For more information e-mail: zgersh2@lsu.edu or davidson@math.lsu.edu

## April 16-20, 2018: Ken Goodearl

Dr. Goodearl has research interests in various areas such as noncommutative algebras, quantum groups, and $C^{*}$-algebras. He received from the University of Washington. He has held positions across the US and in Europe, including research positions at the Universitat Passau in Germany and at MSRI in Berkely, CA. He has written several books and over a hundred journal articles. He is currently a professor at UC Santa Barbara.

**Undergraduate Talk** (239 Lockett; Monday, April 16, 3:30-4:20 PM)

Title: How fast does a group or an algebra grow?

Abstract: An algebraic object "grows" from a set $X$ of generators as larger and larger combinations of those generators are taken. In the case of a group $G$, this means taking longer and longer products of generators and their inverses. For an algebra $A$ (a ring containing a field), it means taking linear combinations of longer and longer products of the generators. The growth rate of $G$ is the rate at which the number of elements that can be obtained as products of at most $n$ generators and their inverses grows with increasing $n$. The growth rate of $A$ amounts to counting dimensions of subspaces spanned by products of at most $n$ generators. These rates of growth provide important measures for the "complexities" of $G$ and $A$, respectively. They may be given by a polynomial function or an exponential function, but there are quite a few surprises -- rates like a polynomial with degree $\sqrt 5$ can occur, or rates in between polynomial and exponential functions, whereas some other potential rates are ruled out. We will discuss the basic ideas of growth for groups and algebras; the distillation of growth rate into a "dimension" for algebras; and the values that this dimension can take.

**Graduate Talk** (239 Lockett; Wednesday, April 18, 3:30- 4:20 PM)

Title: From dimension to Grothendieck groups and monoids

Abstract: In trying to generalize the concept of "dimension'' from finite dimensional vector spaces to structural size measures for other classes of mathematical objects, one quickly arrives at the idea that such "sizes'' should be elements of some abelian group, so that (at the very least) sizes can be added. The natural group to use in linear algebra is $\bf Z$, but in general there is no obvious group at hand. Grothendieck pointed out how to construct an appropriate group as one satisfying a certain universal property. Typically, one wants to not only add but compare "sizes'', in the sense of inequalities. To accommodate comparisons, a combined structure is needed -- an abelian group which is equipped with a (compatible) partial order relation. On the other hand, demanding subtraction for "sizes'' is sometimes asking too much, and "sizes'' should take values in a monoid rather than a group.

We will introduce the above concepts and constructions in the context of modules over a ring, and we will discuss various examples.

## March 19-21, 2018: Renling Jin

Dr. Jin has many research interests including nonstandard analysis, set theory, model theory. After receiving his doctorate in Mathematical Logic from the University of Wisconsin - Madison, he was Charles B. Morrey Jr. Assistant Professor at UC Berkeley and then a National Science Foundation postdoctoral fellow. Dr. Jin has been the editor of the journal Logic & Analysis since 2007. He is currently a professor at the College of Charleston.

**Graduate Talk** (285 Lockett; Monday, March 19, 1:30-2:30PM)

Title: Can nonstandard analysis produce new standard theorems?

Abstract: The answer is yes. Nonstandard analysis which was created by A. Robinson in 1963 incorporates infinitely large numbers and infinitesimally small positive numbers consistently in our real number system. But the strength of nonstandard analysis in the research of standard mathematics has not seemed to be sufficiently appreciated by mathematical community. In the talk, we will introduce two parts of the work done by the speaker and his collaborators on the standard combinatorial number theory using nonstandard analysis. In each of these two parts new standard theorems that were proved by nonstandard methods will be presented.

**Undergraduate Talk** (277 Lockett; Wednesday, March 21, 10:30-11:30AM)

Title: A taste of logic--from the reasoning of a thief to a painless proof of the incompleteness theorem of Gödel

Abstract: We will present some fun part of mathematical logic including a puzzle, a true paradox, and a fake paradox. The discussion will lead to Gödel’s Incompleteness Theorem. Gödel’s Incompleteness Theorem is well-known but difficult to proof. We will present a heuristic proof of the theorem which should be sufficient to understand the idea of the rigorous proof of the theorem.

