The Park City Math Institute 2016 Summer Undergraduate Faculty Program met for the purpose of composing guidelines for undergraduate programs in data science. The group consisted of 25 undergraduate faculty from a variety of institutions in the United States, primarily from the disciplines of mathematics, statistics, and computer science. These guidelines are meant to provide some structure for institutions planning for or revising a major in data science.

We investigate connections between the symmetries (automorphisms) of a graph and its spectral properties. Whenever a graph has a symmetry, i.e. a nontrivial automorphism φ, it is possible to use φ to decompose any matrix M ∈ C n×n appropriately associated with the graph. The result of this decomposition is a number of strictly smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix M. Some of the matrices that can be decomposed are the graph's adjaceny matrix, Laplacian matrix, etc. Because this decomposition has connections to the theory of equitable partitions it is referred to as an equitable decomposition. Since the graph structure of many real-world networks is quite large and has a high degree of symmetry, we discuss how equitable decompositions can be used to effectively bound both the network's spectral radius and spectral gap, which are associated with dynamic processes on the network. Moreover, we show that the techniques used to equitably decompose a graph can be used to bound the number of simple eigenvalues of undirected graphs, where we obtain sharp results of Petersdorf-Sachs type.

We consider the graph Gn with vertex set V pGnq " t1, 2, . . . , nu and ti, ju P EpGnq if and only if 0 ă |i´j| ď 2. We call Gn the straight linear 2-tree on n vertices. Using ∆-Y transformations and identities for the Fibonacci and Lucas numbers we obtain explicit formulae for the resistance distance r Gn pi, jq between any two vertices i and j of Gn. To our knowledge tGnu 8 n"3 is the first nontrivial family with diameter going to 8 for which all resistance distances have been explicitly calculated. Our result also gives formulae for the number of spanning trees and 2-forests in a straight linear 2-tree. We show that the maximal resistance distance in Gn occurs between vertices 1 and n and the minimal resistance distance occurs between vertices n{2 and n{2`1 for n even (with a similar result for n odd). It follows that rnp1, nq Ñ 8 as n Ñ 8. Moreover, our explicit formula makes it possible to order the non-edges of Gn exactly according to resistance distance, and this ordering agrees with the intuitive notion of distance on a graph. Consequently, Gn is a geometric graph with entirely different properties than the random geometric graphs investigated in [6]. These results for straight linear 2-trees along with an example of a bent linear 2-tree and empirical results for additional graph classes convincingly demonstrate that resistance distance should not be discounted as a viable method for link prediction in geometric graphs. DEFINITION 1 (straight linear 2-tree). A straight linear 2-tree is a graph G n with n vertices with adjacency matrix that is symmetric, banded, with the first and second subdiagonals equal to one, and first and second superdiagonals equal to one, and all other entries equal to zero. See Figure 1.Remark 2. We observe that G n is a geometric graph. This can easily be seen by placing the vertices so that all of the triangles in Figure 1 are equilateral.

Abstract. We prove that the total descendant potential functions of the theory of FanJarvis-Ruan-Witten for D 4 with symmetry group J and D T 4 with symmetry group G max , respectively, are both tau-functions of the D 4 Kac-Wakimoto/Drinfeld-Sokolov hierarchy. This completes the proof, begun in [FJR], of the Witten Integrable Hierarchies Conjecture for all simple (ADE) singularities.

Abstract. We prove the Landau-Ginzburg Mirror Symmetry Conjecture at the level of (orbifolded) Frobenius algebras for a large class of invertible singularities, including arbitrary sums of loops and Fermats with arbitrary symmetry groups. Specifically, we show that for a quasi-homogeneous polynomial W and an admissible group G within the class, the Frobenius algebra arising in the FJRW theory [FJR07b]