This is a weekly series of lectures by distinguished visitors and Penn State research faculty. These lectures are instrumental in focusing the interest of MASS participants on various research areas both during their participation in the program and in their future selection of graduate programs.

## 2016

You Think You Know the Pythagorean Theorem? Think Again!

Robert Foote, Wabash CollegeKnowing When to Stop

Theodre Hill, Georgia Institute of TechnologyPartitions, Compositions, and the Excitement of Ramanujan

George Andrews, Penn StateThe length and width of space curves

Mohammad Ghomi, Georgia Institute of TechnologyBilliard table as a mathematician’s playground

Anatole Katok, Penn StateIdentities

Bruce Berndt, University of Illinois Urbana-ChampaignSolving polynomial congruences

Dmitry Fuchs, University of California, DavisHistory of the soliton

Ronald Perline, Drexel UniversityVisual mathematics: From graph partitioning to cellular automata and fair allocation

Yuval Peres, Microsoft ResearchRoots through history

Norbert Schappacher, University of Strasbourg, Francex^n + x + a

Kumar Murty, University of TorontoTwo revealing programs

Richard Schwartz , Brown University

## 2015

A tour of Pritchard Lab

Diane Henderson, Penn StateCongruences for Fishburn Numbers

James Sellers, Penn StateReal enumeration problems

Oleg Viro, Stony Brook UniversityRectangling the square

Aaron Abrams, Washington and Lee UniversityRecurrences for Eisenstein series

Larry Rolen, Penn StateMono-monostatic bodies: the story of the Gömböc

Gábor Domokos, Budapest University of Technology and EcomonicsFlavor of Morse Theory

Augustin Banyaga, Penn StateSphere packing in 2.5 dimensions

Ken Stephenson, University of TennesseeMechanisms of Chaos

Leonid Bunimovich, Georgia Institute of TechnologyLatin squares

Gary Mullen, Penn StateThe mathematics of internet search

Gil Bor, Centro de Investigación en Matemáticas, Mexico

## 2014

Ramanujan, Fibonacci numbers, and Continued Fractions or Why I Took Zeckendorf's Theorem Along On My Last Trip To Canada

George Andrews, Penn State UniversityIntroduction to Entropy

Yuri Suhov, Penn State / University of Cambridge, UKCommutative implies associative?

John Roe, Penn State UniversityThe William Pritchard Fluid Mechanics Laboratory

Vishal Vasan, Penn State UniversityStimulus space geometry and topology from neural activity

Carina Curto, Penn State UniversityRamsey Theory and Dynamics

Vitaly Bergelson, Ohio State UniversityEvolution of resistance to white pine blister rust in high-elevation pines

Simon Tavener, Colorado State UniversityRandom walks: simple and self-avoiding

Greg Lawler, University of ChicagoSolutions to polynomials in two variables

Thomas Tucker, University of RochesterLengthening the edges of a tetrahedron

Richard Schwartz, Brown UniversityPDE models of traffic flow

Alberto Bressan, Penn State University

## 2013

Mathematics through physics

Mark Levi, Penn State UniversitySome applications of geometry to problems in biology

Joel Hass, UC DavisCounting collisions in hard ball gas models and singular geometry of non-positive curvature

Dmitri Burago, Penn State UniversityFrom flapping birds to space telescopes: the mathematics of origami

Robert Lang, Alamo, CAThe William Pritchard Fluid Mechanics Laboratory

Diane Henderson, Penn State UniversityAbout the numbers 12 and 24

Roger Howe, Yale UniversityKirkwood gaps and instability for three body problems

Vadim Kaloshin, University of MarylandPaul Erdos - one of the most influential mathematicians of our times

Krishnaswami Alladi, University of FloridaAdding and counting

Ken Ono, Emory UniversityRamanujan

Ken Ono, N/AMating habits of polynomials

Sarah Koch, University of MichiganWe vote, but do we get what we want?