## November 6-8, 2017: Peter Nelson

Dr. Nelson is a professor at the University of Waterloo in Ontario, Canada. He received his Ph.D. in Combinatorics and Optimization at the University of Waterloo. He was a Postdoctoral Fellow at the Victoria University of Wellington in New Zealand. His research interests include structural and extremal graph theory, and their links with coding theory and extremal combinatorics. Much of Dr. Nelson’s work settles old conjectures in matroid theory.

Squaring the Square

Abstract: Is it possible to decompose a square into smaller squares of different sizes? The solution to this problem, which has surprising links to graph theory, linear algebra and even physics, was discovered by four undergraduate students at Cambridge University in the 1930's. I will tell the interesting mathematical story that led to this discovery.

How to Draw a Graph

Abstract: Given a network of points and edges that can be drawn in the plane without crossing edges, what is the best way to actually draw it? Can such a network always be drawn with just straight lines? I will discuss and (mostly) prove a beautiful theorem of William Tutte that answers this question using intuitive ideas from physics.

## April 3-5, 2017: Dmytro (Dima) Arinkin

Dima Arinkin is a Professor at the University of Wisconsin, Madison. His work focuses on algebro-geometric questions motivated by the geometric Langlands program. He studies the moduli spaces of vector bundles (possibly with additional structures such as connections) on curves and the categories of sheaves on these spaces. Dima Arinkin has received numerous awards, including a Sloan Research Fellowship and a von Neumann Fellowship from the Institute for Advanced Study. He has a history of involvement with mathematics competitions, both in mentoring, and in winning a gold medal himself at the International Mathematics Olympiad.

What makes a space interesting? (Intro to moduli.)

Roughly speaking, geometry is the study of spaces. Here `space' is a placeholder: different flavors of geometry work with differentiable manifolds (differential geometry), topological spaces (topology), varieties (algebraic geometry, my favorite), and so on.

This leads to a question: should we try to study all spaces, or focus on those we consider `interesting'? And what makes a space interesting? One possible answer to this question is that there are interesting spaces called moduli spaces (the word `moduli' goes back to Hilbert and basically means `parameters'). The special feature is that these spaces parametrize objects of some class: e.g., moduli space of triangles parametrizes triangles, moduli space of differential equations parametrizes differential equations, and so on.

In my talk, I will go over the basics of moduli spaces; in the (unlikely) event that there is some time left, I will talk about the Murphy Law for the moduli spaces due to Ravi Vakil.

Connections with a small parameter.

In my talk, I will start with a classical, and relatively easy, statement about differential equations with a small parameter (due to Wasow) and use a geometric point of view to translate it, first, into a claim about connections on a vector bundle on a Riemann surface, and then into a statement about the geometry of the space of such connections (their `moduli space'). The main point of the talk is the interplay between study of `individuals' (differential equations or bundles with connections) and properties of their `community' (their moduli space).

## March 7-9, 2017: Richard Hammack

Richard Hammack is a professor of mathematics at Virginia Commonwealth University in Richmond. A native of rural southern Virginia, he studied painting at Rhode Island School of Design before an interest in computer graphics and visualization led him to a masters in computer science from Virginia Commonwealth University, and then to a Ph.D. in mathematics from UNC Chapel Hill. Before returning to VCU he taught at UNC, Wake Forest University and Randolph-Macon College. He works mostly in the areas of combinatorics and graph theory.

**I**ntegrate *THIS*: The mathematics of planimeters

A planimeter is a mechanical analog device that evaluates definite integrals. A typical planimeter features a dial and a stylus attached to an arm. As the stylus traverses the boundary of a region, the dial reads off the enclosed area. Planimeters have been mostly forgotten since the advent of computers, but at one time they were fairly commonplace.

I will explain the history and mathematics of planimeters, and I will demonstrate one that I made from two pieces of cast-off junk. It has only one moving part, but it can evaluate any definite integral that it can reach.

Not every graph has a robust cycle basis

The cycle space of a graph G is the vector space (over the 2-element field) whose vectors are the spanning eulerian subgraphs of G, and addition is symmetric difference on edges. As any eulerian subgraph is a sum of edge-disjoint cycles, the cycle space is spanned by the cycles in G, so one can always find a basis of cycles. Such a basis is called a cycle basis for G.

