LSU

Mathematics

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 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 2018:** **

**Ken Goodearl**(April 16-20, 2018)

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 accomodate 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.

**Renling Jin**(March 19-21, 2018)

Dr. Jin has many research interests including nonstandard analysis, set theory, model theory. After recieving his doctorate in Mathematical Logic from the University of Wisconsin - Madison, he was Charles B. Morrey Jr. Assistant Professor at UC Berkely 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.

The audience are not assumed to have prior knowledge of nonstandard analysis.

**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 Godel

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 Godel’s Incompleteness Theorem. Godel’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.