Number Theory in the Department of Mathematics & Statistics

Number Theory in the Department of Mathematics & Statistics http://www.uncg.edu/math/numbertheory

Number theory is one of the oldest re- Euclid proves there are infinitely many primes search areas in mathematics. It is con- and proposes the the Euclidean Algorithm for cerned with the study of integers (in par- 300BC computing greatest common divisors. ticular prime numbers) and generaliza- tions thereof.

On the right we give some examples Fermat proves that ap−1 mod p = 1 for all primes from the long history of number theory. p, and integers a not divisible by p (Fermat’s little Some results took centuries to prove af- 1600S theorem). He states that an + bn = cn has no solu- ter they were first stated. tion for positive integers a, b, c, if n > 2 (Fermat’s last theorem). Public key cryptography is an important practical application of number theory in the everyday life of most people. Most Riemann hypothesizes that the non-real zeros of secure digital communication is based the zeta function ζ(s) (pictured on the left) all on number theory and the security de- 1859 1 have real part 2 . This important conjecture re- pends on the difficulty of solving num- mains unproven to this day. ber theoretic problems. The members of the number theory group at UNCG work in several areas, Hadamard and de la Vallée-Poussin indepen- including algebraic, analytic, and com- dently employ ζ(s) in proofs of the prime number putational number theory, and the the- 1896 theorem that states that the function π(n) giving ory of modular forms. the number of prime numbers up to n is asymp- n totic to log n . 66 50 10 80 73 64 57 48 41 34 78 71 62 55 46 39 32 76 69 60 53 44 37 74 67 58 51 35 30 65 49 42 28 72 63 56 47 40 33 77 70 61 54 45 38 31 75 68 59 52 43 36 29 73 66 57 50 41 34 71 64 55 48 32 27 69 62 46 39 25 76 60 53 44 37 30 74 67 51 35 28 65 58 49 42 72 63 56 40 33 26 70 54 47 38 31 68 61 52 45 29 24 66 59 43 36 73 57 50 41 34 27 71 64 48 32 25 69 62 55 46 39 60 53 37 30 23 67 51 44 65 58 42 35 28 21 72 56 49 33 26 70 63 47 40 24 Rivest, Shamir, and Adleman invent the RSA 61 54 38 31 68 52 45 29 22 66 59 43 36 8 57 50 34 27 20 64 48 41 69 62 55 39 32 25 18 60 53 46 23 67 44 37 30 65 58 51 35 28 21 public key cryptosystem. It uses Fermat’s little 56 49 42 19 63 40 33 26 1977 54 47 61 38 31 24 17 66 59 52 45 29 22 15 64 57 50 43 36 20 41 34 27 62 55 48 25 18 60 53 46 39 32 theorem. 16 37 30 23 65 58 51 44 14 63 56 49 42 35 28 21 61 54 47 40 33 26 19 59 52 45 38 31 24 17 57 50 43 36 29 22 15 62 55 48 41 34 27 20 13 60 53 46 39 32 25 18 11 58 51 44 37 16 56 30 23 6 49 42 35 28 21 14 61 54 47 59 40 33 26 19 12 52 45 57 38 31 17 10 50 43 24 55 36 29 8 48 41 22 15 53 46 34 27 20 13 58 51 39 32 56 44 25 18 11 49 37 30 Wiles proves Fermat’s last theorem. 42 23 54 47 35 16 9 52 40 28 1995 21 57 45 33 14 7 50 38 26 55 43 31 19 12 5 48 36 24 53 41 29 17 46 34 22 10 51 39 27 15 44 32 8 49 37 20 54 42 25 13 47 30 18 6 52 40 35 4 45 28 23 11 50 33 16 4 43 38 21 53 48 31 26 9 41 36 14 2 51 46 24 19 34 29 44 39 7 49 22 17 12 37 32 27 52 47 42 The graph on the far left shows the absolute 30 25 20 15 10 5 50 45 40 35 48 43 38 33 28 23 18 13 8 46 41 36 31 26 21 3 value of the Riemann zeta function ζ(σ + it) 16 49 44 39 34 11 29 24 19 6 47 42 37 32 27 14 45 40 22 9 for 0 ≤ σ ≤ 6 and −58 ≤ t ≤ −0.3. The 35 30 17 2 48 43 25 12 38 33 20 46 28 4 41 36 23 15 44 31 7 zeros are the white dots in the valleys. 39 26 18 47 34 21 10 42 29 37 13 32 24 45 40 16 35 27 19 43 8 The other plot gives the distribiution of the 38 30 22 11 46 33 25 14 5 41 28 17 44 36 39 31 20 33 2 14 2 2 zeros of higher derivatives of ζ(s) in the left 12 38 6 37 7 0 5 23 12 6 0 7 12 6 0 0 17 -10 43 9 25 -8 8 1 9 -6 1 9 -4 1 -2 28 24 32 11 24 half plane. 45 3 20 29 13 29 3 13 3 16 40 4 27 19 16 4 21 18 15 21 18 15 10 26 22 26 10 30 35 The Number Theory Faculty