Because their vectors carry combinatorial information, cycle spaces have many applications, and different kinds of cycle bases cater to different kinds of problems. A lot of recent attention has focused on so-called robust cycle bases. Robust cycle bases are known to exist only for a few classes of graphs. Despite this, previously no graph was known to not have a robust cycle basis. We will see that the complete bipartite graphs K_{n,n} have no robust cycle basis when n ≥ 8. This leads to some tantalizingly open questions, particularly for the range 4 < n < 8, but also for general graphs.** **

## November 1-9, 2016: Chelsea Walton

Dr. Walton is a professor at Temple University. She earned her Ph.D. from the University of Michigan, while also working as a visiting student at the University of Manchester. Dr. Walton held postdoc positions at the University of Washington, at the Mathematical Sciences Research Institute, and at MIT. During her time as a postdoc, one of Dr. Walton's focuses was on outreach programs. While at MIT, she taught for the Edge program, was the coordinator for PRIMES circle, and received the Infinite Kilometer Award for outreach. Her mathematical research interests are in noncommutative algebra and representation theory. Traveling has been a big part of Dr. Walton's career, and has sparked her interest in visiting locations all over the world, including Argentina, Morocco, Peru, and Poland.

Hamilton's Quaternions

Abstract: In this talk I will discuss the last great achievement of Sir William Rowan Hamilton- the discovery of the quaternion number system. This discovery was very controversial for its time and nearly drove Hamilton mad! The talk will be full of drama, intrigue, and wonderful mathematics. Some familiarity with complex numbers would help, but is not needed.

Quantum Symmetry

Abstract: Like Hopf algebras? You will after this talk! The aim of this lecture is to motivate and discuss "quantum symmetries" of quantum algebras (i.e. Hopf co/actions on noncommutative algebras). All basic terms will be defined, examples will be provided, along with a brief survey of recent results.

## September 19-20, 2016: Kiran Kedlaya

Dr. Kedlaya is a professor at UC San Diego. He received his Ph.D. in Mathematics from the Massachusetts Institute of Technology. He was awarded an NSF postdoctoral fellowship and held positions at the Mathematical Sciences Research Institute in Berkeley, at the University of California at Berkeley, and at the Institute for Advanced Study in Princeton. A partial list of the prestigious awards he has received include: the Stefan E. Warschawski Endowed Chair, a Alfred P. Sloan Fellowship, a Clay Liftoff Fellowship, a Presidential Early Career Award for Scientists and Engineers, and a Fellowship from the American Mathematical Society. His research interests include p-adic analytic methods, p-adic Hodge theory, algorithms, and applications in computer science. He is also interested in the education and promotion of mathematics. He has been on the USA Mathematical Olympiad committee, the board of directors for the Art of Problem Solving Foundation, and has authored a Putnam Exam problem book.

These talks have been made possible by the Student Government PSIF and by funding from the LSU Math Department.

The ABC Conjecture

Abstract: The ABC conjecture asserts that if A, B, C are three positive integers such that A + B = C, then these three integers cannot between them have "too many" repeated prime factors. The precise statement of the conjecture explains the difference between the fact that there are lots of such triples consisting of perfect squares (Pythagorean triples) but not consisting of higher perfect powers (Fermat's Last Theorem). I'll discuss the precise statement of the conjecture, some appealing consequences of this conjecture in various parts of number theory, and the status of a recent (2012) announcement of a proof.

Computational Number Theory Online: SMC and LMFDB

Abstract: This is more of a demonstration than a talk: I will indicate how to get started with two different but complementary online tools. SageMathCloud (SMC) is a cloud-based version of the Sage computer algebra system, which includes extensive number-theoretic functionality (and plenty of coverage in other areas of mathematics also). The L-Functions and Modular Forms Database (LMFDB) is a website that assembles various tables of number-theoretic objects, like elliptic curves and modular forms, in an easily browsable format that highlights the deep relationships among these objects.

## April 6-8, 2016: Mark Reeder

Mark Reeder is a Professor of Mathematics at Boston College. He received his PhD in 1988 from Ohio State State University, and taught at the University of Oklahoma for nine years before moving to BC in 1997. He studies representation theory and number theory.

Understanding the Hypersphere

A Hypersphere is a sphere in four dimensions. It controls motion in our three dimensional world. Though we feel the effects of the hypersphere, our brains cannot easily visualize the whole of it. In this talk, Algebra, Geometry and Topology will unite to help us understand the Hypersphere.

From the Hypersphere to E8

This talk will be an introduction to compact groups G of continuous symmetries. We will see that Hyperspheres are the bricks in the construction of G, and also that G contains a "principal hypersphere" which knows way more than it should about the topology and group theory of G.

