Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, January 27, 2015.

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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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Tuesday, January 27, 2015

11:00 am in 243 Altgeld Hall,Tuesday, January 27, 2015

#### The Hodge filtration on topological Hochschild homology

###### Saul Glasman   [email] (MIT)

Abstract: In the 80s, many mathematicians independently defined a filtration on the Hochschild homology of a commutative algebra A that recovers the Hodge filtration of the de Rham complex of A in the case where A is a smooth Q-algebra. The subject of this talk is a refinement of this to a filtration by spectra of the topological Hochschild homology of a commutative ring spectrum. I'll give some motivation for this filtration and describe its construction and properties, and then I'll discuss how to lift it to a filtration of topological cyclic homology using techniques of equivariant stable homotopy theory.

1:00 pm in Altgeld Hall 243,Tuesday, January 27, 2015

#### Fully irreducible Automorphisms of the Free Group via Dehn twisting in $\sharp_k(S^2 \times S^1)$

###### Funda Gultepe (UIUC Math)

Abstract: By using a notion of a geometric Dehn twist in $\sharp_k(S^2 \times S^1)$, we prove that when projections of two $Z$-splittings to the free factor complex are far enough from each other in the free factor complex, Dehn twist automorphisms corresponding to the $Z$ splittings generate a free group of rank $2$. Moreover, every element from this free group is either conjugate to a power of one of the Dehn twists or it is a fully irreducible outer automorphism of the free group. View lecture at http://youtu.be/9nSWL4g19lk

3:00 pm in 243 Altgeld Hall,Tuesday, January 27, 2015

#### Motivic Stable Pair Invariants and K3 Surfaces

###### Sheldon Katz (Illinois Math)

Abstract: This talk focuses on the conjectural motivic extension, developed jointly with Klemm and Pandharipande, of the results of Pandharipande and Thomas which prove the KKV formula for enumerative invariants on K3 surfaces. Background will be given on stable pair invariants and motivic invariants. Reference: http://arxiv.org/abs/1407.3181

3:00 pm in Altgeld Hall 241,Tuesday, January 27, 2015

#### Decomposition of Sparse Graphs into Forests: The Nine Dragon Tree Conjecture for $k\le2$.

###### Douglas B. West   [email] (Zhejiang Normal University and UIUC Math)

Abstract: For a loopless multigraph $G$, the *fractional arboricity* $Arb(G)$ is the maximum of $|E(H)|/(|V(H)|-1)$ over all subgraphs $H$ with at least two vertices. Generalizing the Nash-Williams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if $Arb(G) <= k + d/(k+d+1)$, then G decomposes into $k+1$ forests with one having maximum degree at most $d$. The conjecture, which is sharp, was previously proved for $d=k+1$ and for $k=1$ when $d\le6$. We prove it for all $d$ when $k \le 2$, except for $(k,d)=(2,1)$. (Joint work with Min Chen, Seog-Jin Kim, Alexandr Kostochka, and Xuding Zhu.)

4:00 pm in 243 Altgeld Hall,Tuesday, January 27, 2015

#### Interest and Planning Meeting

Abstract: We will be scheduling talks for this semester, making an email list, and possibly going out for dinner afterwards. All fields of analysis including, but not limited to functional analysis, complex analysis, geometric analysis, probability, differential equations and ergodic theory are welcome. If you are unable to attend the meeting, but would like to give a talk please email Colleen Ackermann (ackrmnn2@illinois.edu) with a list of preferred weeks. Younger grad students who have an interest in analysis, but may not feel ready to give a talk are also more than welcome to attend.

4:00 pm in 245 Altgeld Hall,Tuesday, January 27, 2015

#### Insurance risk processes with dynamic changes

###### Shu Li (University of Waterloo)

Abstract: In this talk, we analyze two insurance risk processes which allow dynamic adjustments based on their surplus experience. First, we consider a risk model with varying premiums, where the insurer's decision of the next period's premium rate is based on the surplus increments between successive random review times. Under certain assumptions of the review time and claim arrival process, we derive a matrix-form defective renewal equation for the Gerber-Shiu function, as well as an explicit expression for the joint density of the surplus prior to ruin and the deficit at ruin. Numerical examples are later considered to evaluate certain ruin-related quantities and some risk management conclusions are made. Second, we analyze a drawdown-based regime-switching (DBRS) Levy insurance model in which the underlying drawdown process is used to model an insurer's level of financial distress over time and to trigger regime-switching transitions. By some analytical arguments, we derive explicit formulas for the two-sided exit problem and further examine the conditions under which the survival probability is not trivially zero (i.e., the positive security loading conditions). As a special case of the DBRS model, we investigate a jump diffusion model with regime-switching premiums and build connections with other existing risk models in the literature.