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for events the day of Thursday, September 12, 2013.

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Thursday, September 12, 2013

12:30 pm in 464 Loomis Laboratory,Thursday, September 12, 2013

Supersymmetric Quantum Mechanics and Index Theorems

Onkar Parrikar (Illinois Physics)

Abstract: Anomalies are quantum violations of classical conservation laws in presence of background Electromagnetic, or Gravitational fields. In studying anomalies, we encounter asymptotic expansions of heat kernels for Dirac operators, and N=1 supersymmetric quantum mechanics turns out to be a powerful physicist's tool for computing these. Interestingly, it also provides a simple way to arrive at the Atiyah- Singer index density for the Dirac operator. In this talk, we will review some of the details about how this works for Dirac operators on curved manifolds. If time permits, we will discuss generalizations to include torsion.

1:00 pm in Altgeld Hall 347,Thursday, September 12, 2013

Length Spectral Rigidity for Strata of Euclidean Cone Metrics

Ser-Wei Fu (UIUC Math)

Abstract: When considering Euclidean cone metrics on a surface induced by quadratic differentials, there is a natural stratification by prescribing cone angles. I will describe a simple method to reconstruct the metric locally using the lengths of a finite set of closed curves. However, the main discussion will be on the surprising result that a finite set of simple closed curves cannot be length spectrally rigid when the stratum has enough complexity. This is extending a result of Duchin-Leininger-Rafi.

2:00 pm in 149 Henry Administration Building,Thursday, September 12, 2013

How to write mathematical papers

Bruce Berndt (UIUC Math)

Abstract: When one begins to write mathematical papers, one encounters difficulties that do not arise in writing ordinary prose. We will address those difficulties and many pitfalls, in which not only beginning mathematicians but also experienced mathematicians easily fall.

3:00 pm in 243 Altgeld Hall,Thursday, September 12, 2013

Local cohomology modules over polynomial rings of prime characteristic - Part I

Yi Zhang (UIUC)

Abstract: Let $R=k[x_1,\cdots, x_n]$ be a polynomial ring over a field $k$ of characteristic $p>0.$ If $I$ is an ideal of $R,$ we denote $H^i_I(R)$ the $i$-th local cohomology module of $R$ with support in $I.$ We discuss an adjointness theorem of Frobenius map. Then we use this property to study the dimension of the associated primes of $H^i_I(R),$ the grading on $H^i_{\mathfrak{m}}(H^j_I(R))$ in case $I$ is homogeneous and $\mathfrak{m}=(x_1,\cdots,x_n),$ and an algorithm to determine the vanishing of $H^i_{\mathfrak{m}}(H^j_I(R))$.

4:00 pm in 245 Altgeld Hall,Thursday, September 12, 2013

Co-convex bodies and multiplicities of ideals

Kiumars Kaveh (University of Pittsburgh)

Abstract: The notion of (Samuel) multiplicity of an ideal I in a local ring R generalizes the fundamental notion of intersection multiplicity of subvarieties in an algebraic variety. Focus of much research in commutative algebra has been on its finitions/extensions, properties and computation. In this talk we make a connection between the notion of multiplicity and convex geometry. More precisely to each (primary) ideal in a (large class of) local rings of dim n, we associate a convex set in the positive orthant $\mathbb{R}$+n such that the volume of its complement (a co-convex set) gives the multiplicity of the ideal. As an application one recovers the Brunn-Minkowskii inequality of Teissier-Rees-Sharp for multiplicities. We will also discuss the Alexandrov-Fenchel inequality for co-convex sets and its analogue for multiplicities of ideals. This is a joint work with Askold Khovanskii.