Department of

# Mathematics

Seminar Calendar
for Commutative Ring Theory Seminar events the year of Tuesday, November 14, 2017.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
     October 2017          November 2017          December 2017
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4  5  6  7             1  2  3  4                   1  2
8  9 10 11 12 13 14    5  6  7  8  9 10 11    3  4  5  6  7  8  9
15 16 17 18 19 20 21   12 13 14 15 16 17 18   10 11 12 13 14 15 16
22 23 24 25 26 27 28   19 20 21 22 23 24 25   17 18 19 20 21 22 23
29 30 31               26 27 28 29 30         24 25 26 27 28 29 30
31


Thursday, January 26, 2017

3:00 pm in 243 Altgeld Hall,Thursday, January 26, 2017

#### Organizational Meeting

Thursday, March 2, 2017

3:00 pm in 243 Altgeld Hall,Thursday, March 2, 2017

#### Multidimensional Persistent Homology

###### Hal Schenck (UIUC Math)

Abstract: A fundamental tool in topological data analysis is persistent homology, which allows detection and analysis of underlying structure in large datasets. Persistent homology (PH) assigns a module over a principal ideal domain to a filtered simplicial complex. While the theory of persistent homology for filtrations associated to a single parameter is well-understood, the situation for multifiltrations is more delicate; Carlsson-Zomorodian introduced multidimensional persistent homology (MPH) for multifiltered complexes via multigraded modules over a polynomial ring. We use tools of commutative and homological algebra to analyze MPH, proving that the MPH modules are supported on coordinate subspace arrangements, and that restricting an MPH module to the diagonal subspace $V(x_i-x_j | i \ne j)$ yields a PH module whose rank is equal to the rank of the original MPH module. This gives one answer to a question asked by Carlsson-Zomorodian. This is joint work with Nina Otter, Heather Harrington, Ulrike Tillman (Oxford).

Thursday, October 19, 2017

3:00 pm in 243 Altgeld Hall,Thursday, October 19, 2017

#### Koszul almost complete intersections

###### Matthew Mastroeni (UIUC Math)

Abstract: Let $R = S/I$ be a quotient of a standard graded polynomial ring $S$ by an ideal $I$ generated by quadrics. If $R$ is Koszul, a question of Avramov, Conca, and Iyengar asks whether the Betti numbers of $R$ over $S$ can be bounded above by binomial coefficients on the minimal number of generators of $I$. Motivated by previous results for Koszul algebras defined by three quadrics, we give a complete classification of the structure of Koszul almost complete intersections and, in the process, give an affirmative answer to the above question for all such rings.

Thursday, November 16, 2017

3:00 pm in 243 Altgeld Hall,Thursday, November 16, 2017

#### A Counterexample to the Weitzenböck Conjecture in Characteristics p > 2

###### Stephen Maguire (UIUC Math)

Abstract: Weitzenböck’s Theorem states that a representation $\mu: \mathbb{G}_a \to \mathrm{GL}(V_n)$ has a finitely generated ring of invariants $k[X]^{\mathbb{G}_a}$ if the field $k$ is an algebraically closed field of characteristic zero. In this talk, we produce a representation $\mu : \mathbb{G}_a \to \mathrm{GL}(V_6)$ over an algebraically closed field $k$ of characteristic $p > 2$ such that the ring of invariants $k[x_1, \dots , x_6]^{\mathbb{G}_a}$ is not a finitely generated $k$-algebra. In order to do this, we reduce this problem to a curve counting problem, and then use this reduction to further reduce this problem to a problem about the support of a bi-graded ring.