Chairs
Professor George E Andrews, Pennsylvania State University, USA
Professor George E Andrews, Pennsylvania State University, USA
George E. Andrews was born in Salem, Oregon. He received his B.S. and M.A. degrees from Oregon State University, and his Ph.D. from the University of Pennsylvania. Andrews is Evan Pugh University Professor in Mathematics at the Pennsylvania State University. An expert on q-series, he has written and published more than 300 papers and has just completed (jointly with Bruce Berndt) the fifth and final volume explicating Ramanujan's Lost Notebook. Andrews was elected to the American Academy of Arts and Sciences in 1997, and to the National Academy of Sciences (USA) in 2003. He was awarded an honorary professorship at Nankai University in 2008. In 2009 he became a SIAM Fellow. He holds honorary degrees from Parma, Florida, Waterloo, Illinois and SASTRA University (India). Andrews served as President of the American Mathematical Society from February 1, 2009 to January 31, 2011, and became a Fellow of the AMS in 2012
13:30-14:00
Srinivasa Ramanujan and signal-processing problems
Professor P P Vaidyanathan, California Institute of Technology, USA
Abstract
In 1918, Srinivasa Ramanujan introduced a summation, known today as the Ramanujan-sum. He used this to express several arithmetic functions in the form of infinite series. For many years this sum was used by other mathematicians to prove important results in number theory. In recent years, engineers and physicists have found applications of this sum in digital signal processing, especially in identifying periodic components of signals with integer periods. This has also led to generalization of the Ramanujan-sum decomposition in several directions, opening up new theory as well as applications in the representation and identification of periodic signals. Key to these developments is the so-called Ramanujan subspace which is a space containing a specific class of integer-periodic functions. These subspaces turn out to be fundamental in the representation of arbitrary periodic sequences. The new developments include Ramanujan dictionaries for sparse representation of periodic signals, Farey dictionaries for the same, and Ramanujan filter banks for tracking periodicity as it evolves and changes in time. The well known subspace algorithm for parameter identification called the multiple signal classification (MUSIC) algorithm has also been generalized to obtain the so-called iMUSIC algorithm for integer periodicities. The results have applications in the study of periodic signals with inherently integer periods, such as segments of DNA and protein sequences among others. This talk gives an overview of the theory and outlines some of these applications.
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Professor P P Vaidyanathan, California Institute of Technology, USA
Professor P P Vaidyanathan, California Institute of Technology, USA
P. P. Vaidyanathan is the Kiyo and Eiko Tomiyasu Professor of Electrical Engineering at the California Institute of Technology. He received the B.Tech. and M.Tech. degrees from the University of Calcutta, India, and the Ph.D degree in Electrical and Computer Engineering from the University of California, Santa Barbara. He has authored over 500 papers and four books in the signal processing area, and has received prizes for excellence in teaching at Caltech multiple times. He is a Fellow of the IEEE, recipient of the F. E. Terman Award of the American Society for Engineering Education, and past distinguished lecturer for the IEEE Signal Processing Society. He has received the IEEE Signal Processing Society's Technical Achievement Award (2002), Education Award (2012), and Society Award (2016), and the 2016 IEEE Gustav Robert Kirchhoff Award (a Technical Field award) for ``fundamental contributions to digital signal processing
14:15-14:45
From Ramanujan graphs to Ramanujan complexes
Professor Alex Lubotzky, Hebrew University, Israel
Abstract
Ramanujan graphs are k-regular graphs with all non trivial eigenvalues are bounded (in absolute value) by 2SR(k-1). They are optimal expanders (from spectral point of view). Explicit constructions of such graphs were given in the 80's as quotients of the Bruhat-Tits tree associated with GL(2) over a local field F, by the action of suitable congruence subgroups of arithmetic groups.
The spectral bound was proved using works of Hecke, Deligne and Drinfeld on the 'Ramanujan conjecture' in the theory of automorphic forms.
The work of Lafforgue, extending Drinfeld from GL(2) to GL(n), opened the door for the construction of Ramanujan complexes as quotients of the Bruhat-Tits buildings associated with GL(n) over F. This way one gets finite simplical complxes which on one hand are 'random like' and at the same time have strong symmetries. These seemingly contradicting properties make them very useful for constructions of various external objects.
Recently various applications have been found in combinatorics, coding theory and in relation to Gromov's overlapping properties. The last application is probably the most suprising one, showing that there exist d-dimensional BOUNDED DEGREE simplical complexes X with the following remarkable property: for every continuous map from X to the d-dimensional Euclidean space, there is a point which is covered by a fixed fraction of the d-dimensional faces of X.
