LSU
| Mathematics

The goal of the LSU Mathematics Student Colloquium is to give both undergraduate and graduate students the opportunity to hear and interact with speakers from across the country, providing information and perspective possibly relevant to their graduate and postgraduate careers.

Each invited speaker will spend several days at LSU, giving multiple talks and making himself or herself available to undergraduates.

Talks are not confined to the math department but open to everyone. Those majoring in related fields are encouraged to come.

We are a newly charted LSU student organization (constitution and bylaws). We are munificently funded by the Student Government Programming, Support, and Initiatives Fund (PSIF), and the LSU Mathematics Department. We are grateful for the generous past funding made by grants from the National Science Foundation (a VIGRE grant) and the Board of Regents.

Spring 2017:

**Dmytro (Dima) Arinkin**(April 3--5, 2017)

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.

**Undergraduate Talk** (277 Lockett; Monday, Apr. 3, 2:30-3:30PM)

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

Abstract: 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.

**Graduate Talk** (277 Lockett; Wednesday, Apr. 5, 2:30-3:30PM)

Title: Connections with a small parameter.

Abstract: 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).

**Richard Hammack**(March 7--9, 2017)

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.

**Undergraduate Talk** (244 Lockett; Tuesday, Mar. 7, 2:30-3:30PM)

Title: Integrate *THIS*: The mathematics of planimeters

Abstract: 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.

**Graduate Talk** (277 Lockett; Thursday, Mar. 9, 2:30-3:30PM)

Title: Not every graph has a robust cycle basis

Abstract: 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.** **

Fall 2016:** **

**Chelsea Walton**(November 1--9, 2016)

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 Washingtion, 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.

**Undergraduate Talk** (239 Lockett; Wednesday, Nov. 2, 12:30-1:30PM)

Title: 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.**Graduate Talk** (9 Lockett; Thursday, Nov. 3, 3:30-4:30PM)

Title: 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.

**Kiran Kedlaya**

(September 19--20, 2016)

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.

**Undergraduate Talk** (137 Lockett; Monday, Sept. 19, 12:30-1:30PM)

Title: 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.

**Graduate Talk** (103 Coates; Monday, Sept. 20, 12:30-1:30PM)

Title: 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.

**Budapest Semester Informational Meeting** (The Keisler Lounge in Lockett; Monday, Sept. 19, 2:30PM)

Study abroad opportunities in mathematics: The Budapest Semesters in Mathematics (BSM) program has been providing North American students the opportunity to spend one or two semesters learning mathematics in the "Hungarian style" for over 30 years. Recently, the Budapest Semesters in Mathematics Education (BSME) was launched to provide similar opportunities for those interested in Hungarian pedagogy. This information session will describe both programs, their similarities and differences, and how to participate.