Don Saari, UC Irvine

## 2012

Manhole Covers and Convex Geometry

Tanya Khovanova, Massachusetts Institute of TechnologyMarkov equation and the most irrational numbers

Alexander Veselov, Loughborough University, UK5-dimensional geometry is not like 2-dimensional geometry (and 3 and 4 are somewhere in between)

Mark Meckes, Case Western Reserve UniversityDonald Coxeter: tales of mystery and imagination

Valentin Ovsienko, University of LyonA new approach to CRC processing

Donald Newhart, NSAOn the use of computers in Mathematics and Science: the good, the bad, and what we should do about them

Sheldon Newhouse, Michigan State UniversityNumber of collisions of n balls on the infinite billiard table after Burago-Ferleger-Kononenko.

Anton Petrunin, Penn StateMathematical Puzzles that S-t-r-e-t-c-h Your Intuition

Peter Winkler, Dartmouth CollegeSome polynomial dynamics systems

Alice Medvedev, UC BerkeleySeeing Invisible: Mathematics of Medical Imaging

Peter Kuchment, Texas A&M UniversityA short look at the long history of the lemniscate of Bernoulli

Joel Langer, Case Western Reserve University

## 2011

Patterns of curves in the plane and piecewise translations

Pat Hooper, CUNYVisual Differential Geometry and the Hyperbolic Plane

Tristan Needham, University of San FranciscoCombinatorics of Total Positivity

Michael Gekhtman, Notre DameCutting a square into triangles

Aaron Abrams, MSRI/Emory UniversityThe Real, Complete Story of Conics

Frank Sottile, Texas A&MGeometry and analysis on fractals

Alexandre Kirillov, University of Pennsylvania“The more things change...”

Chaim Goodman-Strauss, University of ArkansasThe Lost Notebook of Srinivasa Ramanujan

George Andrews, Penn StateTrigonometric identities and the Weyl character formula

Charles Conley, University of North Texas

## 2010

Hard disks in a box: topology and complexity

Yu. Baryshnikov, Bell LabsSymmetry, regular solids and quantum field theory

Adrian Ocneanu, Penn StateThe rental harmony theorem

David Futer, Temple UniversityTilings with rational polygons

Richard Kenyon, Brown UniversityAmenability of the Thompson group $F$ and some elementary properties of $Z^n$

Marl Sapir, Vanderbilt UniversityBehind the Hofstadter's butterfly: the competition between order and chaos

Svetllana Jitomirskaya, University of California IrvineSymmetry: From Triangles to Quantum Physics

Kenneth Gross, University of VermontSome reflections on mirrors

Ron Perline, Drexel UniversityOuter Billiards, Polytope Exchange Maps, and Renormalization

Richard Schwartz, Brown UniversityCapillary surfaces

Robert Finn, Stanford UnivrsityAlexandrov's theorem on development of polyhedron

A. Petrunin, Penn State

## 2009

Tropical geometry

Ilia Itenberg, University of Strasbourg and MSRIUsing elliptic and hyperelliptic curves in pairing-based cryptography

Kirsten Eisentraeger, Penn StateAn Invitation to Minimal Surfaces

Matthias Weber, Indiana UniversityDiscovering and proving theorems by physical reasoning

Mark Levi, Penn StateMinimal surfaces in sub-Riemannian geometry

Luca Capogna, University of Arkansas[0,1] is not a minimality detector of [0,1]^2