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Professor Alex Lubotzky, Hebrew University, Israel
Professor Alex Lubotzky, Hebrew University, Israel
Professor Alex Lubotzky is the Weil Professor of Mathematics at the Einstein Institute of the Hebrew University. He has authored 4 books and over 130 research papers. His name appears on the ISI list of the most cited scientists in the world. He has been a visiting professor at Stanford, Columbia, the University of Chicago, Yale, NYU, ETH and the Institute for Advanced Study at Princeton. Lubotzky received a number of Israeli and international prizes, including the Israel prize 2018. He is an Honorary Foreign Member of the American Academy of Arts and Sciences, and of the Israel Academy of Science. In 2006 he received an honorary degree from the University of Chicago for his contributions to modern mathematics. From 1996 - 1999 he served as a member of the Israeli Knesset.
15:30-16:00
Optimal point configurations and Fourier interpolation formulas
Professor Maryna Viazovska, Ecole Polytechnique Federale de Lausanne, Switzerland
Abstract
The sphere packing problem asks for the densest configuration of non-intersecting open unit balls at the Euclidean space. This classical geometric problem is solved only in dimensions 1, 2, 3, 8, and 24. In this talk, Professor Viazovska will present a solution of the sphere packing problem in dimensions 8 and 24. It seems that each dimension has its own features and requires a different approach. One method of estimating the density of a sphere packing from above was suggested by H Cohn and N Elkies in 2003. Their approach is based on Fourier optimization. Namely, they showed that the existence of a function satisfying certain inequalities for the function itself and for its Fourier transform leads to an upper bound of the density of a sphere packing. Using this method Cohn and Elkies were able to prove almost sharp bounds in dimensions 8 and 24. Professor Viazovska will show that functions providing exact bounds can be constructed explicitly. Moreover, Professor Viazovska will present a new type of Fourier interpolation formula.
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Professor Maryna Viazovska, Ecole Polytechnique Federale de Lausanne, Switzerland
Professor Maryna Viazovska, Ecole Polytechnique Federale de Lausanne, Switzerland
Maryna Viazovska was born in Kiev in Ukraine in 1984. She obtained her Bachelor degree in Mathematics in 2005 from Kiev National University and a Master's degree in 2007 from the University of Kaiserslautern. In 2013 she received her PhD from the University of Bonn. After a postdoctoral position at the Humboldt University in Berlin she joined the faculty of the École Polytechnique Fédérale Lausanne, where she became full professor in 2018.
16:15-16:45
Ramanujan conjecture, Ramanujan graphs and Ramanujan complexes
Professor Wen-Ching Winnie Li, Pennsylvania State University, USA
Abstract
The purpose of this talk is to give a panorama of the three topics in the title carrying Ramanujan's name.
In 1916 Ramanujan predicted that the pth Fourier coeffcient of the weight 12 discriminant function is bounded by 2p^{11/2} for all primes p. This conjecture is now proved for classical holomorphic cuspidal Hecke eigenforms by the work of Eichler-Shimura, Deligne, and Deligne-Serre. After the connection between the classical modular forms and representations of GL(2) over Q was understood, the Ramanujan conjecture is expected to hold in great generality, revealing deep arithmetic properties of automorphic forms. In particular, it is established for automorphic cuspidal representations of GL(n) over global function fields by Drinfeld and L La orgue.
The Bruhat-Tits building attached to PGL(n + 1) over a p-adic field F is an n-dimensional simplicial complex on which PGL(n+1, F) acts. The orbit space of a discrete cocompact subgroup Γ of PGL(n+1, F) is a finite n-dimensional simplicial complex X_{Γ}. The validity of the Ramanujan conjecture is used to select infinitely many such Γ so that the spaces X_{Γ} are spectrally optimal, called Ramanujan graphs (for n = 1) and Ramanujan complexes (for n ≥ 2). There are zeta functions counting closed geodesics in X_{Γ} of given dimension and type. The Ramanujan graphs/complexes are those whose zeta functions satisfy the Riemann Hypothesis. The prime geodesic theorems for X_{Γ} follow from the analytic behaviour of the associated zeta functions, and for Ramanujan graphs/complexes, one also obtains a good error estimate, similar to what happens for prime numbers.
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Professor Wen-Ching Winnie Li, Pennsylvania State University, USA
Professor Wen-Ching Winnie Li, Pennsylvania State University, USA
W. Li is a Distinguished Professor of Mathematics at the Pennsylvania State University. Her research focuses on automorphic forms, number theory, and their applications to combinatorics. Her thesis work on the newform theory was cited in Andrew Wiles' proof of Fermat's Last Theorem. She applied results in automorphic forms and number theory to construct efficient communication networks called Ramanujan graphs and Ramanujan complexes. Her work on the arithmetic of modular forms for noncongruence subgroups has revitalized the field. Li earned a B.S. degree at National Taiwan University and a Ph.D. degree from the University of California at Berkeley. Prior to Penn State, she held positions at the University of Illinois at Chicago, the Institute for Advanced Study at Princeton, and Harvard University. She was the Director of Taiwan's National Center for Theoretical Sciences 2009-2014. She received the 2010 Chern Prize in Mathematics and was named a Fellow of the American Mathematical Society in 2013.