## March 14-18, 2016: Tadele Mengesha

Tadele Mengesha is an Assistant Professor of Mathematics at the University of Tennessee Knoxville. He received his Ph.D. in 2007 from Temple University under the guidance of Yury Grabovsky. He has held postdoctoral positions at LSU under the mentorship of Robert Lipton as well as at Penn State under the mentorship of Qiang Du.

Dr. Mengesha works on the mathematical analysis of nonlocal models and partial differential equations. His recent focus is on the study of the peridynamic model, an integration-based model for the deformation of solids, that is found to be effective in capturing the emergence of singular phenomena such as fracture and cracks. His primary interest is developing the basic analytical underpinnings of peridynamics as well as other nonlocal models. That includes the study of associated function spaces, establishing relevant functional inequalities and implementation of standard variational and asymptotic techniques.

Integration by parts, local and nonlocal

In this talk, I will review the impact of integration by parts on our understanding of the "derivative" of rough functions. Once a property of smooth functions, integration by parts can be used as a defining character of regular functions as well as their derivative in the broader sense. As a consequence, a renewed notion of solution to ordinary and partial differential equations, especially those with irregular source term or irregular coefficients can be formulated. I will discuss some frequently used spaces of functions that can be described with the new notion of regularity. Extension of the definition and application of integration by parts to integral equations, also know as, nonlocal equation will be presented.

Averaged directional difference quotients

We study averaged directional difference quotients of vector fields and their continuity property over several function spaces. A third order tensor field will be used to distinguish appropriate directions in which slopes are averaged. The averaged directional derivatives will be shown to approximate classical notions of derivatives. We will use this approximation property to characterize vector fields in the space of Sobolev, bounded variation, and bounded deformation functions in a unified way.

## February 29, 2016: Sara Billey

Prof. Sara Billey is a world-renowned researcher in algebraic combinatorics. After receiving her PhD from University of California, San Diego under Mark Haiman and Adriano Garsia, she went on to do a postdoc under Nolan Wallach and Richard Stanley. She is now a professor of mathematics at the University of Washington. Prof. Billey has co-chaired several international workshops, including FPSAC 2010 and ones at ICERM and Banff. She is the recipient of many outstanding awards, including being the only academic mathematician in 2000 to receive the Presidential Early Career Award for Scientists and Engineers, which she received in the White House after an invitation from President Bill Clinton himself. Prof. Billey also co-authored the book "Singular Loci of Schubert Varieties" with Lakshmibai, which remains one of the best resources in the area. When she is not doing mathematics, Prof. Billey enjoys playing a variety of sports, as well as playing bridge, flute, and riding her unicycle.Trees, Tanglegrams, and Tangled Chains

Tanglegrams are a special class of graphs appearing in applications concerning cospeciation and coevolution in biology and computer science. They are formed by identifying the leaves of two rooted binary trees. We give an explicit formula to count the number of distinct binary rooted tanglegrams with n matched vertices, along with a simple asymptotic formula and an algorithm for choosing a tanglegram uniformly at random. The enumeration formula is then extended to count the number of tangled chains of binary trees of any length. This includes a new formula for the number of binary trees with n leaves. We also give a conjecture for the expected number of cherries in a large randomly chosen binary tree and an extension of this conjecture to other types of trees. This talk is based on recent joint work with Matjaz Konvalinka and Frederick (Erick) Matsen IV posted at https://arxiv.org/abs/1507.04976 .

Enumeration of Parabolic Double Cosets for Coxeter Groups

Parabolic subgroups $W_I$ of Coxeter systems $(W,S)$ and their ordinary and double cosets $W / W_I$ and $W_I \backslash W /W_J$ appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets $w W_I$, for $I \subseteq S$, forms the Coxeter complex of $W$, and is well-studied. In this talk, we look at a less studied object: the set of all parabolic double cosets $W_I w W_J$ for $I, J \subseteq S$.

Each double coset can be presented by many different triples $(I,w,J)$. We describe what we call the lex-minimal presentation and prove that there exists a unique such choice for each double coset. Lex-minimal presentations can be enumerated via a finite automaton depending on the Coxeter graph for $(W,S)$.

In particular, we present a formula for the number of parabolic double cosets with a fixed minimal element when $W$ is the symmetric group $S_n$. In that case, parabolic subgroups are also known as Young subgroups. Our formula is almost always linear time computable in $n$, and the formula can be generalized to any Coxeter group.