Dr. Sebastian Pauli email: [email protected] Associate Professor web page: http://www.uncg.edu/~s_pauli Office: Petty 145

Dr. Pauli wrote his Diplomarbeit (Masters thesis) un- der the supervision of Michael Pohst at Technische Computational Number Theory Universität Berlin. He received his Ph.D. from Con- This area of mathematics deals with cordia University in Montreal in 2001 and as a Post developing algorithms for explicit Doctoral Fellow was the lead developer of the com- computation of the objects under puter algebra system KASH/KANT during the years consideration and the implementa- 2002–2006. He has been at UNCG since 2006. tion of these algorithms. Using these methods, abstract objects are better His research is in computational number theory. He understood and many new examples is particularly interested in algorithms for local fields can be found. These can then be used and computational class field theory. He is also in- to further develop the theory. vestigating the distribution of the zeros of the deriva- tives of the Riemann Zeta function.

Dr. Filip Saidak email: [email protected] Associate Professor web page: http://www.uncg.edu/~f_saidak Office: Petty 104

Dr. Saidak received his B.Sc. at The University of Auckland in New Zealand, and his M.Sc. and Ph.D. Analytic Number Theory in number theory at Queen’s University in Ontario, Analytic number theory is a branch Canada. He then held postdoctoral and visiting po- of number theory that employs sitions at the University of Calgary (Alberta), Uni- methods and techniques from com- versity of Missouri (MO), Macquarie University (Syd- plex analysis in order to investigate ney), and Wake Forest University (NC). arithmetic properties of the integers and the distribution of prime num- He is mainly interested in classical questions con- bers. The Prime Number Theorem cerning prime numbers and their distribution, which (1896) is the central result of this he investigates using analytic and probabilistic meth- subject, while the Goldbach Conjec- ods. A special topic of his interest is the location of ture (1742) and the Riemann Hypoth- zeros of the Riemann zeta function and its deriva- esis (1859) are the most famous open tives; others include problems centering around the problems. differences between consecutive primes, distribution and divisibility properties of primes of special forms, as well as values of various arithmetical functions. Dr. Brett Tangedal email: [email protected] Associate Professor web page: http://www.uncg.edu/~batanged Office: Petty 112

Dr. Tangedal earned his Ph.D. from the University of California at San Diego in 1994 under the direction of Algebraic Number Theory Harold Stark. After holding various positions at the Algebraic numbers were first studied University of Vermont, Clemson University, and the in a systematic way by Gauss and College of Charleston, he joined the faculty at UNCG Kummer in order to formulate and in 2007. prove the laws of higher reciprocity in number theory. The study of alge- His research interests lie in algebraic number theory braic number theory for its own sake, with a particular emphasis on explicit class field the- and for its connections to such areas ory. This involves the constructive generation of rela- as algebraic geometry and cryptogra- tive abelian extensions of a given number field using phy, forms a vast domain of current the special values of certain transcendental complex research. and p-adic valued functions. Almost all of his re- search to date is concerned with a system of conjec- tures, due to Stark and others, that make class field theory explicit in a precise manner using the special values mentioned above.

Dr. Dan Yasaki email: [email protected] Associate Professor web page: http://www.uncg.edu/~d_yasaki Office: Petty 146

Dr. Yasaki has an M.A. (2000) and Ph.D. (2005) from Duke University under the supervision of L. Saper. Modular Forms After a three year post-doc at the University of Mas- ‘Modular forms are functions on the sachusetts, he has been part of the UNCG faculty complex plane that are inordinately since 2008. In 2009, he was a Visiting Scholar at symmetric. They satisfy so many in- the University of Sydney, where he developed a com- ternal symmetries that their mere ex- puter package for computing Bianchi cusp forms in istence seem like accidents. But they Magma. do exist.’ (Barry Mazur) His research interests are in the area of modular The theory of modular forms was forms, particularly the connection between explicit first developed in connection with reduction theory of quadratic forms and the com- the theory of elliptic functions in the putation of Hecke data for automorphic forms. Re- first half of the 19th century. They cent work has focused on producing new examples show up in many different areas of of cusp forms over number fields of small degree. mathematics, revealing connections that are still not fully understood. UNCG Summer School in Computational Number Theory http://www.uncg.edu/math/numbertheory/summerschool

From 2012 to 2016, the number theory group at UNCG has organized five summer schools — hosting a total of 132 participants. External funding was received from the National Security Agency, the National Science Foundation, and the Number Theory Foundation. Further summer schools are planned for the coming years. The summer school aims to complement the traditional training that graduate students receive by exposing them to a constructive and computational approach to objects in number theory.

Posters for the UNCG Summer School in Computational Number Theory 2012, 2013, 2014, 2015, and 2016. On a typical day of the summer school, external and local experts give talks in the morning and in the afternoon students solve problems related to this material. The talks early in the week in- troduce the students to the subject. Talks later in the week cover related areas of current research and unsolved problems.

Mathematics Programs at UNCG

• Ph.D. in Computational Mathematics • B.S. in Mathematics • M.A. in Mathematics • B.A. in Mathematics

Contact [email protected] or visit us on the web at http://www.uncg.edu/math.