Jon Chaika, N/AEverything you wanted to know about 2x2 matrices

Svetlana Katok, Penn StateTopological Network Topology

Robert Ghrist, University of PennsylvaniaTopology and Social Choice

S. Weinberger, University of ChocagoTwo problems in combinatorial plane geometry

Anton Petrunin, Penn StateBraid groups and their applications

Frederick Cohen, University of Rochester

## 2008

Theory of Dynamical Systems on service to Number Theory

A. Gogolev, Penn StateThe four vertex property and topology of surfaces with constant curvature

M. Ghomi, Georgia TechHigher dimensional solutions to low dimensional problems

K. Stolarsky, University of IllinoisThe Devil's Pentagram

R. Schwartz, Brown UniversityFundamental solutions of differential operators and D-module theory

F. Malikov, University of Southern CaliforniaIntroduction to Hypergeometric Functions

V. Retakh, Rutgers UniversityDNA Topology

De Witt Sumners, Florida State UniversityThe discrete square peg problem

I. Pak, University of MinnesotaCantor and Sierpinski, Julia and Fatou: Crazy Topology in Complex Dynamics

R. Devaney, Boston UniversityHow to Tie Your Unicycle in Knots: An Introduction to Legendrian Knot Theory

Josh Sabloff, Haverford College

## 2007

Toy modeling

Tadashi Tokieda, University of Cambridge, N/AAnalysis, Geometry and Arithmetics of fractals

Alexadre Kirillov, University of Pennsylvania, N/APlaying zero-sum games and convex geometry after John Nash

Dmitry Burago, Penn State, N/AGeometry of Polyhedra

Igor Rivin, Temple University, N/APhysical proofs of mathematical theorems

Mark Levi, Penn State, N/AWhat does the future hold for restricted patterns?

Zvezdelina Stankova, Mills College, N/AFrom the Fundamental Theorem of Algebra to astrophysics: a "harmonious" journey

Dmitry Khavinson, University of South Florida, N/ACan understanding cancer be translated into effective anticancer therapy?

Andrei Gudkov, Roswell Park Cancer Institute, Buffalo, NY, N/AHilbert's Tenth over Subrings of Q

Alexandra Shlapentokh, East Carolina University, N/AA Glimpse of Several Complex Variables

Steven Krantz, American Institute of Mathematics, N/AInverse problems in Arithmetic Combinatorics

Alexander Razborov, Institute for Advanced Study, N/A

## 2006

**"Uniform Distribution: from Diophantine approximations to ergodic theorems"**

Vitaly Bergelson, Ohio State University**"Virtual links and ribbon graphs"**

Sergei Chmutov, Ohio State University**"Packing discs in the plane"**

Ronald Graham, University of California, San Diego**Noncrossing partitions, meanders, and the fundamental theorem of algebra**

David Savitt, University of Arizona**Direct methods in the calculus of variations**

Paul Rabinowitz, University of Wisconsin**Classification of Legendrian Knots and Links**

Lisa Traynor, Bryn Mawr College**Bending polyhedra**

Yuri Burago, Steklov Institute (St. Petersburg, Russia)**The Integral Octonions (That Funny Number System)**

Derek Smith, Lafayette College**Topology of algebraic curves and surfaces (Hilbert's 16th problem and beyond)**

Nikita Netsvetaev, St. Petersburg University (St. Petersburg, Russia)

## 2005

**"Gravity-defying phenomena"**

Mark Levi, Penn State University**"How old was Diophantus's son?"**

Don Zagier, MPI-Bonn & College de France**"Bugs, Blobs and Roto-Routers"**

James Propp, University of Wisconsin, Madison**"Schwarzian derivative, projectivity and some applications to rigidity"**

Chengbo Yue, Academy of Mathematical and System Sciences, Chinese Academy of Sciences**"Mathematics of symmetry in 4 dimensions - a sculpture"**

Adrian Ocneanu, Penn State University**"New Solutions to the N-body problem"**

Richard Montgomery, University of California, Santa Cruz**"How many prime factors does a number have?"**

Krishnaswami Alladi, University of Florida**"Venn Diagrams and Symmetric Chain Decompositions"**

Carla Savage, North Carolina State University**"Knots, Tangles, DNA and Quantum Physics"**

Louis Kauffman, University of Illinois at Chicago**"Different facets of linking numbers"**

Boris Khesin, University of Toronto

## 2004

**"Riemann Mapping Theorem via steepest descent"**

Mark Levi, Penn State University**"Volume and area formulas"**

Robert Connelly, Cornell University**"The dynamics of thin flexible things: shaking and breaking"**