This talk is based on joint work with Matjaz Konvalinka, T. Kyle Petersen, William Slofstra and Bridget Tenner.

## January 25-29, 2016: Tom Braden

Tom Braden received his PhD from Massachusetts Institute of Technology in 1995. He held postdoctoral appointments at the Institute for Advanced Study and Harvard University before moving to the University of Massachusetts, Amherst in 2001, where he is now a full professor. Dr. Braden has also held visiting positions at the Hebrew University, Reed College, and Stony Brook University. He is an expert in the topology of singular algebraic varieties, particularly as they are applied to representation theory and combinatorics. In his spare time he enjoys studying and playing gamelan music of central Java.

Geometry of Machines

One interesting kind of space whose geometry we can study is a configuration space: a space whose points represent possible states in a mechanism or other physical system. Navigating along a path inside the space is then represented by motions of the machine. Some quite complicated and high dimensional spaces which cannot be visualized directly can be explored very concretely in this way.

I will focus mainly on configuration spaces of planar bar-and-joint machines, which are two-dimensional machines made from rigid bars, hinges, and anchors. Amazingly, a theorem of Kapovich and Milson says roughly that any manifold can appear as (part of) the configuration space of such a machine.

Deformations in Topology and Algebra

The idea of deformation appears all over mathematics. In its most basic form, one takes an object and fits it into a family of related objects parametrized by some auxiliary variables. When the family varies nicely enough, the entire family can have nicer properties than the original object did.

This talk will present a few interesting settings from topology and algebra in which this idea works nicely. In the realm of topology, the equivariant cohomology ring of a space with a torus action is a deformation of the ordinary cohomology ring. In algebra, one can view polynomial differential operators as a deformation of ordinary polynomial functions, and a useful way to study certain representations of Lie algebras is by deforming the action of a Cartan subalgebra. These ideas will be presented via concrete examples and a minimum of technical machinery.

## November 18-19, 2015: Mark Goresky

Mark Goresky received his Ph.D. from Brown University in 1976. He spent two years as a C.L.E. Moore Instructor at MIT, followed by three years as Assistant Professor at the University of British Columbia. He moved to Northeastern University in 1980 where he eventually became a full professor with a joint appointment in the Department of Mathematics and in the College of Computer Science. He has been a "long term member" at the Institute for Advanced Study in Princeton since 1994.

Professor Goresky is best known for his joint work with Robert MacPherson, including their discovery of intersection cohomology and their development of stratified Morse theory. His recent work involves automorphic forms and Langlands' program. He also maintains an interest in electrical engineering and computer science. He recently published a book "Algebraic Shift Register Sequences" (Cambridge University Press) together with Andrew Klapper, who is currently at the University of Kentucky.

Professor Goresky has received a number of academic honors and awards. He has served on a wide range of national and international scientific panels and on the Editorial boards of a number of top journals. In his spare time he flies radio controlled electric powered model airplanes and helicopters.

A glamorous Hollywood star, a renegade composer, and the mathematical development of spread spectrum communications

During World War II Hedy Lamarr, a striking Hollywood actress, together with George Antheil, a radical composer, invented and patented a secret signaling system for the remote control of torpedoes. The ideas in this patent have since developed into one of the ingredients in modern digital wireless communications. The unlikely biography of these two characters, along with some of the more modern developments in wireless communications will be described.

Modular forms and beyond

Several examples of the spectacular coincidences in number theory that can be "explained" using elliptic curves and modular forms will be described.

A plan to find (and prove) higher dimensional generalizations of these phenomena was mapped out 40 years ago by Robert Langlands. Since then, Langlands "program" has occupied the attentions of hundreds of talented mathematicians in what surely must be one of the grandest mathematical gestures in history. Today, much of Langlands' plan is nearing completion, but many mysteries still remain. Some of the ingredients in this vast circle of ideas will be described.

## November 2-3, 2015: Clayton Shonkwiler

Clayton Shonkwiler is an Assistant Professor of Mathematics at Colorado State University. He participated in the 2002 REU at LSU as an undergraduate and received his Ph.D. in 2009 from the University of Pennsylvania under Dennis DeTurck and Herman Gluck. He was a visiting researcher at the Isaac Newton Institute and the recipient of the University of Georgia's 2014 Postdoctoral Research Award before coming to Colorado State. Dr. Shonkwiler is an expert in the study of random knots and more generally of random walks subject to topological constraints, which are used to model biologically significant ring and network polymers, and his research synthesizes differential and symplectic geometry with topology, probability, and computation. He also enjoys creating mathematical art.

