Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, April 27, 2017.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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1  2  3  4                      1       1  2  3  4  5  6
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Thursday, April 27, 2017

11:00 am in 241 Altgeld Hall,Thursday, April 27, 2017

#### A formula for the partition function that "counts"

###### Andrew Sills (Georgia Southern University)

Abstract: A partition of an integer n is a representation of n as a sum of positive integers where the order of the summands is considered irrelevant. Thus we see that there are five partitions of the integer 4, namely 4, 3+1, 2+2, 2+1+1, 1+1+1+1. The partition function p(n) denotes the number of partitions of n. Thus p(4) = 5. The first exact formula for p(n) was given by Hardy and Ramanujan in 1918. Twenty years later, Hans Rademacher improved the Hardy-Ramanujan formula to give an infinite series that converges to p(n). The Hardy-Ramanujan-Rademacher series is revered as one of the truly great accomplishments in the field of analytic number theory. In 2011, Ken Ono and Jan Bruinier surprised everyone by announcing a new formula which attains p(n) by summing a finite number of complex numbers which arise in connection with the multiset of algebraic numbers that are the union of Galois orbits for the discriminant -24n + 1 ring class field. Thus the known formulas for p(n) involve deep mathematics, and are by no means "combinatorial" in the sense that they involve summing a finite or infinite number of complex numbers to obtain the correct (positive integer) value. In this talk, I will show a new formula for the partition function as a multisum of positive integers, each term of which actually counts a certain class of partitions, and thus appears to be the first truly combinatorial formula for p(n). The idea behind the formula is due to Yuriy Choliy, and the work was completed in collaboration with him. We will further examine a new way to approximate p(n) using a class of polynomials with rational coefficients, and observe this approximation is very close to that of using the initial term of the Rademacher series. The talk will be accessible to students as well as faculty, and anyone interested is encouraged to attend!

12:30 pm in 464 Loomis Laboratory,Thursday, April 27, 2017

#### Entanglement branes in a two-dimensional string theory

###### Gabriel Wong (Virginia Physics)

Abstract: There is an emerging viewpoint that classical spacetime emerges from highly entangled states of more fundamental constituents. In the context of AdS/CFT, these fundamental constituents are strings, with a dual description as a large-N gauge theory. To understand entanglement in string theory, we consider the simpler context of two-dimensional large-N Yang-Mills theory, and its dual string theory description due to Gross and Taylor. We will show how entanglement in the gauge theory is described in terms of the string theory as thermal entropy of open strings whose endpoints are anchored on a stretched entangling surface which we call an entanglement brane.

1:00 pm in 243 Altgeld Hall,Thursday, April 27, 2017

#### Compressed Learning II

###### Marius Junge (UIUC)

Abstract: Part II

2:00 pm in 241 Altgeld Hall,Thursday, April 27, 2017

#### MacMahon's partial fractions

###### Andrew Sills (Georgia Southern University)

Abstract: A. Cayley used ordinary partial fractions decompositions of $1/[(1-x)(1-x^2)\ldots(1-x^m)]$ to obtain direct formulas for the number of partitions of $n$ into at most $m$ parts for several small values of $m$. No pattern for general m can be discerned from these, and in particular the rational coefficients that appear in the partial fraction decomposition become quite cumbersome for even moderate sized $m.$ Later, MacMahon gave a decomposition of $1/[(1-x)(1-x^2). . .(1-x^m)]$ into what he called "partial fractions of a new and special kind" in which the coefficients are "easily calculable numbers" and the sum is indexed by the partitions of $m$. While MacMahon's derived his "new and special" partial fractions using "combinatory analysis," the aim of this talk is to give a fully combinatorial explanation of MacMahon's decomposition. In particular, we will observe a natural interplay between partitions of $n$ into at most $m$ parts and weak compositions of $n$ with $m$ parts.

4:00 pm in 245 Altgeld Hall,Thursday, April 27, 2017