningful connections between fields that might initially seem quite separate, profoundly enriching both his research endeavors and his teaching. Through analysis of slopezero multiplicities in Dieudonné modules, they determined patterns in prime factorizations of power sums, yielding explicit formulas for p-ranks and insights into the ordinariness of these Jacobians as the prime ideal varies.[25]Applications extend to arithmetic structures in function fields.
Furthermore, Professor Thakur's insightful exploration into elliptic Carmichael numbers has opened up exciting new avenues within the study of number theory, firmly establishing his reputation as a leading and influential figure in the field.
Beyond his significant research contributions, Professor Thakur holds a deep and genuine commitment to both education and mentorship.
Thakur's work emphasizes how these functions encode arithmetic data, such as the class number hK of extensions K/Fq(t), which grows like qg where g is the genus, contrasting with the logarithmic growth in number fields.[18]Key concepts in Thakur's contributions involve Carlitz modules, which analogize the complexexponential and multiplication in the functionfield setting, providing tools for studying units and regulators similar to complexmultiplication in elliptic curves.
Unlike number fields, where analytic continuation relies on complex analysis, function field arithmetic uses algebraic geometry over finite fields, with the zeta function factoring as ζK(s)=∏(1−αiq−s)−1 by Weil's theorem, allowing explicit computations of class numbers via point counting on curves. Thakur proved that products of single zeta values expand into linear combinations of multiple zeta values with coefficients in Fp, mirroring the classical shuffle product but adapted to the function field context.[24] For instance, ζ(a)ζ(b)−ζ(a+b)−ζ(a,b)−ζ(b,a)=∑fiζ(ai,a+b−ai), where the fi are explicit in terms of binomial coefficients modulo p.[24] This was later refined for non-rational infinite places, where classical shuffle relations hold under sign conditions, such as ζ(a)ζ(b)=ζ(a,b)+ζ(b,a)+ζ(a+b).[24]Key results include the study of zeta-like multizeta values, which are ratios involving the rational zeta function.
His dedication in the classroom is beautifully mirrored by his commitment to nurturing the next generation of mathematicians; he has guided numerous PhD and master's students, helping them to cultivate their own unique research projects in diverse areas, including complex topics like arithmetic geometry and the mathematical underpinnings of social interactions.
Professor Thakur's academic curiosity gracefully extends beyond the traditional confines of pure mathematics, venturing into realms such as political economy and the study of the African Diaspora.
These values, defined for Fq, arise as sums over effective divisors and capture arithmetic phenomena analogous to those in number fields, such as periods from iterated extensions of Carlitz motives.[23]A foundational aspect is the establishment of shuffle relations for these multizeta values. This includes the study of zeta functions, L-functions, and class number problems, which provide insights into the distribution of primes and ideal class groups in these settings.
Thakur wrote a research monograph Function Field Arithmetic.
Thakur has been serving on the editorial boards of Journal of Number Theory, International Journal of Number Theory, and P-adic Numbers, Ultrametric Analysis and Applications.
Achievements
He was the founder-director of ANK productions, a Mumbai-based theatre company, established in 1976.