Andrew Belmonte, Pennsylvania State University**"Spontaneous symmetry breaking in a 1-D process with variable length"**

Andre Toom, Universidade Federal de Pernambuco, Brazil**"What does the limit shape mean in geometry and combinatorics?"**

Anatoly Vershik, St. Petersburg State University and Steklov Institute**"How many roots does a matrix polynomial equation have?"**

Vladimir Retakh, Rutgers University**"The dynamics and geometry of microscopic structures in piecewise rotations"**

Aroek Goetz, San Francisco State University**"Experiments with triangular billiards"**

Richard Schwartz, University of Maryland**"Polynomials and Knots"**

Walter Neumann, Columbia University**"On the filtering problem of stationary processes"**

Mariusz Lemanczyk, Nicholas Copernicus University, Torun, Poland

## 2003

**"Is it possible to cut a tetrahedron into pieces and to assemble a cube? (Hilbert's 3rd Problem)"**

by Dmitry Fuchs, University of California at Davis**"The Geometry of Spiral Circle Packings"**

by Kenneth Stephenson, University of Tennessee**"Double bubbles"**

by Frank Morgan, Williams College**"The SIAM 100 Digit Challenge: Adventures in Numerical Computing"**

by Stanley Wagon, Macalester College**"Braids, graphs, and robots"**

by Aaron Abrams, University of Georgia**"Irrationality measures using integration by parts"**

by Krishna Alladi, University of Florida**"Pascal Triangles in Modular Arithmetic, Entropy and Algebraic Functions"**

by Hillel Furstenberg, Yale University and Hebrew University of Jerusalem**"Fermionic Integration"**

by A. Vaintrob, University of Oregon

## 2002

**"Laurent phenomenon"**

by Andrei Zelevinsky, Northeastern University**"Topological problems in combinatorics"**

by Robin Forman, Rice University**"On projective duality"**

by Serge Lvovski, Independent University of Moscow**"Asymptotic lines and the Griffiths-Harris rigidity of compact Hermitian symmetric spaces"**

by Joseph Landsberg, Georgia Institute of Technology**"The exponential in mathematics and physics"**

by Adrian Ocneanu, Penn State University**"Impossible Crystals"**

by Roger Penrose, Oxford University and Penn State University**"Some connections between combinatorics, group theory, topology and computer science"**

by Mark Sapir, Vanderbilt University**"The Nash Equilibrium"**

by Mikhail Shubin, Northeastern University**"Iterated exponentials"**

by J. Anderson, Penn State University**"Enumerative geometry"**

by Mira Bernstein, Wellesley

## 2001

**"Family Algebras."**

by Alexandre Kirillov, University of Pennsylvania.**"Integral Lexicographic Codes"**

by John Conway, Princeton University**"Klein's Erlanger Programme"**

by Augustin Banyaga, Penn State University**"Virasoro group and projective structures on the circle: a hidden face of an old coin"**

by V. Ovsienko, CNRS, Centre of Theoretical Physics, Luminy, France**"Vector Calculus and the Topology of Domains in 3-space"**

by Herman Gluck, University of Pennsylvania**"Minimal surfaces, discrete harmonic analysis, and random walks"**

by Alexei Sossinsky, Independent University of Moscow**"Physical proofs of mathematical theorems"**

by Mark Levi, Penn State University**"One hundred years of Hilbert's fourth problem"**

by Juan Carlos Alvarez, New York Polytechnic University**"Loops in**`R`^{3}: New Angles on an Old Topic"

by Bruce Solomon, Indiana University

## 2000

**"Knotted Flowlines"**

by Robert Ghrist, Georgia Tech**"EULER, JACOBI, RAMANUJAN: Interesting Formulae, Beautiful Results and Some Recent Variations."**

by Irwin Kra, State University of New York at Stony Brook**"Aperiodic Tilings and Computation in the Hyperbolic Plane."**