15 Views of the Hypersphere

The usual sphere in 3-dimensional space---meaning a hollow sphere, like a basketball--- is such a simple and familiar object that we don't really need to think about how to visualize it. The corresponding shape in 4-dimensional space, called the hypersphere, is a very natural mathematical object which is also important in general relativity and other areas of physics. Of course, it's a bit harder to visualize without 4-dimensional eyes, but I'll present 15 ways of thinking about the hypersphere which will hopefully make it a little more familiar and understandable. There will be lots of pictures and animations.

A Geometric Perspective on Random Walks with Topological Constraints

A random walk in 3-space is a classical object in geometric probability, given by choosing a direction at random, taking a step, and repeating n times. Random walks with topological constraints are collections of random walks which are required to realize the edges of some predetermined multigraph. The simplest nontrivial example is a random polygon, which is just a random walk which is required to form a closed loop.

Since random walks are used to model polymers, random walks with topological constraints give models for polymer conformations with nontrivial topologies. Examples range in complexity from plasmids and viral DNA, which form simple closed loops, to rubbers and gels, which form large networks. Consequently, there is keen interest both in analyzing the geometric probability theory of such walks and in developing fast simulation algorithms.

In this talk I will describe some of the most exciting recent developments in the mathematics of random walks with topological constraints, focusing on the case of random polygons. These developments arise from a purely geometric understanding of the space of possible polygon configurations. Despite being based on rather deep theorems from symplectic/algebraic geometry, this geometric perspective is surprisingly elementary and has already yielded both exact theoretical statements and powerful new algorithms. This includes joint work with Jason Cantarella (University of Georgia) and Tetsuo Deguchi and Erica Uehara (Ochanomizu University, Tokyo).

## April 1-2, 2015: Congming Li

Congming Li is Professor of Applied Mathematics at University of Colorado, Boulder. He obtained his PhD under Louis Nirenberg at the Courant Institute in 1989. He has made important contributions to problems arising in Nonlinear Partial Differential Equations, Calculus of variations, Geometric Analysis and Dynamics of Fluids.

The maximum principle-from simple calculus to frontier research

In this talk, we present the following topics:

- the calculus of the local maxima
- the maximum principle for partial differential equations
- the method of moving planes.

We hope to help our students learn the frontier research on the method of moving planes from the very simple characteristics of local maxima: f'(x_0) = 0 and f''(x_0) <= 0.

In this talk, I will give a brief introduction on the method of moving planes, show some interesting applications of this method, and present some key points in making new advancements.

## March 25-27, 2015: Peter Kuchment

Peter Kuchment is a distinguished professor at Texas A&M, and one of the leading mathematicians working in PDE and mathematical physics. He received his Ph.D. in Mathematics at Kharkov State University, USSR in 1973 and Science Doctor in Mathematics at Academy of Sciences, Kiev, USSR in 1983. Professor Kuchment became Fellow of the American Mathematical Society in 2012, and was the featured speaker at CBMS-NSF conference "Mathematical techniques of medical imaging," University of Texas, Arlington, May 2012. He is also the author and coauthor of many books such as “Floquet Theory for Partial Differential Equations”, and “Introduction to Quantum Graphs”.

Tomography: mathematics of seeing invisible

Everyone has heard of CT, MRI, and Ultrasound medical scanners. Not many, though, know that mathematics plays a major role in obtaining the corresponding images. I will introduce basics of the mathematics of tomographic imaging. No prior knowledge of the subject is assumed.

The nodal count mystery

The beautiful nodal patterns of oscillating membranes, usually called by the (incorrect) name Chladny patterns, have been known for several centuries (Galileo, Leonardo, Hooke) and studied in the last hundred years by many leading mathematicians. In spite of that, many properties of these patterns remain a mystery. We will present the history and a recent advance in the area of counting the nodal domains. No prior knowledge of the subject is assumed.

## March 16-17, 2015: Aaron Lauda

Aaron Lauda is one of the leading mathematicians working in representation theory. He has made major contributions to the subject of categorifications of quantum groups and their representations, and to the study of higher categories. The Khovanov-Lauda-Rouquier algebras are among the most fundamental families of algebras studied nowadays. Prof. Lauda graduated from the University of Cambridge in 2006 under Martin Hyland and took a Ritt Assistant Professorship in Columbia where he won a Sloan Research Fellowship. He joined USC in 2011 as an Assistant Professor where he won an NSF CAREER award in 2012 and was promoted to a full professor in 2013. He co-authored the e-book "Higher-dimensional categories: an illustrated guidebook," which provides an excellent treatment of the techniques used in the field.