by Chaim Goodman-Strauss, Princeton University**"Search Engines and Measurements on the Modular Group."**

by Leonid Polterovich, Tel Aviv University**"Internet Search and Markov Chains."**

by Michael Brin, University of Maryland**"The Peano Kernel: Convolving Abstract and Applied Mathematics."**

by Michael Gage, University of Rochester**"Flexible Polyhedra"**

by Dmitry Fuchs, University of California at Davis**"The Story of the Rogers-Ramanujan Identitites"**

by G. Andrews, Penn State University**"The Future of Mathematics"**

by Doron Zeilberger, Temple University

## 1999

**"The Meaning of Chaos"**

by Greg Swiatek, Penn State**"When Do Polynomials Have Common Zeros?"**

by Dale Brownawell, Penn State**"Knots and Knot Invariants"**

by Arkady Vaintrob, University of New Mexico**"Hilbert's 16th Problem Near Its Centenary"**

by Yulij Ilyashenko, Cornell**"Replicating Tiles"**

by Viorel Nitica, Notre Dame**"Nightmares and Dreams of Lyapunov: Stability and Semigroups"**

by Yuri Latushkin, University of Missouri**"On the Hilbert-Smith Conjecture"**

by Alexander Dranishnikov, Penn State**"The DNA Geometric Inequality: An Open-ended Story"**

by Serge Tabachnikov, University of Arkansas**"The Banach-Tarski Paradox and Amenable Groups"**

by Misha Guysinsky, Tufts University**"Group Reconstruction"**

by Conrad Plaut, University of Tennessee

## 1998

**"Rational numbers, right triangles, and elliptic curves"**

by Karl Rubin, Stanford University**"Bicycle wheels, Gauss-Bonnet formula and Berry's phase"**

by Mark Levi, Penn State University**"Why representation theory is interesting and useful?"**

by Alexandre Kirillov, University of Pennsylvania**"Playing pool with pi"**

by Gregory Galperin, Eastern Illinois University**"The Life and Work of India's Greatest Mathematician Srinivasa Ramanujan"**

by Bruce Berndt, University of Illinois**"Mathematics of Fractal Images"**

by Yakov Pesin, Penn State University**"An Introduction to K-theory"**

by Paul Baum, Penn State University**"Arithmetic Curves and Triangulations"**

by Fedor Bogomolov, Courant Institute**"Social Life of Curves"**

by Simon Gindikin, Rutgers University

## 1997

Professor Kirillov delivers his colloquia talk,

Why the symplectic and contact geometry are so important?, MASS-97

**"Connecting the dots: the theory and practice of interpolation"**

by Douglas Arnold, Penn State

(This lecture was accompanied by extensive computer demonstrations.

The computer related materials for this lecture are available at

http://www.math.psu.edu/dna/interpolation/interpolation.html.)**"Why the symplectic and contact geometries are so important?"**

by Alexandre Kirillov, University of Pennsylvania**"Mountaineering and quantum mechanics"**

by John Roe, Jesus College, Oxford UK**"Platonic solids, their symmetry groups and Klein's invariant theory"**

by Adrian Ocneanu, Penn State**"Probabilistic strategies in games"**

by Sergei Ivanov, Steklov Institute for Mathematics, Russia**"Invariants of curves by evolution of frames"**

by Richard L. Bishop, University of Illinois at Urbana-Champaign

## 1996

**"The man who loved numbers, Biography of Ramanujan"**

by George Andrews, Penn State**"Stability of the inverted pendulum and topology of the symplectic group"**

by Mark Levi, Rensselaer Polytechnic Institute

(This lecture featured a number of beautiful mechanical experiments and their theoretical explanations.)**"On classification of the finite simple groups"**

by Edward Formanek, Penn State.**"Dynamics of the logistic family"**

by Greg Swiatek, Penn State**"Playing billiard with a help of mathematics"**

by Yuri M. Suhov, University of Cambridge, UK.**"Unprovable theorems and fast-growing functions"**

by Stephen Simpson, Penn State