Diagrammatic algebra

In this talk we will introduce a calculus of planar diagrams that can be used to represent algebraic structures in a wide variety of contexts. We will start by introducing a diagrammatic framework for studying linear algebra. In this framework, familiar notions such as trace and dimension take on a diagrammatic meaning. We will see how the notion of duality transforms algebraic notions into intuitive manipulations of diagrams. Finally, we will see how this diagrammatic reformulation of linear algebra can be used to study invariants of tangled pieces of string (knot theory).

From ladder diagrams to knot theory

The star of this talk will be an algebraic object called a quantum group. This is an algebraic object closely connected to Lie theory. We will review some basic facts about the representation theory of quantum groups before turning our attention to representations one can construct on certain planar diagrams called 'ladder diagrams'. Our aim is to provide a hands-on introduction to representation theory by exploring how quantum groups impose structure on these ladder diagrams. Though this example may seem somewhat trivial, we will show that viewing ladder diagrams as representations of quantum groups allows us to construct diagrammatically defined knot invariants in an elementary way.

Our aim is to demonstrate that using only the definition of the quantum group and one additional ingredient, one can produce a wealth of diagrammatically defined knot invariants including the Jones polynomial. We do not assume any previous background in topology or Lie theory.

## March 10-12, 2015: Ben Webster

Ben Webster is an Assistant Professor in Mathematics at the University of Virginia. His areas of expertise include geometric representation theory, diagrammatic algebra and low-dimensional topology. He received his BA in 2002 from Simon's Rock College after attending Budapest Semesters in Mathematics and an REU at LSU in 2001. He received his Ph.D. in 2007 and went on postdoctoral positions at the Institute for Advanced Study and MIT as well as tenure track positions at the University of Oregon and Northeastern University before coming to Virginia in 2013. He's been the recipient of an NSF CAREER award and an Alfred P. Sloan fellowship, as well as spending Spring 2014 as a Junior Chair at the Université Denis Diderot (Paris 7).

Untying knots: topology, DNA and coloring

I'm sure you've all seen a very tangled rubber band in your life. You know logically that it must be possible to untangle the band without breaking it but this can be hard to put this knowledge into practice. In fact, if your obnoxious roommate had cut the rubber band, tangled its ends together, and then carefully glued the ends together (isn't he always doing stuff like that?), how would you know? Of course, you'd know you hadn't untangled it yet, but you can try every possible way.

Luckily, mathematicians have your back. They've worried precisely this for a century (they can't trust their roommates either, apparently). Bacteria have worried about it a lot longer (with isomerase playing the role of the prankster roommate). I'll give a basic introduction to the theory of knots and their use in mathematics and biology. In particular, I'll show you how to wise up to your roommate's tricks (though it may take a while).

I'll give an expository talk about the representation theory of symmetric groups over the rational numbers or finite field. While this is a very classical subject, it's one which is very rich, and where there is still a lot to say. I'll emphasize the connection to the emerging field of categorification, and explain in what sense a Lie algebra acts on the category of representations of S_n (for all n).

## February 9-10, 2015: Lihe Wang

Lihe Wang is Professor of Mathematics at the University of Iowa. He obtained his PhD under Luis Caffarelli at the Courant Institute of Mathematics in 1989. After spending several years at Princeton University and UCLA, he took up his current position at the Univ of Iowa. He was awarded a Sloan fellowship in 1994. Some of his areas of expertise include Fully Nonlinear equations, Homogenization, Geometric Analysis, Parabolic PDE's and Fluid Mechanics.

Mathematics: its origin, meaning, and applications

What is behind the common feature of modern sciences? What is the driving force behind the modern productivity? What are the methods to discover information within random numbers?

In this talk, we will try to present the miracle power of calculus with algebra and geometry and make clear the ubiquitous presence of mathematics throughout the discovery of calculus, linear algebra and differential equations.

We will talk about a unified approach to the regularity theory of elliptic and parabolic equations. The introduction of Sobolev spaces, the meaning of embedding theorems, and the geometric and probabilistic meaning of some of the regularity and singularity theory will be discussed.